External Ballistics (taken from the 5th Edition of Sierra Manuals)
1.0 Introduction Since the mid-1980s, shooters have begun to use personal computers with ballistics software programs to calculate bullet trajectories and to explore variations in trajectories caused by changes in shooting conditions. During this period a number of ballistics software programs have been developed, especially by suppliers of bullets for reloading rifle and handgun cartridges, and have appeared on the market. Sierra introduced the Infinity exterior ballistics program in 1997. Infinity is the latest version of Sierra’s continuing exterior ballistics program development effort, which began with our first exterior ballistics software in 1967. Today, there are many other exterior ballistics programs, and a few interior ballistics programs, available to shooters. From our point of view, all these available programs are quite good. Their capabilities vary, and their computational accuracies differ a little. But on the whole they are quite acceptable for almost all shooting purposes. Today’s hand-loaders are fortunate that a wide selection of software programs is available at very attractive prices. We hope the reader will forgive us when we say that we like Infinity best. These authors take great pride in having had important roles in the development of the Infinity software program. Actually, the development of Infinity was a full team effort. The ballistics experts at Sierra helped greatly to establish the functional requirements for Infinity, based on their personal expertise and their interactions with shooters throughout the world. Then, these authors took the major responsibilities for the physics, mathematics, functional design and scientific encoding of the program. Finally, the software professionals at 305 Spin in Sedalia, MO, integrated the scientific program into Microsoft Windows and performed all the functions necessary to make Infinity ―user friendly.‖ The same team has been responsible for implementing continual updates to Infinity, but the principal criticisms and suggestions leading to those updates have been received from shooters using Infinity. We are very grateful to them. This article on Exterior Ballistics has been written specifically for shooters who use an exterior ballistics software program on a personal computer. This is a departure from the Exterior Ballistics articles that we have contributed to previous editions of the Sierra Reloading Manuals. The historical and mathematical approach to ballistics used in the previous articles has been omitted. This article instead concentrates on using ballistics software to determine ballistic coefficients of bullets, calculate bullet trajectories under a full range of shooting conditions, and answer questions about effects on trajectories caused by shooting conditions. Infinity has been used to support discussions throughout this article, but most of the calculations described should be able to be performed with one or more of the other available programs. Section 2.0 of this article describes the ballistic coefficient. This section explains what the ballistic coefficient is, how it is related to a drag function, why it must be referenced to sea level altitude and standard atmospheric conditions, how it affects a bullet trajectory, and why in a practical sense a ballistic coefficient changes with bullet velocity. Section 2.0 also describes how ballistic coefficients are measured. It presents lessons we have learned from more than 30 years of practical experience with measurements, and it provides examples of ballistic coefficient measurements we have made.
Section 3.0 describes effects of shooting conditions at the firing point on bullet trajectories. Included are effects of altitude above sea level, atmospheric conditions, winds, and shooting uphill or downhill. Some problems about sighting in a gun are discussed, i.e., using a short target range to sight in at a longer zero range, determining the zero range from where bullets group at a known range, and sighting in at a local target range to be zeroed in at some other shooting location. The concept of point blank range is described, and how to select a zero range in order to maximize the point blank range of any gun for game or silhouette targets. The maximum range of a bullet is discussed, as is the bore elevation angle necessary to achieve that maximum range. Finally in Section 3.0, the maximum height a bullet will reach if fired straight up is described. These last two topics are of great interest in designing and operating outdoor shooting ranges. Section 4.0 is completely new material not treated anywhere in our previous articles. In the past few years, we have received questions from an increasing number of target shooters about small effects they have observed in bullet trajectories — effects that cannot be explained by available exterior ballistics software programs. All the software programs generally available to shooters use a three degree-of-freedom dynamical model for a flying bullet — that is, a point mass with a ballistic coefficient. The small, unexplained effects can be attributed to rotational motions of spin-stabilized bullets. Rotational motions of a bullet are modeled only in six degree-of-freedom ballistics programs. Such programs are used in the military. Accuracy of target rifles has continually improved through the years, particularly in long-range target shooting. The small effects of bullet rotational motions have become observable because rifle accuracy has improved to a point where these effects can be seen under some conditions. In Section 4.0 we attempt to explain these effects and their causes. These include the yaw of repose of a bullet and an associated cross-range deflection, turning of a bullet to follow a cross-wind and an associated vertical deflection, and turning of a bullet to follow a vertical wind and an associated cross-range deflection. These seem to be the most observable effects of the rotational motions of sporting bullets. Sections 5.0 and 6.0 relate specifically to Sierra’s Infinity program. Section 5.0 describes the content and format of the printout records from Infinity, that is, the trajectory parameters, their physical units, and other information communicated to shooters by the printout records. Section 6.0 is an overview of the capabilities of Infinity, describing its major features, operating modes, and how to use the program for trajectory computations and to answer questions concerning bullet trajectories. It is always a pleasure to hear from users of Sierra products and a special pleasure to hear from those interested in ballistics. Please do not hesitate to contact us with questions or comments.
2.0 The Ballistic Coefficient ―So, just what is a ballistic coefficient, and what does it do for a bullet’s trajectory?‖ These are questions we have been asked many, many times, and they are not easy questions to answer. We will try to answer the first question in this subsection and then proceed to the second question in the next subsection. In later subsections we will describe methods used to measure ballistic coefficients, and give some examples of measured BC values for Sierra’s bullets. Henceforth, ballistic coefficient will be abbreviated as BC.
2.1 The Ballistic Coefficient Explained
There are at least three ways to describe the BC. First, it is widely recognized as a figure of merit for a bullet’s ballistic efficiency. That is, if a bullet has a high BC, then it will retain its velocity better as it flies downrange from the muzzle, will resist the wind better, and will ―shoot flatter.‖ But this description is qualitative, rather than quantitative. For example, if we compare two bullets and one has a BC 25% higher than the other, how much is the improvement in bullet ballistic performance? This question can be answered only by calculating the trajectories for the two bullets and then comparing velocity, wind deflection, and drop or bullet path height versus range from the muzzle. So, the figure of merit approach really gives only a qualitative insight into bullet performance, and sometimes this insight is not correct. It often happens that the bullet with the smaller BC is lighter than the bullet with the higher BC. The lighter bullet therefore can be fired at a higher muzzle velocity, and it can then deliver better ballistic performance just because it leaves the muzzle at a higher velocity. We will talk more about this later. The second way to describe the BC is to use its precise mathematical definition. Mathematically, the BC defined as is the sectional density of the bullet divided by the form factor. This definition emerges from the physics of ballistics and is used in mathematical analysis of bullet trajectories. But in a practical sense, this definition is not satisfactory to most people for at least two reasons. The first is the question of a bullet’s form factor. The form factor is a property of the shape of the bullet design, but it is no easier to explain than the BC. The second reason is that this mathematical definition can lead to an erroneous conclusion. Assume for the moment that the form factor is just a constant property of the bullet design (not always true). The sectional density of a bullet is its weight divided by the square of its diameter. (The square of any number is the number multiplied by itself). So, to get a large BC we need a large sectional density. It appears from the mathematics that a bullet with a very small diameter should have a very large sectional density because its weight is divided by a very small number, and this should give it a very high BC. In other words, this line of reasoning would lead us to expect that small caliber bullets should have very large BC values. But this is not true because when the diameter of the bullet is small, the volume also is small. The weight of the bullet then is small, and the sectional density is necessarily small also. The net result is that small caliber bullets generally have lower BC values than larger caliber bullets. The third way to describe the ballistic coefficient traces back to the historical development of the science of ballistics in the latter half of the 19th century. This explanation is lengthier, but it provides a better understanding of what the BC is and what its role is in trajectory calculations. The latter half of the 19th century and the early part of the 20th century was a period of very intensive and fruitful development in the science of ballistics. The developments in ballistics were driven by technological advances in guns, projectiles, propellant ignition, and propellants throughout the 19th century, and by warfare, particularly in Europe and America. Warfare was almost an international sport among the kings, emperors, Kaisers and tsars in Europe throughout the 1800’s. The United States experienced the War of 1812, the Mexican War, the Civil War, the Indian wars in the West, and the SpanishAmerican War within that same century. Governments were eager to fund research, development and manufacturing of improved guns and gunnery, because battles were generally won by the forces that had superior arms. Percussion ignition was invented in 1807 by the Rev. Alexander Forsythe in Scotland. In 1814, Joshua Shaw, an artist in Philadelphia, invented the percussion cap. In 1842, the U.S. Army adopted the percussion lock for the Model 1842 Springfield Musket, replacing flintlock ignition in earlier shoulder arms. Rifled muskets and handguns began to replace smoothbore military weapons in the mid-1850’s after a French Army officer, Capt. Claude Minie, developed a means to expand a bullet upon firing to cause it to fit the grooves of a rifled barrel. This advancement combined the rapidity and ease of loading of round balls—which had been the standard military projectile for over a century—
with the increased range and deadly accuracy of rifled arms. The range and precision of military weapons, for both small arms and artillery, was increasing dramatically. The period between 1855 and about 1870 witnessed much research and development in breech loading rifles and handguns. The first metallic self-contained, internally primed cartridge (the 22 Short rimfire cartridge) was introduced by Smith & Wesson in 1857 in their Model No. 1 breech-loading revolver. Breech-loading rifles firing self-contained cartridges appeared in the 1860’s, and some were used during the U.S. Civil War (e.g., the Spencer carbine and the Henry rifle). In 1866 in the United States, Hiram Berdan obtained a patent on a primer that was suitable for centerfire cartridges. That same year in England, Col. Edward Boxer patented a full cartridge for the British Snider Enfield rifle, which was a centerfire cartridge utilizing the Boxer primer. (It is interesting to note that later the Berdan primer was widely adopted on the European continent, while the Boxer primer became standard in the United States.) In 1873, the U.S. military adopted the Model 1873 Trapdoor Springfield rifle with the 45-70 centerfire cartridge. In the space of just 31 years the U.S. Army changed from smoothbore muskets with flintlock ignition to rifles with self-contained metallic centerfire cartridges. Just 11 years later in 1884, a French physicist named Paul Vielle developed the first smokeless propellant that was stable and loadable for military purposes. Earlier powder developments had led up to Vielle’s discovery, but they were useful only for sporting purposes. The French Army loaded Vielle’s smokeless propellant in the 8mm Lebel cartridge for the Model 1886 Lebel rifle, the very first military rifle firing a smokeless propellant cartridge. Smokeless propellant was quickly adopted by other nations, including the U.S., and caused significant advancements in bullet performance and design. Muzzle velocity in military rifles, which was less than 1400 fps in the 45-70 and most other black powder cartridges, increased to more than 2000 fps in the earliest smokeless propellant cartridges. This led to the development of jacketed bullets of smaller caliber and lighter weights, i.e., 7mm, 30, and 8mm calibers, which could be fired at even higher velocities and not deposit lead in the barrels at those velocities. Before the end of the 19th century, pointed bullets and boat-tail bullets were also developed to significantly improve bullet ballistic performance. With all these developments in guns and ammunition, the need to understand the ballistics of projectiles became more acute. It was no longer sufficient to target a gun by hit-and-miss methods. Of course, graduated sights had existed on both smoothbore and rifled muskets for many years, but the elevation marks on the sights had been determined by firing tests of these weapons with a specific projectile at a specific muzzle velocity, at a specific altitude, and with a specific set of weather conditions. As warfare grew in intensity and mobility, it became vitally necessary to understand the physics of bullet motion. In other words, it was necessary to find a way to calculate bullet trajectories as well as the changes in those trajectories caused by changes in bullets, muzzle velocities and firing conditions. An immense problem thwarted this objective for many years. This problem was understanding the physics and mathematically describing the aerodynamic drag force on a projectile. The invention of the ballistic pendulum by the English ballistician Benjamin Robins in 1740 had led to the astounding discovery (at that time) that the drag force on a bullet was many times more powerful than the force due to gravity, and that it changed markedly with bullet velocity. That event started a chain of firing tests, instrumentation developments, and theoretical investigations that lasted at least 200 years. Progress was slow because aerodynamic drag is a very complex physical process, and mathematics had to be developed to make accurate computation of trajectories possible long before the age of computers.
An early observation was that the drag force was different on every type of projectile, so that measurements of drag deceleration seemed to be necessary on each type of projectile over the full velocity range between muzzle velocity and impact velocity. However, around 1850 Francis Bashforth in England proposed a practical idea that greatly simplified things and is used in the present day. He proposed a model bullet, or ―standard‖ bullet, on which comprehensive measurements of drag deceleration versus velocity could be made. Then, for other bullets this ―standard‖ drag deceleration could be scaled by some means, so that exhaustive drag measurements could be avoided for those bullets. Bashforth could not have known how successful his suggestion would be. Ballisticians and physicists were working intensively to mathematically describe the aerodynamic drag force and derive the equations of motion of bullet flight. They had recognized in the equations of motion a theoretical scale factor for aerodynamic drag that would adjust the standard drag model to fit a nonstandard bullet. This scale factor turned out to be the form factor of the nonstandard bullet divided by the sectional density, that is, the reciprocal of the BC. The form factor was a number that accounted for the different shape of the nonstandard bullet compared to the standard bullet. Bashforth’s suggested standard bullet had a weight of 1.0 pound, a caliber of 1.0 inch, and a point with a 1.5 caliber ogive. Firing tests on projectiles of approximately this shape and weight were conducted in England and Russia between about 1865 and 1880. However, the definitive drag deceleration tests were performed by Krupp at their test range in Meppen, Germany, between 1875 and 1881. In 1883 Col. (later General) Mayevski in Russia formulated a mathematical representation of the drag force for the standard bullet. In the 1880’s, an Italian Army team led by Col. F. Siacci formulated an analytical approach and found analytical closed form solutions to the equations of motion of bullet flight for level-fire trajectories. This meant that trajectory calculations for shoulder arms could be performed algebraically, rather than by the more tedious methods of calculus. The Siacci team’s results also showed that not only could the standard drag deceleration be scaled by using the BC of the nonstandard bullet, but also the standard trajectory computed for the standard bullet could be scaled by the same factor to compute an actual trajectory for the nonstandard bullet. This was a very important breakthrough that greatly reduced the amount of work in trajectory computations. The Siacci approach was adopted by Col. James M. Ingalls of the U.S. Army Artillery. His team produced the Ingalls Tables, first published in 1900, which in turn became the standard for small arms ballistics used by the U.S. Army in World War I. So, the ballistic coefficient actually is a scale factor. The BC scales the standard drag deceleration of the standard bullet to fit a nonstandard bullet. However, the BC works in a reciprocal manner. That is, the higher the BC of a nonstandard bullet, the lower the drag is compared to the standard bullet. This is alright, because it means that the higher the BC of a bullet, the better will be its ballistic performance. The physical units of the BC are pounds per square inch (lb/in2). The BC value for the standard bullet then is 1.0 (weight 1.0 lb, diameter 1.0 inch, and form factor 1.0 by definition for the standard bullet). Ballistic coefficients of most sporting and target bullets have values less than 1.0, and generally BC values increase as caliber increases. A bullet can have a BC higher than 1.0. For example, some heavy 50 caliber bullets have BC values greater than 1.0. Military agencies in different nations developed many standard bullets over the years. This was done because of fundamentally different shapes in military projectiles, such as sharper points and boat tails. The purpose was to establish better standards for these classes of bullet shapes. In recent years, however, this practice has largely been abandoned in the military. With modern instrumentation and computers, it has become possible to measure the drag deceleration of every individual projectile type used by the military. Thus, there is no longer a need for a standard projectile for military applications. Or, we might say that every type of military projectile is its own standard.
This is not true for commercial bullets, however. Ballistic coefficients are used for all commercial bullets for sporting and target-shooting purposes, mainly because the BC of each is relatively easy and inexpensive to measure, compared to measuring the drag deceleration. The standard projectile for commercial bullets is still nearly identical to Bashforth’s standard bullet. The standard drag model, also called the standard drag function, for this projectile is known as G1. The BC values quoted by all producers of commercial bullets are referenced to G1. It is important to note that BC values quoted by commercial producers cannot be used with any drag model other than G1. It is possible to find other standard drag models by looking up historical military ballistics data. But, if a standard drag model other than G1 is used, the BC values of bullets must be measured with reference to that drag model in order to calculate accurate trajectories. The values are likely to be very different from the values referenced to G1.
2.2 Bigger Is Not Always Better It is absolutely true that, if two bullets are fired with the same muzzle velocity at the same firing point and with the same weather conditions, the bullet with the higher BC will arrive first at a specific target, will have higher velocity and energy when it arrives, will suffer less wind deflection, and will have less drop than the bullet with the lower BC. This will happen regardless of caliber, bullet weight, or bullet type. But notice the big IF condition in this statement. If the bullets are fired with different muzzle velocities, at different altitudes, or under different weather conditions, any conclusions from the comparison may not be entirely correct. What this means is that when comparing the ballistic performance of different bullets, all the firing conditions must be taken into account. Another very important consideration is that when comparing bullet ballistic performance, the performance must be evaluated for the purpose and objectives to be accomplished. For example, the usual purpose may be target shooting or hunting. If we wish to choose a bullet for the purpose of target shooting, the main objectives are maximizing accuracy and minimizing wind deflection. A very important advantage of handloading is that the muzzle velocity produced by the cartridge can be adjusted so that the gun delivers its best accuracy. Such a muzzle velocity is usually a little less than the maximum safe velocity of the cartridge in that gun, and this velocity usually must be discovered by trial at the shooting range. Target shooters with modern highpower target rifles generally are achieving accuracies on the order of 0.2 minutes of angle (MOA) or less. On the other hand, ―shooting flat‖ is not very important, because the ranges to the targets are fixed distances, sighting shots are usually allowed, and sights on guns are allowed to be adjusted between stages of the matches. However, sensitivity to crosswind is very important, because matches take place under variable wind conditions. Even if a shooter is highly experienced in estimating windage corrections, it is best that the bullet selected for the match have the smallest practical sensitivity to crosswind. For hunting purposes ―shooting flat‖ is a major objective, as are adequate retained energy and momentum over the effective range of the gun for the intended game, adequate accuracy, and low sensitivity to crosswinds and vertical winds. A gun that ―shoots flat‖ produces small bullet drop within the effective range for the intended game. This is very important, because it is difficult for a hunter to estimate range to a game animal under practical conditions in the field. Of course, there are several optical and electro-optical range finders available, but most hunters cannot be assured that their game will be patient and stand perfectly still while they attempt to use a range finder under nonhunter-friendly field conditions. The concept of point blank range is one very practical way to ease
the range-estimating problem for hunters in the field (more about this in a later section). A ―flat shooting‖ gun inherently has more point blank range than a gun with a ―rainbow‖ trajectory. ―Flat shooting‖ requires high muzzle velocities and bullets with high ballistic efficiency. Adequate accuracy for hunting usually means groups of 1.0 to 1.5 MOA for medium and large game, and 0.5 to 0.75 MOA for varmints. Wind sensitivity is very important, because field environments typically have windy conditions. Crosswinds may happen anywhere, and vertical winds are experienced in hilly or mountainous regions. The wind deflection sensitivity of a bullet to a vertical wind is exactly the same as its sensitivity to a crosswind. In other words, suppose that a 1.0 mph crosswind from the shooter’s right to left will cause a deflection of the bullet of, say, 6 inches to the left at the target. Then a vertical wind of 1.0 mph upward will also cause a 6 inch deflection of the bullet in the upward direction. This is very important when shooting across a canyon or along a hillside when the wind is blowing. To illustrate the points made above in this subsection, we will consider two simplified examples, shown in Tables 2.2-1 and 2.2-2. Two popular cartridges are used for these examples. The first (see Table 2.2-1) is the 308 Winchester, also known as the 7.62 x 51 mm NATO cartridge. This cartridge is very popular for hunting medium game, such as deer and antelope. It is also used for target shooting. For example, in Service Rifle competitions, they are fired in the M14 and M1A rifles, and in Match Rifle competitions they are fired in bolt action rifles. The second cartridge (see Table 2-2-2) is the 300 Winchester Magnum. This cartridge is very popular for both hunting large game, such as elk, moose and bear, and for target shooting, particularly the Long Range target competitions. These cartridges have been selected as examples because there is such a wide variety of 30 caliber bullets available for handloading. The examples are simplified in that the numbers in Tables 2.2-1 and 2.2-2 are for sea level altitude and standard atmospheric conditions. So, the performance of each bullet is not calculated for realistic field conditions, but these examples validly illustrate our key points. The numbers in the tables have been calculated using Sierra’s Infinity Exterior Ballistics Program. Each table is for one of the two example cartridges. Furthermore, each table is separated into two sections: one for hunting purposes and the other for target purposes. In the hunting purposes section, the first column in the table lists each bullet selected for comparison. The next column contains the muzzle velocity of each bullet. Each listed velocity is at or near the top end of the velocity range for that bullet recommended in the Reloading Data Section of this Manual for the cartridge in each table. The third column lists the ballistic coefficient of each bullet at the muzzle velocity. Later in this section we will describe how BC varies with bullet velocity. The number in the table compares the bullets at the muzzle velocity level for each. The BC variations as the bullet flies are taken into account in the trajectory calculations. The fourth column lists the energy of each bullet at the muzzle. For hunting purposes it has been assumed that the effective range of fire is 400 yards, that the rifle is zeroed in at 250 yards, and that a telescope sight is used, with the centerline of the telescope 1.5 inches above the cen-terline of the bore. Then for these assumptions, the fifth column shows the maximum bullet path height (sometimes called the maximum ordinate) above the hunter’s line of sight through the telescope, together with the downrange position from the muzzle at which this maximum bullet path height occurs. The remaining four columns in this section of the table show ballistics properties at the 400 yard maximum effective range point. Column 6 lists the remaining velocity; column 7 lists the bullet energy; column 8 lists the distance the bullet passes below the hunter’s line of sight at 400 yards;
and the last column lists the wind deflection sensitivity — that is, the inches of deflection per mile per hour of either crosswind or vertical wind. In the second section of Tables 2.2-1 and 2.2-2 for target shooting purposes, the first column lists the target bullets selected for comparison; the second column shows the muzzle velocity for each bullet; and the third column lists the BC value at the muzzle velocity. Again, each listed velocity is at or near the top end of the velocity range for that bullet recommended in the Reloading Data Section of this Manual for the cartridge in each table. For purposes of comparison, it is assumed that a nearmaximum load delivers the best accuracy, which is not always true. Two range distances are considered for target shooting — 600 yards and 1000 yards. Column 4 shows the remaining velocity of each bullet at 600 yards, and column 5 lists the wind deflection sensitivity at 600 yards. Columns 6 and 7 show these same two parameters at 1000 yards from the muzzle. Note that the wind deflection sensitivity values listed in the tables are per mile per hour of crosswind or vertical wind. In other words, if the wind speed is 10 mph, the bullets will deflect 10 times the amount shown in the tables. Consider first the 308 Winchester cartridge in Table 2.2-1. For hunting purposes, the table shows that the bullet that shoots flattest is the 150 grain SBT (Spitzer Boat Tail) at 2800 fps muzzle velocity. However, the bullet with minimum wind deflection sensitivity is the 200 grain SBT at 2400 fps muzzle velocity. So, the lighter 150 grain bullet with smaller BC passes about 4 inches closer to the line of sight at 400 yards than does the 200 grain bullet with a significantly higher BC. On the other hand, the heavier bullet deflects considerably less in a crosswind or vertical wind. If a crosswind or vertical wind speed were 10 mph, the 150 grain bullet would be deflected 16.2 inches, while the 200 grain bullet would be deflected just 12.8 inches. So, the choice of hunting bullets depends on which is more important to the shooter: a flatter trajectory or sensitivity to wind conditions. For target shooting purposes, the bullet with the minimum wind deflection sensitivity in the 308 Winchester cartridge is the 200 grain MatchKing fired at 2450 fps muzzle velocity. In this case, the 200 grain MatchKing is better than the 220 grain MatchKing just because the heavier bullet cannot be fired at a high enough muzzle velocity. If the muzzle velocity of the 220 grain bullet could be increased to around 2300 fps, then its higher BC would give it less wind sensitivity than the 200 grain bullet. But the cartridge does not have enough powder capacity to safely allow the velocity increase. For the 300 Winchester Magnum cartridge, Table 2.2-2 shows that for hunting purposes the 180 grain SBT GameKing bullet loaded to 3100 fps muzzle velocity shoots the flattest of all five bullets listed. The 200 grain SBT GameKing bullet loaded to 2900 fps has the least wind deflection sensitivity, but it is just a tiny bit better than the 180 grain bullet. So, in this example the numbers in the table indicate that the 180 grain SBT GameKing bullet is probably the best choice for hunting. However, some hunters
Table 2.2-1 Ballistic Coefficient Effects for the 308 Winchester Cartridge HUNTING PURPOSES
At 400 yds Range (Zero at 250 yds) Selected
Velocity Energy (fps)
(ft-lbs) Path (in) (in/mph)
125 gr SPT 3000 Pro-Hunter
3.43 @ 143 yds
150 gr SBT 2800 GameKing
3.68 @ 143 yds
165 gr SBT 2600 GameKing
4.32 @ 139 yds
180 gr SBT 2500 GameKing
4.49 @ 138 yds
200 gr SBT 2400 GameKing
4.66 @ 136 yds
TARGET SHOOTING PURPOSES
At 600 yds Range
At 1000 yds Range
155 gr HPBT Palma
175 gr HPBT MatchKing 180 gr HPBT MatchKing 190 gr HPBT MatchKing 200 gr HPBT MatchKing 220 gr HPBT MatchKing
Table 2.2-2 Ballistic Coefficient Effects for the 300 Win Magnum Cartridge
HUNTING PURPOSES At 400 yards Range (Zero at 250 yards) Selected Bullet
Mzzl. BC at Vel. Mzzl. (fps) Vel.
Mzzl. Max Bullet Velocity Energy Energy (ft-lbs)
(ft-lbs) Path (in) (in/mph)
165 gr SBT 3200 GameKing
2.57 @ 144 yds
180 gr SBT 3100 GameKing
180 gr SPT 3100 Pro-Hunter
2.79 @ 142 yds
200 gr SBT 2900 GameKing
3.06 @ 141 yds
220 gr RN 2750 Pro-Hunter
4.05 @ 140 yds
TARGET SHOOTING PURPOSES At 600 yds Range At 1000 yds Range Selected Bullet 168 gr HPBT MatchKing 180 gr HPBT MatchKing 190 gr HPBT MatchKing 200 gr HPBT MatchKing 220 gr HPBT MatchKing 240 gr HPBT MatchKing
Mzzl. Vel. (fps)
Wind Drift (in/mph)
Wind Drift (in/mph)
prefer a flat base rather than a boat tail bullet shape, and others prefer a round nose bullet for some game in some terrain. So, Table 2.2-2 includes the 180 grain SPT (Spitzer) Pro-Hunter bullet for direct comparison with the 180 grain SBT GameKing, as well as the 220 grain RN (Round Nose) Pro-Hunter bullet. One can see the 180 grain Spitzer flat base bullet has about 20% lower BC than the 180 grain Spitzer boat tail bullet, and it loses velocity and energy faster, and has about a 25% increase in wind sensitivity. The 220 grain round nose bullet is the heaviest in the table, but it has the lowest BC, loses velocity and energy rapidly as it flies, has the worst trajectory curvature, and
has the worst wind sensitivity. This is a fine bullet for heavy game, but it rapidly loses its advantages at longer ranges. For target shooting with the 300 Winchester Magnum, Table 2.2-2 shows that the best bullet is the 240 grain MatchKing at 2800 fps muzzle velocity. This is the heaviest bullet and has the largest BC. This illustrates a principle that many target shooters have found generally true with magnum cartridges for target shooting. That is, select the bullet with highest BC value and load it as fast as it will go and still deliver maximum accuracy considering recoil sensitivity of the shooter as well as accuracy capability of the rifle. So, considering BC, bigger is sometimes better, but not always. The purposes of the shooter, the shooting situation, and the limitations of the gun and cartridge must be taken into account in choosing a bullet. The best tool to use to examine all the possibilities is one of the ballistics computation software programs for the personal computer. All types of ―what if‖ questions can be explored at the keyboard.
2.3 How the Ballistic Coefficient is Measured We first began to investigate just how to determine the BC values of various bullets in 1969. We learned very quickly that BC values for all bullets need to be measured by firing tests; there is no other way to make an accurate determination. It is true that in 1936, E. I. Du Pont de Nemours & Company, Inc., published a brochure prepared by two ballistics engineers on their staff, Wallace H. Coxe and Edgar Beugless. This brochure, titled Exterior Ballistics Charts, described a method of finding the form factor of a bullet by matching the point shape against a set of ogive contours, and then looking up the form factor value in a table of values. With the form factor known, the BC could then be obtained from a chart in the brochure. The brochure also contained several pages of nomographs and simple computational techniques to determine trajectory variables, such as remaining velocity, maximum trajectory height, wind deflection, etc., versus range from the muzzle. The work of Coxe and Beugless was a great step forward at the time. They presented the first method of BC estimation available to the general shooting community, and their nomographs presented a useful method of calculating ballistics parameters long before the age of computers. Their methods were used by handload developers and wildcatters until several years after the end of World War II. Today, however, the work of Coxe and Beugless is mainly of historical interest. We found in 1969 and 1970 that BC values determined by their method simply are not accurate enough by modern standards. BC values really must be measured by firing tests. We have used three methods of measuring BC values from firing tests. The first two of these methods can be used by shooters equipped with a pair of chronographs, a computer, and exterior ballistics software such as the Sierra Infinity program.
2.3.1 Initial Velocity and Final Velocity Method 18.104.22.168 Measurement Procedure
This first method is illustrated in Figure 2.3-1. This method uses two chronographs for each bullet fired to measure an initial velocity and a final velocity at a measured range distance between the chronographs. The initial velocity chronograph is usually placed near the muzzle of the gun, as shown in Figure 2.3-1. A blast shield with a small hole for bullet passage usually is used to keep the muzzle flash or blast from disturbing the screens of the initial velocity chronograph. If the screens are photoelectric types, as is the usual case, the muzzle flash may trigger screen 1 before the bullet arrives. Also, because the powder gases exit the muzzle at about 1.5 times the bullet velocity, the gases can trigger screen 1. Or, the muzzle blast can cause screen 1 or 2 to bend or vibrate. Any of these effects will cause an erroneous measurement of the initial velocity, which will, in turn, cause an error in the measured BC value. The final velocity chronograph is placed downrange at a carefully measured distance from the initial velocity chronograph. This range distance is measured from the center point between screens 1 and 2 to the center point between screens 3 and 4. This is because each chronograph really measures the bullet travel time between the two screens to which it is connected (i.e., between screens 1 and 2 or between screens 3 and 4). Then, the velocity is obtained by dividing the precisely measured distance between the pair of screens by the measured bullet travel time. This calculation is performed within the electronics of the chronograph. Because the bullet slows down a tiny bit as it travels between the two screens, the measurement of velocity is considered to be valid at the center point between the pair of screens. Of course, if the separation distance between screens 1 and 2 is the same as between screens 3 and 4, the range distance between the two chronographs may be measured from screen 1 to screen 3. The measurement procedure for each bullet fired is to record the initial and final velocities as well as the range distance between the two chronographs. The altitude, temperature, barometric pressure, and relative humidity at the firing point must also be recorded. If the shooting range is not level, the elevation angle must be recorded, especially if it exceeds about 3 degrees either upward or downward. If there is any appreciable wind at the firing point, BC measurements should not be attempted. With these data for each bullet fired, an exterior ballistics software program for a personal computer can be used to calculate the ballistic coefficient. Some exterior ballistics programs contain an optional routine for computation of the BC value, but that is not necessary. The normal trajectory computation routine can be used in an iterative fashion for each bullet fired. That is, first initialize the program by entering the altitude, temperature, barometric pressure, humidity and range elevation angle at the firing point. Then, for each round fired enter the measured initial velocity as the ―muzzle velocity‖ for the trajectory calculation. Then, guess a BC value, calculate a trajectory over the measured range distance, and examine the calculated final velocity. If the calculated final velocity is higher than the measured number, the BC value is too high. (Conversely, if the calculated final velocity is lower than the measured value, the BC value is too low.) Then, reduce (or increase) the BC value a little, calculate another trajectory over the measured range distance, and examine the calculated final velocity. The calculated final velocity should be nearer the measured value. If the calculated value from this second iteration is higher (or lower) than the measured value, reduce (or increase) the BC value in the program and perform another iteration of the calculations. After a few iterations, this method will ―home in‖ on a correct value for the measured BC for the first round fired. Of course, this is a BC value for which the calculated final velocity matches the measured final velocity as closely as possible. This resulting BC value is considered valid for a bullet velocity midway between the initial and final velocities for that round. For the other rounds fired, the resulting BC for the first round can be used as the initial guess for the BC value, and fewer iterations will be required to reach a correct BC value for each of those rounds.
Figure 2.3-1 Test Range Setup for Initial and Final Velocity Method for BC Measurement An example has been prepared to illustrate this method of determining the BC. Suppose we have developed a load for the 308 Winchester (7.62 x 51 mm NATO) cartridge in a bolt-action rifle that pushes a certain .308 diameter 160 grain bullet at a muzzle velocity of about 2750 fps. We do not know the BC of this bullet type and want to measure it on our local shooting range. (Actually, we do not need to know the weight of the bullet or even the caliber to determine the BC. We need only the firing test data as described below.) Suppose that our shooting range is located at an altitude of 790 feet above sea level, and we perform the shooting tests on a day when the temperature is 78° F, the barometric pressure is 30.15 inches of Mercury, and the relative humidity is 80%. Note that the barometric pressure is obtained from a barometer at the range or from a local weather report for the time of day when the firing tests take place. The test range is level, and the range distance between chronographs is 103 yards, which is measured precisely and accurately when we set up the range for the tests. We fire, say, ten rounds to obtain an average BC value for this bullet type at velocities in the vicinity of 2750 fps. We record the altitude, atmospheric conditions, range distance between the chronographs, and the initial and final velocities for each round fired. Then, we retire to our computer at home. We start up the Sierra Infinity program, and it comes up automatically in the ―Trajectory‖ mode of operation. Suppose that for the first round fired at the range, the initial velocity was 2742 fps and the final velocity was 2549 fps, as read from the initial and final velocity chronographs. In the Infinity program we select any 30 caliber bullet in the ―Load Bullet‖ library, and transfer it to the ―Active Bullets‖ list in the upper right corner of the blank part of the screen. Then, we initialize the trajectory computation as follows:
Trajectory Parameters Units: Full English (since we are working in the English system of units) Muzzle Velocity: 2742 fps (for the first round fired) Maximum Range: 103 yds (this is as far as we need the trajectory to be computed) Range Increment: 1 yd (because the distance between chronographs is 103 yds, which is not divisible by any number other than 1) Zero Range: 103 yds (the distance between chronographs) Elevation Angle: 0 (because the test range is level) Sight Height: 1.5 inches (choice for telescope sight on the rifle)
Environment Parameters Barometric Pressure: 30.15 in Mercury (from a barometer at the range or a local weather report) Temperature: 78° F (from a thermometer at the range or local weather report) Altitude: 790 ft (can be obtained from a topographical map or other source) Humidity: 80 % (relative humidity from a weather station at the range or from the local weather report) Wind Direction: Any number between 0 and 12 o’clock is OK Horizontal Wind Velocity: 0 mph (no wind is very important) Vertical Wind Velocity: 0 mph (no wind is very important) At this point we have initialized a trajectory computation for some 30 caliber bullet (we don’t care which one) with the correct muzzle velocity, range distance between the chronographs, trajectory calculation parameters for our purposes, and environmental conditions at the firing point. But, we haven’t performed any trajectory computation yet, so there is nothing on the monitor screen yet. Now, although we will not use it explicitly, we must ―Calculate‖ a trajectory so that the ―Trajectory Variations‖ menu item will be available to us. We then go to the Infinity toolbar at the top of the monitor screen and select ―Trajectory Variations.‖ From the dropdown menu that appears, we select ―Ballistic Coefficients.‖ In the sidebar at the right side of the monitor screen, we then see five values of ballistic coefficient listed. These values mean nothing to us since they are for the bullet that we chose to load into the ―Active Bullet‖ list, not for the bullet that we are testing. It is necessary now to make an initial guess for the BC value of our test bullet. If we guess well, we will not have to make many computation iterations to find the correct BC value. For a 30 caliber bullet that weighs 160 grains and has a Spitzer (sharp pointed) shape, the BC value at around 2600 fps should be somewhere near 0.5. So, let us choose this value as the initial guess. We then change the five numbers in the right-hand sidebar on the monitor screen to the value 0.5. We change all the BC numbers for a particular reason. As we will explain later, Infinity allows the ballistic coefficient of each bullet type to change with bullet velocity as it flies downrange and slows down. This is because the measured BC of a bullet does change with velocity, and accounting for such changes can increase the accuracy of trajectory computations within Infinity. We use five velocity regions for this purpose. Within each velocity region there is a single value of BC valid for that region, and there is a value for each of the five regions. There are then four velocity boundaries separating these regions. When the velocity of a bullet falls through one of these boundaries, Infinity automatically changes the BC to the value for the new region. In our current case, we do not know whether our test bullet starting at 2742 fps and ending up at 2549 fps crosses a velocity boundary for the bullet we are using.Yes, we could look to see and make a more educated selection of the one or two BC values that we would need to change, but if we change all five of the values in the sidebar we will be sure to be safe. After we change the BC numbers in the sidebar to 0.5, we are ready to begin the iterative search procedure for the correct BC value for the first round fired. Table 2.3-1 summarizes the computations in the search procedure. To begin the procedure, click the ―Calculate‖ button on the bottom of the monitor screen. Infinity performs the first trajectory computation, the trajectory parameters appear on the screen, and we immediately scroll down to the final parameter values at 103 yds range. We find that the computed final velocity at 103 yds is 2564.8 fps for this first iteration (see Table 2.3-1). This is higher than the measured final velocity for this round 2549 fps, so the next guess for BC needs to be lower than 0.5.
At this point we have no idea how much to lower the next BC guess, but let’s try 0.4. We set the five BC numbers in the sidebar to 0.4 and click the ―Calculate‖ button for the second trajectory computation. The computed final velocity at 103 yds for the BC equal to 0.4 is 2521.5 fps, which is too low compared to the measured 2549 fps. So for the third iteration, the guess for BC needs to be raised. Let’s try something halfway between 0.4 and 0.5; that is 0.45. We change all five BC numbers in the sidebar to 0.45 and click the ―Calculate‖ button for the third trajectory computation. For this third iteration, we find that the computed final velocity at 103 yds is 2545.5 fps, which is closer but still lower than the measured 2549 fps. So, for the next iteration let’s raise the BC guess to 0.46. Again, we change all five BC numbers in the sidebar to 0.46 and click the ―Calculate‖ button for the fourth trajectory computation. For this fourth iteration, we find that the computed final velocity at 103 yds is 2549.7 fps (as shown in Table 2.3-1). This is just a little higher than the measured 2549 fps. So, for the next iteration we must lower the BC guess just a little. Table 2.3-1 shows the final three iterations, each of which follows the same procedure. In each iteration we change the BC guess by a smaller amount so that the computed final velocity approaches the measured final velocity. The seventh iteration, which has a BC of 0.4583, produces a final computed velocity equal to the measured velocity, and this BC therefore is the correct value for this first fired round. Table 2.3-1 Example of Ballistic Coefficient Iterative Search Procedure
Test Range Parameters: Distance between chronographs: 103 yds Range altitude: 790 ft above sea level Temperature: 78º F Barometric pressure: 30.15 in Mercury Relative humidity: 80% Exterior Ballistics Program: Sierra infinity
Test Round 1: Initial velocity 2742 fps; final velocity 2549 fps Iteration
Computed final velocity
1 2 3 4 5 6 7
0.5 0.4 0.45 0.46 0.458 0.459 0.4583
2564.8 2521.5 2545.5 2549.7 2548.9 2549.3 2549.0
Test Round 2: Initial velocity 2751 fps; final velocity 2556 fps 1 2
3 4 5
0.455 0.454 0.4545
2556.2 2555.8 2556.0
Table 2.3-1 also shows the iterations for the second fired round. In this example we suppose that the second round has a measured velocity of 2751 fps and a measured final velocity of 2556 fps. To initialize for the second round, we momentarily return to the ―Operations‖ selection on the Infinity toolbar at the top of the monitor screen, select the ―Trajectory‖ mode of operation, and change the ―Muzzle Velocity‖ entry in the ―Trajectory Parameters‖ sidebar to 2751 fps. Again, we must ―Calculate‖ so that the ―Trajectory Variations‖ menu item is available. Then, we return to the ―Trajectory Variations‖ selection on the Infinity toolbar, again select ―Ballistic Coefficients‖ on the dropdown menu. We verify that all five entries in the BC sidebar on the monitor have the value 0.4583 from the first round. Note that we have not changed any of the other Trajectory Parameters or Environment Parameters, since they all have the same values for our firing tests. Table 2.3-1 shows the sequence of iterations for the second round. The first iteration with a BC guess of 0.4583 produces a computed final velocity of 2557.6 fps, higher than the measured final velocity of 2556 fps. The second iteration with a BC guess of 0.457 produces a computed final velocity of 2557.1, closer but still higher than the measured 2556 fps. The third iteration with a BC guess of 0.455 produces a computed final velocity of 2556.2 fps, even closer but still a little higher than the measured 2556 fps. The fourth iteration with a BC guess of 0.454 produces a computed final velocity of 2555.8 fps, which is lower than the measured 2556 fps. Since the results of the third and fourth iterations equally straddle the measured 2556 fps, the fifth BC guess is chosen as 0.4545, halfway between 0.455 and 0.454. This final iteration produces a computed final velocity equal to the measured 2556 fps, so this value is the correct value for the second bullet fired. The same procedure should be followed for each of the remaining eight bullets in the test series that we fired at the test range. In this way, we will derive the measured BC values for all ten bullets and can apply statistical analysis to this limited sample of test bullets for this type of bullet at velocities between the initial and final values. The computations in this example have been explained in some detail, so that the reader can repeat these calculations step by step if he or she uses Sierra’s Infinity program. If a different exterior ballistics program is used, the detailed steps of the procedure should be changed because any other program will function a little differently, but the basic method will not change. The idea is to find a BC value that makes the computed final velocity equal to the measured final velocity for each bullet tested. This will be an iterative, trial and error procedure. This example will still be useful as a guide even if a different exterior ballistics program is used, because BC values very close to the ones produced by using Infinity should result. This can serve as a check on the procedure developed for any other program.
22.214.171.124 Important Precautions and Points to Consider There are several important precautions and points to consider when measuring ballistic coefficients by this method.
1. Errors in the measurements. There are many sources of errors that can affect BC measurements. They can be separated into two categories: random errors and systematic errors. A random error is an error that may occur in any test round, but that changes in magnitude or direction (i.e., an erroneous increase or decrease in BC value) from one test round to the next. Typical sources of random errors are round-to-round variations in bullet weight, jacket thickness, core homogeneity, etc. A property of random errors is that they can be effectively removed by averaging the BC measurements of several test rounds. We typically fire at least 10 test rounds for each bullet type at each velocity level that we choose for measuring the BC value. The other category of errors is systematic errors. A systematic error is a consistent error that occurs in every test round fired and is nearly the same magnitude and always of the same direction (i.e., always an erroneous increase or always an erroneous decrease in BC value) from one round to the next. Systematic errors are very bad, and every effort must be made to eliminate their sources. The most important sources of systematic errors are errors in the measured distances between chronograph screens and in the measured range distance from the initial velocity chronograph to the final velocity chronograph. For example, suppose the separation distance between screens 1 and 2 in Figure 2.3-1 is supposed to be 10.0 feet. However, when we set up the screens, we make a measurement error of 1/16 inch, so that the true distance is 10.0 feet plus 1/16 inch (10.0052 ft). This error is consistent for every round fired, so that it is a systematic error source for the BC value. Consider round 1 in Table 2.3-1. The chronograph really measures a travel time between screens 1 and 2, and then divides the erroneous distance 10.0 feet by this travel time to give an initial velocity of 2742 fps, which would contain a systematic velocity error. Then, the travel time between screens must have been 3647 microseconds (that is, 10.0 ft divided by 2742 fps). If the true distance of 10.0052 feet had been divided by this travel time, the true initial velocity should have been 2743.4 fps. Now, most chronographs do not read to tenths of a fps; they round off to the nearest whole fps. So our initial velocity chronograph should have indicated a velocity of 2743 fps, instead of 2742 fps. Using Infinity, it is easy to verify that the BC value for round 1 should then have been 0.4560 for this example, instead of 0.4583. [Note that the roundoff imprecision in the chronographed velocity results in a random error, not a systematic error, in BC value. This error can be effectively removed by averaging over several test rounds.] This example illustrates that a measurement error of about 1 part in 2000 in the separation distance between screens 1 and 2 will cause a systematic error in BC value of about 1 part in 200. That is, the small error in separation distance causes an error in BC value that is ten times larger – a very high sensitivity. A similar analysis will show that a measurement error of 1 part in 2000 in the separation distance between screens 3 and 4 of the final velocity chronograph (see Figure 2.3-1) will cause another systematic error of about 1 part in 200 in the measured BC value. Again, this is a very high error sensitivity. The multiplication factor of 10 in these error sensitivities applies to this particular example. In another situation, the error sensitivities would still be high, but the factor of 10 might change upward or downward. This high sensitivity of systematic errors in BC to errors in the separation distances between the screens of the chronographs is primarily why we use a separation distance of at least 10 feet between screens in Sierra’s test range. Chronographs are available with screen separation distances of 1 or 2 feet. These chronographs are convenient because of their light weight and portability, and they are adequate for the purposes of load development where velocity measurement errors of 10 or 20 fps are tolerable. However to measure ballistic coefficients, systematic errors must be no more than 1 part in 2000, and hopefully less than that. For a screen separation distance of 1.0 foot, the maximum separation distance error would then need to be no more than 0.006 inch. Mechanical tolerances in mounting the screens on their supporting structure and positioning the active sensor within each screen are likely to be greater than this number. So, these chronographs should never be used for BC measurements, despite their convenience.
The situation is not as critical for the separation distance between the two chronographs. If somehow we made an error of one part in 2000 in measuring the 103 yard separation distance (i.e., an error of about 2 inches), this error source would contribute a systematic error of about 1 part in 2000 in the BC value. This is a one-to-one error sensitivity, but it shows that we must carefully measure the range separation distance between the two chronographs. We cannot afford an error of a yard, or even a foot. 2. The measured BC value must be calculated for each round individually, and then statistical analysis can be applied to the results. We need to obtain an average BC value from our measurements, to reduce or eliminate random errors, and to obtain a proper value for trajectory computations. To do this, we must calculate the BC of each test round individually, and then average the resulting values to obtain the average BC for the bullet type at each velocity level chosen for the measurements. We cannot first average all the initial velocity values, then average all the final velocity values, and then calculate a BC value from these average velocity values. This approach would lead to an erroneous average BC value, because of the laws of mathematical statistics. Also, there is useful information in the standard deviation and extreme spread of the individual BC values. These statistical parameters yield some knowledge of the stability of the bullets as they fly, as well as the quality provided by the manufacturing process. 3. The BC value for a bullet is likely to vary with bullet velocity as the bullet flies. We have made this point in all of the previous editions of Sierra’s Reloading Manuals. The reason is that the standard drag model (called the G1 drag model) is not a perfect representation of the aerodynamic drag on sporting bullets over the full range of bullet velocities. Therefore, Sierra follows a policy of measuring the BC of each bullet at several different velocity levels and then publishing BC values for each bullet within up to five velocity ranges that together span the total velocity range for the bullet. A glance at the table of BC values for Sierra bullets elsewhere in this manual will illustrate how these values are published. Sierra’s Infinity exterior ballistics program uses all five BC values for each bullet to compute trajectories for any Sierra bullet. 4. The measured BC value is valid for a certain range of bullet velocity. When the range distance between the two chronographs (see Figure 2.3-1) is relatively short — like 100 yds for rifle bullets or 50 yds for handgun bullets — the difference between the initial velocity and final velocity of each round should be no more than 10 percent of the initial velocity. For example, for round 1 in Table 2.31, the difference between the initial velocity (2742 fps) and the final velocity (2549 fps) is 193 fps, which is about 7 percent of the initial velocity. In this situation the BC value derived for each round characterizes the bullet performance in the range between the initial and final velocities. An alternative point of view is that the BC value is valid for a velocity that is midway between the initial and final velocity values. Thus, for round 1 in Table 2.3-1 the BC value 0.4583 is considered valid for a velocity of 2646 fps. This point of view is justified because the difference between the initial and final velocities is a small fraction of the initial velocity. We use this approach when we measure BC values. Another situation that sometimes arises is that a single BC value is needed for a certain hunting or target shooting situation. For example, if you are a hunter and use a particular cartridge load for certain game, and you use a bullet for which the BC is not known, you may need a single value of BC valid for range distances out to a maximum of 400 yds. You can use the procedure explained above to measure an effective BC by placing the initial velocity chronograph near the muzzle of your rifle and the final velocity chronograph 400 yds downrange (carefully measured). This BC value will serve for all ballistic calculations, such as finding effects of changing altitude, changing weather, winds, uphill/downhill shooting, etc. However, you must think of this BC as valid for (a) the muzzle velocity of your cartridge and (b) a range distance of no more than 400 yards. If you change either of these parameters, the BC value may not be valid for accurate trajectory calculations.
5. BC values determined by using Sierra’s Infinity software using the process described are referenced to sea level standard atmospheric conditions. We have made the point in previous editions of the Sierra Reloading Manuals that measured BC values must be reduced to sea level altitude and standard atmospheric conditions at sea level, and we explained how to perform the necessary calculations. This is effectively done in the Sierra Infinity program (and we believe it is done also in other ballistic software programs). When using Infinity, simply enter the altitude, temperature, barometric pressure, and relative humidity at the firing point when beginning the computations. The BC used by Infinity for each round fired then is assumed to be the value for sea level standard conditions. Thus when the calculated velocity or time of flight values equal the measured values (after being corrected for the defined altitude, pressure, etc. during computations) the input BC is referenced to sea-level standard atmospheric conditions. Note that the barometric pressure entered into Infinity is from a barometer at the range or a local weather report. It is NOT the absolute pressure for the range altitude. The absolute pressure necessary for trajectory computations is calculated within Infinity from the altitude and atmospheric data for the firing point. 6. Protect the chronographs from stray bullets. This is a practical consideration when measuring ballistic coefficients. It is extremely embarrassing (especially when the equipment does not belong to you) and very expensive when a stray bullet destroys a screen or an electronics enclosure. The final velocity chronograph is especially vulnerable because it is far from the muzzle. We use a paper target at the downrange location before the final velocity chronograph is moved into position, and fire several shots to make sure the rifle is properly sighted to put bullets through the ―window‖ in the screen. Then, the final velocity chronograph is moved into position for the firing tests. At Sierra’s test range, armor plates also are used to protect the structure and electronics of both chronographs. Even though all firing is done from machine rests, these plates bear the scars of some accidental stray bullets.
2.3.2 Initial Velocity and Time of Flight Method This method, illustrated in Figure 2.3-2, uses two chronometers (time measuring instruments), which measure the time of flight t12 between screen 1 and screen 2, and the time of flight t13 between screen 1 and screen 3 for each round fired. The screen separation distances d12 and d13 are measured precisely and very accurately, and these distances remain the same for all rounds tested. Screens 1 and 2 are located close to the muzzle of the gun and are separated by at least 10 feet. These screens then provide an accurate measurement of the initial velocity of each round, which is obtained by dividing separation distance d12 by time of flight t12 for each round. This initial velocity is valid at a point halfway between screens 1 and 2, and this point is then the reference point for computations performed by Infinity. As in the previous method, a blast shield with a small hole for bullet passage usually is used to protect screens 1 and 2 from muzzle blast, muzzle flash and powder gases exiting the muzzle.
Figure 2.3-2 Test Range Setup for Initial Velocity and Time of Flight Method for BC Measurement
For each round fired the recorded measurements are the times of flight t12 and t13. Because the reference point for the trajectory calculations to be performed in Infinity is halfway between screens 1 and 2, corrections must be applied to the range distance and the bullet time of flight for each round. The corrected range distance for all trajectory calculations is the distance d13 minus half the distance d12; that is, the distance from the center point between screens 1 and 2 to screen 3. The time of flight for the trajectory calculation for each round is the measured time t13 minus half the measured time t12; that is, the bullet time of flight from the center point between screens 1 and 2 to screen 3. When the firing test data have been obtained, Infinity is initialized with the measured altitude, atmospheric conditions and the corrected range distance for all rounds. Then, for each round fired, an iterative search for the correct BC value takes place in the same manner explained in the previous method (Section 126.96.36.199). In each iteration a BC value is guessed, a trajectory is calculated using the measured initial velocity out to the corrected range distance, and the calculated time of flight is inspected. The correct value of BC has been found for the test round when the calculated time of flight matches the corrected measured time of flight as closely as possible. The precautions and points for consideration described in Section 188.8.131.52 for the initial velocity and final velocity method also generally apply to the present method. There is one significant difference. There is one less systematic error source in the initial velocity and time of flight method. There is no final velocity chronograph in this method, so the very sensitive systematic error source associated with measuring the screen separation distance for the final velocity chronograph does not occur. However, any measurement error in the separation distance between screen 1 and screen 2 will cause a highly sensitive systematic error in the BC values. A measurement error in the distance between screen 1 and screen 3 will cause a less sensitive systematic error in the BC values determined by this method. Any digital instrument that measures the travel times between the screens should contribute little or no errors in the measured BC values, because elapsed times can be measured very precisely and accurately.
2.3.3 Doppler Radar Method Christian Johann Doppler was an Austrian physicist and mathematician who first described the Doppler effect in 1842. He found that when a radio wave, light wave or sound wave is transmitted between objects moving with respect to each other, the frequency of the wave is shifted in proportion to the speed of one object relative to the other. In a Doppler radar system, a transmitting antenna transmits a radar beam toward a moving object. The moving object reflects the beam back to a receiving antenna, which is co-located with the transmitting antenna. Because the object is moving, the reflected beam arriving at the receiving antenna has a frequency that is shifted a small but measurable amount from the frequency of the transmitted beam. This frequency shift is proportional to the speed of the moving object relative to the antennas. In our case, the moving object is the bullet, and the radar antennas are located at the firing position. Doppler radar tracks the bullet as it flies and provides measurements of the radial velocity of the bullet with respect to the antennas; that is, with respect to the firing point. The data from the radar are processed mathematically in a computer using very sophisticated software. At any point in the bullet trajectory, the results of these computations are bullet position coordinates (down range, cross range, and vertical directions), bullet velocity components in these directions, and even drag deceleration, all versus time of flight from the firing point. These data are available almost continuously as the bullet flies from the firing point until it impacts the ground. A firing elevation angle of several degrees can be used so that each bullet is tracked continuously as its velocity decreases from the muzzle through the supersonic, transonic and subsonic velocity regions before impact. Knowing the position and velocity of the bullet at any two points along the trajectory makes possible the calculation of a BC value for bullet performance between those two points. Infinity can be used for the BC calculation. The Doppler radar method is far and away the best method of measuring ballistic coefficients, mainly because it provides measurements of bullet performance throughout bullet flight from supersonic velocity levels through subsonic velocity levels. However, Doppler radars are just not readily available. The radar system is very expensive, and a large computer complex is necessary to process the radar data to produce position and velocity data. A crew of several experts is required to operate the instrumentation and process the data. The cost of these capabilities exceeds the affordability limits of all sporting bullet manufacturers, and Doppler radar facilities are available only at some military sites. For the past several years, these authors and other Sierra representatives have been privileged to participate annually for two days in a series of tests conducted at the U.S. Army Yuma Proving Ground near Yuma, Arizona. The Gun Position (shooting site) used for these tests is equipped with a high performance Doppler radar. The facilities are provided by the U.S. Army for tests planned and conducted by the Association of Firearm and Toolmark Examiners (AFTE), which is an association of forensic criminalists from U.S. and international law enforcement crime laboratories. The authors are technical advisory members of AFTE and have suggested tests to be conducted at the Yuma Proving Ground. Measurements of ballistic coefficients versus velocity for a number of bullets of different shapes have been performed over the past three years, and examples will be described in a later subsection.
2.4 Lessons Learned from Ballistic Coefficient Testing Much has been written in previous editions of Sierra’s Reloading Manuals about our BC measurements. We now have more than 30 years of experience in measuring BC values for Sierra’s
line of sporting bullets, as well as some bullets from other manufacturers, and we have learned a great deal. Our observations and lessons learned through this experience are enumerated and summarized below. 1. Ballistic coefficients must be measured by firing tests. We have tried to determine BC values by the method of Coxe and Beugless. We also have tried to determine BC values using bullets of similar shapes to scale the values based on bullet weights and diameters. But we have never been successful in accurately predicting BC values, or determining these values by any method other than firing tests. 2. The ballistic coefficient of each bullet changes with velocity of the bullet as it flies. The ballistic coefficient of a bullet is not constant with bullet velocity. The reason that the BC changes with velocity is that the standard drag function (the G1 drag function) does not characterize the aerodynamic drag on any sporting bullet throughout the full range of its velocity from the gun muzzle to impact. When a bullet is fired with a supersonic muzzle velocity, as its velocity falls there can be a gradual change in ballistic coefficient until the bullet reaches a velocity near 1600 fps (which is in the upper transonic velocity region). When the bullet velocity falls below 1600 fps, radical changes in ballistic coefficient begin to occur. In the next subsection, we will show some examples of this phenomenon for both rifle and handgun bullets. When the bullet velocity is greater than 1600 fps, the G1 drag function is a reasonable model from which to compute the aerodynamic drag on a bullet. The gradual changes in BC value with velocity can be handled in trajectory calculations by adjusting the BC values used in those calculations by changing the BC stepwise as the bullet traverses four or five velocity regions. The trajectory will start with the bullet velocity in one of those velocity regions. As the bullet velocity decreases and crosses the boundary between that initial velocity region and the next lower region, the BC is changed to the value corresponding to the next lower region. This process is repeated as the bullet velocity falls through successively lower velocity regions. When the bullet velocity is less than 1600 fps, the G1 drag function just does not characterize the aerodynamic drag on the bullet. This causes the BC values to vary widely as the bullet velocity falls through the speed of sound (about 1120 fps) and to lower subsonic velocities. The step change method of adjusting BC values is, at best, a crude approximation. This situation is mitigated somewhat by the fact that aerodynamic drag on a bullet diminishes dramatically in the lower transonic and subsonic velocity regions. Consequently, the effect of large ballistic coefficient errors on bullet trajectories is much less than when the bullet velocities are above 1600 fps. For handgun bullet trajectories, the effect is also lessened by the fact that ranges to the targets or the game animals are considerably shorter than for rifles. But at the present time, accurate long-range trajectories simply cannot be calculated for bullets that travel at lower transonic and subsonic velocities. This affects the ballistics of rifle cartridges such as the 30-30 Winchester, 35 Remington, 444 Marlin, 45-70, and the ―Whisper‖ class of cartridges, as well as all handgun cartridges chambered in rifles. This is an area of continuing research for these authors. Ballistic coefficient data have been gathered for a variety of rifle and handgun bullets at transonic and subsonic velocities. Investigations are under way to find modifications to the G1 drag function at velocities below 1600 fps that will enable ballistic coefficients to remain reasonably constant in this velocity region. We hope to be able to report successfully on this research effort at a later date. 3. The G1 drag function is the best standard drag model to use. We have tested several drag functions (G1 for sporting bullets; GL for lead bullets; G5 for boat tail bullets; and G6 for flat base, sharp pointed, fully jacketed bullets). For each drag function we have measured BC values referenced to that function and observed how those BC values change with bullet velocity. We have
chosen G1 because the changes in BC values with bullet velocity are least, and because there is a vast database in the literature on BC values referenced to the G1 standard. Also, to our knowledge all projectile manufacturers refer their published BC values to the G1 drag function, which facilitates comparisons among bullets of different calibers, weights, shapes and manufacturers. 4. Any of the firing test methods measures a ballistic coefficient of the bullet as it flies through the air, including effects imparted by the gun, the cartridge, and firing point environmental conditions, as well as imperfections in the bullet. Theoretically, the BC of a bullet depends only on its weight, caliber and shape. But in a practical sense, the measured BC of a bullet also depends on many other effects. The gun can affect the measured BC value in two important ways: spin stabilization and tipoff moments. A bullet is gyroscopically stabilized by its spin, which is imparted by the rifling in the barrel. If a bullet is perfectly stabilized by its spin, then its longitudinal axis (which is also its spin axis) is almost perfectly aligned with its velocity vector. If a bullet is not perfectly stabilized (which is usually the case), the bullet may not be tumbling, but its point undergoes a precessional rotation as it flies. In previous editions of Sierra’s Reloading Manuals we have described this precessional rotation and have called it ―coning‖ motion to aid in mental visualization of the motion. As the bullet flies, the point rotates in a circular arc around the direction of the velocity vector. Coning motion results in increased drag on the bullet, and any firing test method then yields an effective BC value for the bullet that is lower than the theoretical value. The rifling twist rate in the gun barrel and the muzzle velocity together control the spin rate of the bullet, and therefore control its degree of stability. When a bullet exits the barrel, it generally has a small angular misalignment, which ballisticians call ―yaw.‖ Yaw is caused by tipoff moments of torque applied to the bullet by powder gases exiting the barrel non-symmetrically around the bullet, or by barrel whip or vibrations. This angular misalignment will cause coning as the bullet begins to fly downrange. Coning can also be caused by an abrupt exit of the bullet from the barrel into a crosswind, although BC measurements should never be attempted when winds exist at the firing point. The cartridge used in the firing tests affects the measured BC values mainly through the muzzle velocity it produces. As noted above, muzzle velocity combines with the twist rate in the rifling to produce the bullet spin rate, which in turn controls stability. In addition, BC values change with the instantaneous velocity of the bullet, and so the muzzle velocity directly affects the measured BC value of the bullet. For example, a 180 grain 30 caliber bullet can be fired at a much higher muzzle velocity in the 300 Winchester Magnum than in a 308 Winchester cartridge. The same is true for a 240 grain 44 caliber bullet from a 44 Magnum compared to a 44 Special. So, the measured BC values can be expected to be different just because of the different starting velocities. Altitude and atmospheric conditions at the firing point affect the mass density of the air through which the bullet flies, in turn affecting aerodynamic drag on the bullet. Measured values of BC will depend on the actual conditions at the firing point, unless special pains are taken to convert those measurements to sea level altitude and standard atmospheric conditions at sea level. Unless this is done, the BC of one bullet cannot be compared to the BC of another, because the test conditions may be different. Measurements of BC values must then be reduced to sea level altitude and standard atmospheric conditions at sea level. Using Sierra’s exterior ballistics software program Infinity in the procedures described in Section 2.3 will perform this reduction to sea level standard conditions automatically. Otherwise, measured BC values at nonstandard conditions must be reduced by manual calculations. Reducing measured values to sea level standard conditions by manual calculations has been described in preceding issues of Sierra’s Reloading Manuals, and these procedures are available from Sierra upon request.
The coning motion caused by the initial yaw of a bullet when it exits the muzzle generally damps out as the bullet flies — that is, it decreases in amplitude as the bullet travels downrange. This is because the causes of initial yaw are transient in nature. In other words, these causes occur only at the muzzle and do not persist as the bullet flies. Also, the aerodynamic forces caused by the coning motion are restoring forces (tend to improve stability of the bullet) as long as the amplitude of the coning motions is not large enough to cause loss of stability (tumbling). This is the fundamental cause of many anecdotes heard by these authors that ―my rifle shoots 1.5 MOA groups at 100 yards, 0.8 MOA groups at 200 yards, and 0.6 MOA groups at 300 yards.‖ However, some causes of coning motion are not transient in nature, and can cause sustained coning motions throughout the flight of the bullet. Any imperfection in bullet structure leading to a small center of gravity offset from the bullet longitudinal axis can cause sustained coning motions of the bullet as it flies. Also, any small aberrations in bullet shape, such as a small imperfection in point shape or tail shape, can cause sustained coning motions as the bullet flies. This a very strong reason to shoot bullets of high manufacturing quality.
2.5 Examples of Ballistic Coefficient Measurements This subsection presents a few examples of ballistic coefficient measurements that we have made by the methods described in Section 2.3. As you examine the figures presented here, please bear in mind that they are engineering-type graphs. Also, these examples have been selected to illustrate some of the points made in the preceding discussions of ballistic coefficients. Figure 2.5-1 shows BC measurements for Sierra’s 6.5 mm (.264 inch) 160 grain Semi-point (SMP) bullet versus velocity. This bullet is very long compared to its caliber, and it has a flat base, long bearing surface and a rounded point. Each dot on the figure is a BC measurement made by the initial velocity and time of flight method for a round fired.
Figure 2.5-1. BC measurements for the Sierra 6.5 mm 160 grain Semipoint bullet
The measurements shown in Figure 2.5-1 were made by reducing the cartridge powder load in each ammunition round to achieve successively lower velocities. These measurements were made specifically to illustrate how BC value changes with velocity for this particular type of bullet. Our usual approach to BC measurements is to select three or four discrete velocity levels within the appropriate velocity range for the type of bullet being tested. For example, for this bullet we would select one level at about 2800 fps, the next at about 2400 fps, the next at about 2000 fps, and the final level at about 1600 fps. Ten rounds would be fired at each of these levels, and the average BC value and statistical variations for that velocity level would be determined. Then, we would determine recommended BC values, velocity subranges and subrange velocity boundaries for the bullet type being tested. Figure 2.5-1 shows convincingly that the BC for this particular bullet type varies continuously with velocity, increasing in value as the bullet flies down-range and its retained velocity drops. The three velocity subranges shown in the figure are recommended for use in computing ballistic trajectories for this bullet type. Within each subrange a constant BC value is used, and when the bullet velocity crosses a subrange boundary, the BC is changed to the new value. This approach permits very accurate trajectory computations. The trend in BC values shown in Figure 2.5-1, to increase in value as velocity decreases in the range above 1600 fps, seems to be common to hollow point and blunt nose bullets. Spitzer pointed bullets seem to have BC values that vary little with velocity or have BC values that decrease as bullet velocity decreases in the velocity range above 1600 fps. Another observation from the figure is that the scatter in magnitude of the BC value is quite small at all velocity levels, indicating that this bullet type is highly stable at all velocities above 1600 fps. This is generally true of flat base bullets with long bearing surfaces. There is one point in Figure 2.5-1, a low BC value at about 2200 fps, which does not conform to this observation. Such ―wild points‖ happen occasionally. When the average characteristics of any bullet are being measured, it generally is justifiable to ignore such wild points if there are very few. If there are more than a few such points, some investigation is necessary to determine the cause. It is well known that bullet stability is critical for accuracy, but it is not well understood that there are different degrees of bullet stability. BC measurements give us some insights into varying degrees of bullet stability. Figure 2.5-2 shows BC measurements for Sierra’s 22 caliber (.224 inch diameter) 69 grain Hollow Point Boat Tail MatchKing bullet as a function of rifling twist rate. The rifling twist rates in the test barrels varied from one turn in 7 inches (1 x 7) to one turn in 12 inches (1 x 12), except that we did not have a test barrel with a 1 x 11 twist rate. All BC measurements were made by the initial velocity and time of flight method. All rounds were fired at around 2800 fps, which is about a maximum load for this bullet in the 223 Remington cartridge in a bolt action rifle. The figure shows the number of rounds fired at each rifling twist rate and the individual BC measurements for each group, together with the average value, the standard deviation (SD), and the extreme spread (ES) of the group. Looking first at the group for the 1x 7 twist rate, the average BC values for this group of 10 rounds is 0.297 when rounded to three significant figures, sufficient for trajectory computations. The standard deviation (SD) of the measurements, 0.0022, is less than 1.0 % of the average BC value for the group, and the extreme spread (ES), 0.0079, is less than 4.0% of the average BC value. These figures illustrate the criteria that we use (SD no more than about 1.0% of average value, and ES no more than about 5.0% of average value) to determine whether the measured data are ―good.‖ If either of these criteria is seriously exceeded, we look for a reason or repeat the measurements.
The 12 round group for the 1x8 twist rate also has an average BC value of 0.297. The standard deviation for the group is 0.0039, which is about 1.3% of the average BC value, and the ES, 0.0129, is about 4.3% of the average BC value. This group obviously is not quite as ―tight‖ as the previous group, but we would not call this ―bad‖ because the SD does not seriously violate our standard deviation criterion. The groups for the 1x9 and 1x10 rifling twist rates also satisfy the stanFigure 2.5-2. BC measurements versus barrel twist rates for Sierra’s
dard deviation and extreme spread criteria, but the average BC values are beginning to decrease. For the 1x9 twist rate, the average BC is 0.295, and for the 1x10 twist rate, the average BC value is 0.294. The group for the 1x12 rifling twist rate shows a striking decrease in average BC value and increase in the scatter in the measurements. We attribute these changes to a decrease in stability of the bullets fired from the barrels with the slower rifling twist rates. We emphasize that none of the bullets tumbled during flight; all bullets printed point-first on paper targets just behind screen 3 in the test setup (see Figure 2.3-2). Our interpretation of the data is as follows. All bullets have some coning motion just after they leave the barrel, as described in Section 2.4. When the rifling twist rate is fast (e.g., the 1x7 and 1x8 twist rates in Figure 2.5-2), the coning motion is small, and the dominant causes of the scatter in the BC measurements are random sources of error such as those described in Section 184.108.40.206. When the rifling twist rate in the barrel is slower (e.g., the 1x9 and 1x10 twist rates in Figure 2.5-2), coning motion increases in magnitude, and it becomes a systematic source of BC measurement error. This systematic effect causes the average value of the BC measurements to decrease, while the scatter in the measurements, caused by random sources of error, does not increase dramatically. In other words, the increased coning motion causes all the bullets to experience increased drag, and on
average, they experience the same increase in drag, which causes a reduced average BC for the group. The random causes of BC error are not overwhelmed by the coning motion, so that the scatter in the BC measurements is about the same. When the rifling twist rate is very slow for the bullet (e.g., the 1x12 twist rate in Figure 2.5-2), we believe that the coning motion increases dramatically. It certainly has a systematic effect on measured BC, and it also has a random round-to-round variation, which overwhelms the random errors associated with small variations the bullet shape or construction. In this situation, fired bullets are only marginally stable, and accuracy is usually very poor. When long, slender, heavy bullets are used in any caliber, fast rifling twist rates are necessary for good bullet ballistic performance and accuracy. Figure 2.5-3 shows BC measurements made for Sierra’s 30 caliber (.308 inch diameter) 190 grain Hollow Point Boat Tail MatchKing bullet as a function of rifling twist rate. The twist rates in the test barrels varied from one turn in 8 inches (1x8) to one turn in 14 inches (1x14), except that we did not have a test barrel with a 1x13 twist rate. All BC measurements were made by the initial velocity and time of flight method. All rounds were fired at around 2350 fps using the 308 Winchester cartridge. Fifteen rounds were fired for each rifling twist rate. Figure 2.5-3 shows the same characteristics for the 190 grain 30 caliber bullet as were observed in Figure 2.5-2 for the 69 grain 22 caliber bullet. The average BC values for the groups are relatively consistent for rifling twist rates from 1x8 through 1x11. The criteria for standard deviation and extreme spread are satisfied very well for these groups, and the scatter patterns are tight. The group for the 1x12 rifling twist rate has a lower average BC value, and with the exception of one ―wild‖ round, the scatter pattern is tight. However, when the rifling twist rate is 1x14, a dramatic decrease in average BC value occurs, with a large increase in the scatter of the BC measurements. This bullet could be used in a barrel with a 1x14 twist rate only if it were fired at a considerably higher velocity to improve stability, such as in one of the 300 Magnum cartridge types. Bullet coning motions usually tend to damp out as the bullet travels down-range. That is, the coning motion of a bullet is largest when it leaves the muzzle and grows smaller as the bullet flies downrange, basically because of air friction. Some shooters refer to this effect as the bullet ―going to sleep,‖ and it can be observed in BC measurements. The effective BC of a bullet is often higher if the measured range between the initial and final chronographs (for the measurement method of Section 2.3.1) or between the initial chronograph and the time of flight screen (for the measurement method of Section 2.3.2) is closer to 200 yards rather than 50 or 100 yards. This effect is illustrated in Figure 2.5-4 for Sierra’s 30 caliber 190 grain Hollow Point Boat Tail MatchKing bullet. Two separate sets of BC measurements for this bullet are shown — one made by the initial velocity and time of flight method with a 50-yard measured distance between the initial chronograph and the time of flight screen, and the other made by the initial and final velocity method with a 250 yard measured distance between the two chronographs.
Figure 2.5-3. BC measurements versus barrel twist rates for Sierra’s .308‖ inch diameter 190 grain Hollow Point Boat Tail MatchKing bullet
The two groups of measurements were made at different muzzle velocities (about 135 fps different in average values), but the velocities were close enough that a valid comparison between BC values can still be made. It is evident that the average BC value of 0.532 for the measurements made over the 250 yard distance is almost 10% higher than the average BC value of 0.485 for the measurements made over the 50 yard distance. This is attributed to the coning motion damping out over the longer measurement range. Note that the scatter pattern for the 250 yard measurements is slightly worse than the scatter pattern for the 50 yard measurements. However, recall that we believe the difference in the average BC values is caused by systematic coning motions, while the scatter pattern in each case is caused by random round-to-round variations in bullet characteristics. It is often neither possible nor practical to have large measurement range distances, such as 200 yards, and this can be a disadvantage in both of these methods of measuring ballistic coefficients. To begin with, the downrange screens are at greater risk of being struck by stray bullets because of aiming errors or cartridge loading errors. If a stray bullet strikes a screen or an electronics box, the result is both embarrassing and expensive. The test sequence is interrupted, and a new screen must be purchased. If a final velocity chronograph is located downrange, it must be read for each round fired. Often this necessitates walking the 200 yards or so downrange to read the chronograph. Figure 2.5-4.
Of course, the final velocity instrument can be placed at the firing point, but then two coaxial cables must be routed from the final velocity screens back to the firing point to conduct the start and stop signals for the chronograph. Electrical pulses travel on coaxial cables at speeds of 60 to 80 percent of the speed of light, or 0.6 to 0.8 foot per nanosecond. This may cause a significant time delay for pulses traveling over those cables, and the rise and fall times of the electrical pulse signals are also lengthened. These effects can cause systematic errors in both methods of measuring BC values when long runs of electrical cables are necessary. When setting up to measure ballistic coefficients, great care must be taken to minimize these effects. Figure 2.5-5 shows BC measurements for Sierra’s 22 caliber 80 grain Hollow Point Boat Tail MatchKing bullet. Five groups of rounds were fired at muzzle velocities ranging from about 2800 fps to about 1600 fps. Note first that this bullet has a surprisingly high BC for a 22 caliber bullet. In fact, it is higher than the BC values of some 30 caliber bullets with weights up to 150 grains. The two groups of measurements at about 2000 fps and 1600 fps have average BC values that are lower than the measurements at the higher velocity levels. Also, the scatter pattern of the group fired at about 1600 fps is somewhat larger than the scatter patterns of the groups fired at higher velocities. We believe that these effects are caused by coning motions of the bullets. The rifling twist rate in the barrel (1x7) was just not fast enough to well stabilize the bullets fired with muzzle velocities near 2000 or 1600 fps. The increased coning motion at 2000 fps causes a systematic decrease in the BC, while the coning motion at 1600 fps is severe enough to cause both a systematic decrease and random variations in the BC values. This illustrates a disadvantage of measuring ballistic coefficients using the initial and final velocity method or the initial velocity and time of flight method. The only way to get BC measurements at low bullet velocities with either of these methods
is to fire the bullets at low muzzle velocities where rifling twist rates are not fast enough to stabilize the bullets sufficiently well to get highly accurate measurements. When ballistic coefficients can be measured by the method described in Section 2.3.3 using a Doppler radar system, the disadvantages of the other two methods are completely avoided. The measurements are more accurate and complete, and important characteristics of ballistic coefficients are fully revealed. Figures 2.5-6 and 2.5-7 are BC measurements made using the Doppler radar at the Yuma Proving Ground for two Sierra bullets, the .338 inch diameter 300 grain MatchKing and the .224 inch diameter 77 grain MatchKing. [Note in both these figures that the velocity axis has been reversed from the previous graphs. Bullet velocity starts at a high value at the left end of the axis and decreases toward the right end of the axis.] The 338 MatchKing rounds were fired at about 2950 fps in 338-378 Weatherby cartridges at an elevation angle of 20 degrees at the firing point. Each round was tracked downrange until each bullet was ―lost‖ by the radar as it sank into ground clutter (low brush and other objects interfering with radar signal trans-
Figure 2.5-6. BC measurements by the Doppler radar method for
mission/reception). This occurred when the velocity of each bullet was about 700 fps, well below the speed of sound. The BC values were calculated for three rounds and are plotted in Figure 2.5-6. The dots in the graph generally indicate very close BC measurements for all three bullets, except where the dots are separated a small distance, where they indicate values for individual bullets. The vertical bars indicate scatter in the BC values for the three rounds where these BC values were calculated at the same velocity. The BC values shown in the figure are typical for all the test rounds fired with this bullet. For the 22 caliber 77 grain MatchKing in Figure 2.5-7, all rounds were fired at about 2600 fps in 223 Remington cartridges at about 20 degrees elevation angle. Each round was tracked until velocity fell to about 600 fps, where the radar signal was lost in ground clutter. Again, BC values for three rounds were calculated and plotted in Figure 2.5-7, and the vertical bars indicate the scatter in BC values for the three bullets. The BC values shown in the figure are typical for all the test rounds fired. The Doppler radar method of BC measurement is clearly the best for several reasons. First, each bullet is fired at the maximum muzzle velocity obtainable from the gun and cartridge, and then is observed by the radar almost throughout its entire flight. There is no need to download cartridges to measure BC values at low velocity and suffer the errors caused by reduced bullet stability, in turn caused by the reduced spin rate. A second significant reason is that each bullet can be allowed to travel downrange from the
Figure 2.5-7. BC measurements by the Doppler radar method for Sierra’s .224 inch diameter 77 grain Hollow Point Boat Tail MatchKing bullet muzzle for 150 or so yards before BC measurements begin, so that the initial coning motion of the bullet at the muzzle can damp out or at least damp to its minimum value. BC measurements can then be computed for each round from the radar-tracking data as frequently as desired along the bullet trajectory. The third major reason is that each bullet can be observed throughout its range of velocities, as it slows from supersonic velocities through transonic velocities, through the speed of sound (about 1120 fps), and then on down to low subsonic velocities. Figures 2.5-6 and 2.5-7 show some remarkable BC characteristics for these rifle bullets. At supersonic velocities, the BC of each type of bullet is nearly constant, showing that the G1 drag model is appropriate for these sporting bullets in this velocity range. When bullet velocity falls below about 1600 fps in the transonic velocity range, the BC of each bullet type decreases dramatically. A minimum BC value is reached just above the speed of sound. A dramatic increase in BC value occurs just below the speed of sound. A maximum BC value is reached when the bullet velocity is about 1000 fps, and then the BC value decreases as bullet velocity falls to lower subsonic levels. A similar type of BC variation has been observed for handgun bullets. An
Figure 2.5-8. BC measurements by the initial velocity and time of flight method for Sierra’s .44 caliber 240 grain Jacketed Hollow Cavity Sports Master handgun bullet example is shown in Figure 2.5-8 for Sierra’s 44 caliber (.4295 inch diameter) 240 grain Jacketed Hollow Cavity Sports Master bullet. The BC for this stubby, hollow-point bullet behaves differently compared to the rifle bullets in Figures 2.5-6 and 2.5-7. It rises dramatically just above the speed of sound, falls dramatically just below the speed of sound, and then rises to a peak value at about 1050 fps, decreasing from this peak at lower subsonic velocities. These measurements were made by the initial velocity and time of flight method. This same ballistic coefficient behavior has been observed for Sierra’s 9mm 115 grain Full Metal Jacket Tournament Master bullet, and for a 41 caliber 220 grain Full Patch Jacketed bullet (no longer in production). This behavior appears to be characteristic of many, if not all, handgun bullets. As mentioned earlier, this ballistic coefficient behavior implies that the G1 drag model does not characterize sporting bullets for rifles or handguns very well at velocities lower than about 1600 fps. This in turn means that we cannot calculate highly accurate trajectories for bullets at low velocities. This situation is somewhat mitigated by the fact that the total aerodynamic drag on a bullet decreases dramatically as bullet velocity falls through the speed of sound to subsonic velocity levels. This is an area of intensive research by these authors. We are privileged to have access to Doppler radar tracking data for a large number of bullets tested at the Yuma Proving Ground, through the courtesy of the YPG and the Association of Firearm and Toolmark Examiners. Our research has two prime objectives. The first is to better understand BC measurement techniques. We must find a way to measure bullet ballistic coefficients in Sierra’s test range with the limited capabilities of that approach, and then to correct those measurements to true bullet BC values based on what we learn from the Doppler radar method. The Doppler radar method, while clearly the best method, has overwhelming practical disadvantages that prevent its use by commercial bullet manufacturers such as Sierra. The cost of the instrumentation is several hundred thousand dollars, a trained crew is
necessary to operate and maintain the radar, and a test range several miles in length is necessary for testing bullets. Our approach is to use the other two methods of BC measurement in Sierra’s test range and to use the Doppler radar data for selected bullets to learn how to interpret the test data from Sierra’s range to obtain true values of bullet BC. The second prime objective of our research is to determine a modification of the G1 drag function for velocities below 1600 fps. This change must make the modified G1 function better characterize rifle and handgun bullets at low velocities, so that BC values referenced to this modified function do not change so radically with bullet velocity. Then, we can compute accurate long-range trajectories for rifle and handgun bullets traveling at low velocities.
3.0 Exterior Ballistic Effects on Bullet Flight When a bullet flies through the air, two types of forces act on the bullet to determine its path (trajectory) through the air. The first is gravitational force; the other is aerodynamics. Several kinds of aerodynamic forces act on a bullet: drag, lift, side forces, Magnus force, spin damping force, pitch damping force, and Magnus cross force. The most important of these aerodynamic forces is drag. All the others are very small in comparison when the bullet is spin-stabilized. To a very good approximation, the drag force and the gravitational force together determine the trajectory of any spinstabilized bullet. The other small aerodynamic forces cause only small variations from the trajectory that is determined by drag and gravity acting alone on the bullet. For trajectory computation purposes, this allows the bullet to be modeled as a point mass with a ballistic coefficient. In other words, the bullet motion is modeled as a three degrees of freedom (3 DOF) physics problem, that is, as a point mass with three translational degrees of freedom. The bullet trajectory calculated with this approach is almost exactly correct. If we wished to treat the small variations caused by the other aerodynamic forces, it would be necessary to model the bullet as a spinning body with six degrees of freedom (6 DOF), three translational and three rotational degrees of freedom. This approach, while more exact, is very complex in both a physics and a mathematical sense, and it is neither practical nor necessary to use this approach for sporting purposes. In Section 4.0 we will discuss a few 6 DOF effects that can be observed on long range sporting bullet trajectories. However, it will be obvious that we can treat these effects only qualitatively. The only quantitative observation we will be able to make is that these effects are small. The 3 DOF model of bullet flight is used in all exterior ballistics programs available to handloaders, including Infinity. In the 3 DOF model, the gravitational force always acts vertically downward at the location of the bullet, regardless of the bullet’s orientation relative to the vertical direction. Aerodynamic drag always acts opposite to the bullet’s direction of travel through the air. That is, drag always tends to slow the bullet down. As the direction of bullet travel changes during the trajectory, so does the direction of the drag force. Drag is a very complex function of bullet velocity relative to the air, and it depends critically on the density of the air and on the speed of sound in the air through which the bullet is moving. Air density, in turn, depends on true barometric pressure, temperature of the air, and relative humidity at the location of the bullet as it flies. These atmospheric parameters depend critically on the altitude of the bullet above sea level. The speed of sound in the air depends primarily on temperature of the air, and so it also depends on altitude at the bullet location. In addition to gravity and drag, there are other strong effects on the path of a flying bullet. Wind, which is any motion of the air mass through which the bullet is flying, is one such effect. A headwind
or tailwind will cause the bullet to experience more or less drag, respectively, than it would if it traveled in still air. A crosswind causes the bullet to turn in the direction that the crosswind is blowing. A vertical wind causes the bullet to turn upward or downward, following the vertical wind that blows upward or downward, respectively. And so, the path of the bullet (trajectory) is changed by winds compared to the path it would have in still air. The 3 DOF model of bullet flight permits the wind effects to be computed almost exactly for a spin-stabilized bullet in an exterior ballistics program such as Infinity. Shooting uphill or downhill also causes significant changes to the trajectory of a flying bullet compared with its trajectory on a level firing range. In fact, if a gun is sighted in on a level range, a bullet fired either uphill or downhill always shoots high relative to the shooter’s line of sight through the gun sights. Two effects contribute to the trajectory changes. One is geometrical, and we will describe this effect in a later subsection. The other is the fact that when a bullet travels upward or downward relative to the firing point, the density of the air changes, affecting the drag. These trajectory changes also are computed in Sierra’s Infinity program. The temperature of the propellant (powder) in a cartridge at the instant of ignition can have a strong effect on chamber pressure. If a gun is sighted in on a target range at some temperature, and then is fired in another environment in which the propellant temperature is different, the muzzle velocity of a bullet can be significantly different compared to what it was when the gun was sighted in. The difference in muzzle velocity, of course causes a change in bullet trajectory. The temperature sensitivity of different powders varies from one type to another. Only the manufacturer of any type of powder can quantify the temperature sensitivity of that powder to a handloader. But, an exterior ballistics program can be used to explore the sensitivity of the trajectory of any bullet to changes in muzzle velocity. Sierra’s Infinity program is designed to make this computation especially easy. All the effects mentioned above are described in more detail in the following subsections of this Section 3.0. They are treated one at a time, so that the reader can determine which are most important for his or her particular shooting application. However, one of the most significant advantages of an exterior ballistics software program like Infinity is that any combination of these effects can be explored for any one or more cartridges. For example, the effect of a wind and altitude change can be computed, or the effect of a change in atmospheric conditions can be determined. Likewise, the effect of a change in muzzle velocity when shooting uphill or downhill can be determined, or a 44 Magnum handgun can be compared with a 44 Magnum rifle. Any other imaginable combination of cartridges and shooting conditions can be explored. Four more interesting topics are described later in this section. The first is sighting in, or zeroing, a gun at a selected range distance under various shooting conditions. This is a familiar process to almost all shooters, but a number of interesting questions frequently arise. The second topic is Point Blank Range (PBR), which is a very interesting concept for hunters and shooters participating in the silhouette games. PBR is the range distance out to which a shooter can hold his sights directly on a target, without holding over or under, and be assured of a hit. A technique is described to maximize the PBR of any gun/cartridge combination for a target of a given vertical size. A third topic which is of frequent interest to shooters is the maximum range to which a gun/cartridge combination will shoot and the elevation angle of the bore that must be used to achieve that maximum range. Designers of public shooting ranges often address this question. The final topic is the maximum altitude a bullet will reach if a gun is pointed straight up and fired. This is often a question for safety reasons. As a reference for discussions in the following subsections, we must describe the important parameters of a bullet trajectory. Figure 3.0-1 has been prepared as an aid to this description. A bullet exits the muzzle of a gun in the direction of the extended bore line. After it exits the muzzle, gravity causes the bullet to begin to fall away from the extended bore line. The bullet trajectory then arcs
downward, as shown by the bold line in Figure 3.0-1. ―Drop‖ is a term used to denote the vertical distance from a point on the extended bore line to a corresponding point on the trajectory, at any range distance from the muzzle. It is very important to understand that ―drop‖ is measured in the vertical direction (direction of gravity), regardless of whether the barrel of the gun is level (as shown in Figure 3.0-1) or tilted upward or downward. In Figure 3.0-1 the drop (do) at range (Ro) is illustrated.
Figure 3.0-1. Illustrating the Parameters of a Bullet Trajectory The shooter, however, aims the weapon at a target using the line of sight through the gun sights. The line of sight is not parallel to the extended bore line; there is a small angle between the two lines, which is greatly exaggerated in Figure 3.0-1. From the shooter’s point of view through the gun sights, the trajectory begins below the line of sight by a distance equal to the sight height, then rises to cross the line of sight, and then arcs over to again cross the line of sight at a second point called the zero range Ro. The sight height is a very important parameter. With iron sights, it is the distance from the centerline of the bore to the tip of the front sight, as shown in the figure. With telescope sights, the sight height can be taken as the distance from the center-line of the bore to the axis of the telescope, which also is the centerline of the objective lens. ―Bullet path‖ is a term used to denote the perpendicular distance between a point on the line of sight and the corresponding point on the bullet trajectory at any range distance from the muzzle. It is very important to understand that ―bullet path‖ is measured perpendicular to the line of sight regardless of whether the barrel of the gun is level or tilted upward or downward. Bullet path is then a measure of where the bullet would be as ―seen‖ by the shooter, if that were possible. At the muzzle, the bullet path is negative by an amount equal to the sight height, because the bullet starts out below the line of sight. The bullet then rises to first cross the line of sight, and then the bullet path is positive, reaching a maximum value at a distance of about 55% of the zero range distance Ro. The bullet path then decreases to a zero value at the zero range and goes negative at ranges greater than the zero range.
3.1 Effects of Altitude and Atmospheric Conditions The effects of altitude and atmospheric conditions on aerodynamic drag are very closely coupled and must be treated together. This was not understood very well by ballisticians until about the beginning of the 20th century. Many firing tests took place in Europe in the latter half of the 19th century, especially in England, Germany, France and Italy, in an effort to understand aerodynamic drag and develop theoretical models for drag. Ballisticians found it difficult to compare measured data when the firing tests were made at locations having different altitudes and different atmospheric conditions. Ballisticians gradually came to realize that drag measurements made in different locations, or even at the same location under different atmospheric conditions, could not be compared unless the measurements were somehow referenced to a set of standard altitude and atmospheric conditions. This led to the adoption of a standard set of altitude and atmospheric conditions to which measurements could be referenced. At the same time, analytical methods were developed to convert data measured at nonstandard altitude and atmospheric conditions to their standard values. Data from different locations and/or different atmospheric conditions could then be compared. In the United States, standard altitude and standard atmospheric conditions were adopted by the U.S. Army Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland at about the beginning of the 20th century. These conditions, called the Standard Metro conditions, are used for ballistics computations.
The Standard Metro conditions are: Altitude: Barometric Pressure: Temperature: Relative Humidity:
Sea Level 750 mm Hg = 29.53 inches Hg 59°F = 15°C 78 percent
(Hg denotes the chemical element mercury)
The values of air density and speed of sound corresponding to these conditions are: Air Density: Speed of Sound: 0.0751265 lb/ft3 = 1.2030 kg/m3 1120.27 fps = 341.46 m/s Also, the acceleration due to gravity used for ballistics computations is:
Acceleration due to Gravity: 32.174 fps = 9.80665 m/s The drag function G1 is referenced to these standard conditions, and ballistic coefficients are therefore referenced to the same conditions. Of course, these standard conditions are used for reference only; it would be a very rare event if anyone were to shoot a gun under these standard conditions. So, the question and the problem is how to calculate real world trajectories at different altitudes and under different atmospheric conditions. The historical approach to this problem has been to first extend the Standard Metro atmospheric conditions to altitudes higher than sea level, that is, to create a ―Standard Metro atmosphere‖ versus
altitude. Table 3.1-1 shows the Standard Metro atmospheric conditions versus altitude up to an altitude of 15,000 feet above sea level, which is sufficient for hunting and target shooting on the North American continent. The next step is to treat the differences between actual atmospheric conditions at any altitude point and the standard atmospheric conditions at that altitude as small variations from the standard conditions. This approach has been successful for several reasons. The main reason is that air density decreases dramatically with altitude, while it changes much less dramatically with small differences between actual atmospheric conditions and standard conditions at any given altitude. Furthermore, the small change in air density caused by a small difference between actual air temperature and standard air temperature at any altitude point tends to be offset by the change in air density caused by a small difference between actual barometric pressure and standard barometric pressure at that altitude point. This is because a higher-than-normal temperature (a warm, balmy day) tends to be accompanied by a higher-than-normal barometric pressure of the atmosphere. That is, high temperature tends to decrease air density, while high pressure tends to increase air density. The air density ratio column in Table 3.1-1, which is the ratio of standard air density at altitude to the standard air density at sea level, shows that the air density decreases rapidly as altitude increases. Air density is a direct
Table 3.1-1 Standard Metro Atmospheric Parameters versus Altitude
Altitude (Feet) Sea Lev 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000
Air Density Ratio (at Alt./at Sea Lvl) 1.0000 0.9702 0.9414 0.9133 0.8862 0.8598 0.8342 0.8094 0.7853 0.7619 0.7392 0.7172 0.6959 0.6752 0.6551 0.6356
Temp. (Deg F) 59.0 55.4 51.9 48.3 44.7 41.2 37.6 34.1 30.5 26.9 23.4 19.8 16.2 12.7 9.1 5.5
Baro Pressure (mm Hg) (in Hg)
Speed of Sound Ratio (at Alt./at Sea Lvl)
750.0 722.7 696.3 670.9 646.4 622.7 599.8 577.8 556.6 536.1 516.3 497.3 478.9 461.1 444.0 427.6
1.0000 0.9873 0.9744 0.9614 0.9483 0.9350 0.9216 0.9080 0.8943 0.8805 0.8666 0.8525 0.8383 0.8239 0.8094 0.7948
29.53 28.45 27.41 26.41 25.45 24.52 23.62 22.75 21.91 21.11 20.33 19.58 18.85 18.16 17.48 16.83
multiplier in the equation for the drag force on a bullet, and because of this, the drag force also decreases rapidly as altitude increases. This decrease in air density with altitude has by far the largest effect on a bullet trajectory, compared to the actual atmospheric conditions and the speed of sound versus altitude. As mentioned in the preceding paragraph, the differences between actual temperature and standard temperature, and between actual barometric pressure and standard barometric pressure, have small
effects on a bullet trajectory compared to the effect of decreasing air density, and these effects usually tend to offset each other due to weather patterns. The speed of sound ratio column in Table 3.1-1, which lists the ratio of the standard speed of sound at altitude to the standard speed of sound at sea level, shows that the speed of sound also decreases quite rapidly with altitude. However, the speed of sound is not a direct multiplier in the equation for drag force. In fact, it enters the equation in such a way that its effect on the drag force is much smaller than the effect of the decrease in air density. The true speed of sound does vary slightly from the standard value because of actual weather conditions, but the effect of the variation is considerably smaller than the small effect of the standard speed of sound. Humidity also has a small effect on a bullet’s trajectory, and at all altitudes. Humidity affects the air density, tending to decrease the air density a small amount, depending on the relative humidity in the atmosphere and the vapor pressure of water at the temperature of the atmosphere. The effect of humidity is generally worst at locations near sea level on very hot days, but even under these conditions, the effect is small. For example, for a location near sea level on a 90°F day with barometric pressure the same for both situations, absolutely dry air (zero relative humidity) is not quite 0.02 percent MORE dense than air saturated with water vapor (fog, meaning 100 percent relative humidity). This seems strange; wet air feels ―heavier‖ than dry air. But it is true because a water molecule weighs less than a nitrogen molecule, which it displaces if the pressure and temperature remain the same. This tiny change in air density is not completely negligible for long-range shooting. For example, under these same atmospheric conditions, the drop at 1000 yards for Sierra’s .308‖ diameter 168 grain MatchKing bullet fired at 2700 fps muzzle velocity will be about 2.4 inches more for absolutely dry air than for saturated wet air.
A word about barometric pressure. In this country, the National Weather Service and local weather bureaus report sea level-referenced barometric pressures regardless of location. For example, if you were in New York City (at sea level) on a balmy day, the barometric pressure might be reported near 30 inches of mercury (in Hg). If you were in Denver, CO, (5200 ft altitude) on a balmy day, the barometric pressure might also be reported near 30 in Hg. Now, the true barometric pressure at the altitude of Denver should be about 25 in Hg, not 30 in Hg. Our weather bureaus report sea level-referenced barometric pressures so that citizens can compare the weather in Denver with the weather in New York, or Los Angeles, or Fairbanks, AK, or Salt Lake City, or anywhere else in this nation. Also, the barometer instruments that we can purchase in stores are designed to read out sea level-referenced barometric pressures. Now, of course, the trajectory of a bullet at any location depends on the true atmospheric pressure at that location, not at sea level. Sierra’s Infinity program takes this into account. It is important to realize that Infinity is designed so that the user must enter the altitude of the shooting location and the sea level-referenced barometric pressure at that location, as well as the temperature and the relative humidity (if known). These parameters can be obtained from TV, a local weather station, or portable instruments. Then, Infinity will calculate the true barometric pressure at the firing point from atmospheric variation laws coded into the program. A great advantage of the standard atmospheric conditions is that, based only on altitude, bullet ballistics can be calculated for locations where the true atmospheric conditions are unknown or unpredictable, and the resulting trajectories will be accurate enough for most practical purposes. To illustrate this, let’s consider an example. Suppose that a hunter living near St. Louis, MO, has a Model 70 Winchester rifle in 300 Winchester Magnum that he uses to hunt mule deer and elk in western Colorado at an altitude near 8500 feet above sea level. His gun is telescope sighted. He loads Sierra’s .308" dia 200 grain Spitzer Boat Tail (SBT) GameKing bullet at 2800 fps muzzle velocity for hunting. He sights his gun in at a target range near St. Louis that is located at an altitude near 500 feet above sea level. The question is, if he sights his rifle in at the target range near St. Louis, where will his gun shoot in western Colorado where he intends to hunt? Sierra’s Infinity program will be used to answer this
question. Suppose he sights his gun in on a late summer day in St. Louis when the temperature at the target range is 92°F, and a local weather report lists the barometric pressure at 30.25 in Hg and the relative humidity at 90 percent. For the 300 Winchester Magnum, he uses a zero range of 300 yards. When in Colorado he will use a laser rangefinder, and he will limit his shots at mule deer or elk to no more than 500 yards. After he finishes sighting his gun in, he returns to his home and performs the following calculations on his personal computer using the Infinity program. He calculates three trajectories for the 200 grain SBT GameKing bullet in the 300 Winchester Magnum cartridge and carefully examines the bullet path parameter from the output data. [Bullet path is the trajectory variable that locates the bullet relative to the shooter’s line of sight through the gun sights as the bullet travels downrange. It is most important because it tells the shooter how high or low his bullet will strike the target, or how much he has to hold over or hold under a target at any downrange location.] The first trajectory is a reference trajectory for the environmental conditions at the target range near St. Louis. Then, he uses the ―Trajectory Variations‖ feature inInfinity to calculate a trajectory in his hunting location, first based on standard atmospheric conditions only, and then based on atmospheric conditions that he predicts based on his previous experiences in the hunting area. So, for the .300 Winchester Magnum cartridge, he selects the Sierra .308" dia 200 grain Spitzer Boat Tail GameKing bullet from the ―Load Bullet‖ library in Infinity, and selects the ―Normal Trajectory‖ mode of operation of the program. In the ―Trajectory Parameters‖ list, he sets the muzzle velocity at 2800 fps, maximum range at 500 yards, range increment at 50 yards, zero range at 300 yards, the elevation angle at 0 degrees, and the sight height at 1.75 inches because his telescope sight has a large objective bell. In the ―Environmental Parameters‖ list he sets the conditions for the target range near St. Louis, that is, barometric pressure at 30.25 in Hg, temperature at 92°F, altitude at 500 ft, humidity at 90 percent, and the wind speeds to 0 mph. He then commands Infinity to calculate the reference trajectory for the St. Louis environs. The bullet path numbers versus range are listed in Table 3.1-2. It is evident that between the muzzle and the zero range the bullet rises a little more than 5 inches maximum above the line of sight, but at 500 yards. the bullet is nearly 30 inches low. The next trajectory calculation is made using the ―Trajectory Variations‖ capability in Infinity. The hunter selects the ―Environmental Parameters‖ option in that mode and makes the following changes to calculate the trajectory variations based on standard atmospheric conditions at the hunting location. The standard conditions are barometric pressure at 29.53, temperature at 59, and humidity at 78. He sets the altitude at 8500 ft. [Recall that Infinity automatically adjusts the standard atmospheric conditions at sea level to the values appropriate for 8500 ft altitude.] He again commands a calculation, and Infinity outputs the bullet path differences shown in the third column of Table 3.1-2. It can be seen that the 300 Winchester Magnum always will shoot high compared to the reference trajectory at St. Louis, but the hunter really needs to make no sighting correction unless possibly when the game animal is close to 500 yards away. The third trajectory for the 300 Winchester Magnum is calculated again by using the ―Trajectory Variations‖ capability. From previous experiences in western Colorado, the hunter believes that the weather will be fair with low humidity, but cold. So he adjusts the barometric pressure to 29.90, the temperature to 20, and the humidity to 20, leaving the altitude at 8500 ft. After the calculation is commanded, Infinity outputs the bullet path differences in the fourth column of Table 3.1-2. Note that these bullet path differences are relative to the bullet path values in the second column of Table 3.1-2 for the reference trajectory at St. Louis, and not to the numbers in the third column. It can be seen that the trajectory calculated for the non-standard atmospheric conditions is very close to the trajectory calculated
with only standard atmospheric conditions at the hunting location. The data in Table 3.1-2 support two observations. The first is that this 300 Winchester Magnum cartridge has a trajectory that is quite flat. The reference bullet path at 500 ft above sea level stays between a little over 5 inches above the line of sight and does not fall more than 5 inches below the line of sight until the range exceeds a little more than 350 yards. At 8500 ft above sea level the bullet path stays within this band until about 365 yards. This is excellent performance, as expected for this very popular magnum cartridge for western hunting. The second observation is that calculating a trajectory for the hunting location based on standard atmospheric conditions gives an accurate representation of the trajectory for actual atmospheric conditions, as pointed out above. Comparing columns three and four in Table 3.1-2 shows that the bullet path changes based on the predicted actual atmospheric conditions are very close to those based on standard conditions. The largest difference between the bullet paths is at 500 yards, and it is just 0.3 inches. This observation holds true for the vast majority of cartridges and atmospheric conditions at all shooting locations. We recommend that when the actual atmospheric conditions are unknown or unpredictable at any shooting location, standard atmospheric conditions be used for the altitude of the location. The altitude of any location usually can be estimated from a topographical map, a local weather station, or an atlas of North America.
Table 3.1-2 Reference Bullet Path and Changes for the 300 Winchester Magnum Cartridge loaded with Sierra’s 30 caliber 200 grain SBT Bullet at 2800 fps. Range
Bullet Path Changes (2)
Bullet Path Changes (3)
Path (1) (inches)
at Hunting Location (inches)
at Hunting Location (inches)
0 50 100 150 200 250 300 350 400 450 500
-1.75 1.72 4.01 5.06 4.80 3.14 0.0 - 4.71 - 11.10 - 19.28 - 29.35
0.0 0.0 0.02 0.07 0.17 0.34 0.60 0.98 1.52 2.23 3.18
0.0 0.0 0.02 0.06 0.15 0.30 0.54 0.89 1.37 2.03 2.88
(1) (2) (3)
Reference trajectory from sighting the rifle in near St. Louis, 500 ft altitude and actual atmospheric conditions at the shooting range (see text). From trajectory calculated for the hunting location, 8500 ft altitude and standard atmospheric conditions (see text). From trajectory calculated for the hunting location, 8500 ft altitude and predicted atmospheric conditions (see text).
Note that a positive bullet path change in columns 3 and 4 means that the gun will shoot higher than the reference trajectory.
3.2 Effects of Winds A ―wind‖ is a movement (relative to the ground) of the air mass through which the bullet flies. The effect of a wind on a bullet’s trajectory depends on the speed of the wind and the direction in which it blows. Every shooter is familiar with headwinds, tailwinds and crosswinds. A headwind is a wind that blows from the target toward the shooter. A tailwind is a wind that blows in the opposite direction, from the shooter toward the target. A crosswind, of course, blows from right to left or from left to right across the line between the shooter and the target. These winds are always considered to blow horizontally. Some shooters are not aware that there also can be vertical winds. These are winds that blow vertically upward or downward across a line between the shooter and the target. Vertical winds are especially important for hunters in hilly or mountainous country. Anytime a wind is blowing against a hillside or mountainside, a vertical component of wind must occur. Long-range target shooters also may suffer from vertical air currents. Mirage is an effect caused by vertical air movements. Mirage makes a distant target ―hazy and jumpy‖ when viewed through the gun sights, in turn making aiming extremely difficult. And the vertical currents also move the bullets upward or downward as they fly downrange. A wind can blow from any direction, and the wind needs to be separated into components in order to compute its effect on a bullet trajectory. Infinity characterizes any wind as a combination of a horizontal wind from any direction together with a vertical wind, if there happens to be one. For a horizontal wind, Infinity makes it particularly easy for a user to enter the wind conditions. A horizontal wind is specified in the ―Environmental Parameters‖ list inInfinity by two parameters, wind direction and the wind speed. The horizontal wind direction is specified as an hour angle on an imaginary clock face denoting the direction from which the wind is blowing. The clock is imagined as lying in the horizontal plane with 12 o’clock being the direction from the shooter to the target, 6 o’clock being the opposite direction (from the target to the shooter), 3 o’clock being the direction from the shooter’s right to left as he or she views the target, and 9 o’clock being the direction from the shooter’s left to right as he or she views the target. Infinity recognizes minutes as well as hours in the data input to specify the horizontal wind direction. For example, a horizontal wind direction entered as 1.30 (denoting 1:30 o’clock) signifies a wind blowing from a direction halfway between 1 o’clock and 2 o’clock, that is a quartering wind blowing toward the shooter. After a horizontal wind direction and speed are entered, Infinity will compute a headwind (or tailwind) component blowing toward (or away from) the shooter, and a crosswind component blowing from the shooter’s right or left. The speed of the horizontal wind must be entered into Infinity in statute miles per hour in English units, or in kilometers per hour in metric units. When the wind has a vertical as well as a horizontal motion, separating the wind into vertical and horizontal components is a more complex task. Every situation is different. If the wind is simply blowing up or down a hillside with a known slope angle, then simple trigonometry can be used to separate the total wind speed into horizontal and vertical components. But such a simple case is not usual. The vertical wind analysis feature of Infinity is most widely used to determine the sensitivity of any trajectory to vertical winds. This knowledge will help a hunter or target shooter to understand the effect of a vertical wind, and to compare performances of different cartridges in windy shooting
situations. Hunters should be especially aware that vertical winds are encountered when hunting in hilly or mountainous terrain, in ravines and close to steep hillsides. A vertical wind component will cause a bullet to shoot high or low, just as a crosswind will cause a bullet to shoot left or right. The direction of a vertical wind component, of course, is known, up or down. So, in Infinity, the vertical wind speed is entered as positive for an upward wind or negative for a downward wind. The vertical wind speed also is entered in statute miles per hour in English units, or kilometers per hour in metric units. The effect of each of the three components of any wind is quite different. This is because the aerodynamic drag on a bullet is a function of the bullet’s speed with respect to the air through which it flies. Therefore, if the air is moving, the drag on the bullet is different than it would be if the air were still. For example, in the case of a headwind acting alone and blowing from the target toward the shooter, the speed of the bullet relative to the air would be greater than it would be if the air were still. Then, the drag on the bullet would be higher, and the bullet would travel slower relative to the ground and drop more than it would if the air were still. On the other hand, for a tailwind acting alone and blowing from the shooter toward the target, the speed of the bullet relative to the air would be less than it would be if the air were still. Then, the drag on the bullet would be lower, and the bullet would travel faster relative to the ground and drop less than it would if the air were still. Generally, unless the wind speed is high and the range is very long, a headwind or tailwind causes only a small deflection of the bullet relative to the still air trajectory. A crosswind acting alone would cause primarily a horizontal deflection (wind drift) of the bullet relative to the trajectory in still air. [Later in Section 4 we will describe how a bullet turns to follow a crosswind because it is spin-stabilized, and how a small vertical deflection of the bullet also occurs in the presence of a crosswind.] The deflection caused by a crosswind is quite large, even for moderate ranges and high velocities of the bullet. In a similar manner, a vertical wind acting alone causes a vertical deflection of the bullet that is quite large relative to the still air trajectory, and also a small horizontal deflection. [These deflections also occur because the bullet turns to follow the wind, as explained in Section 4.] The sensitivity of vertical bullet deflections to vertical wind speeds is just equal to the sensitivity of horizontal bullet deflections to crosswind speeds. A wind that blows from any direction can always be resolved into not more than three components, (1) a headwind or tailwind, (2) a crosswind, and (3) a vertical wind. If a wind blows such that two or all three components occur and act simultaneously on the bullet trajectory, the net effect is somewhat different than simply combining the effects of the components acting separately. Two examples, one for a rifle and one for a handgun, have been prepared to illustrate the effects of the separate wind components and the combined effect of all three components of a wind acting simultaneously. Table 3.2-1 has been prepared, using Infinity, for a 308 Winchester (7.62 NATO) cartridge loaded with Sierra’s 30 caliber 175 grain MatchKing bullet at 2550 fps for a High Power target match with military service rifles, and at the 600-yard stage of competition. Suppose that the rifle has been sighted in at 600 yards under still air conditions (no wind), and the firing range is located at an altitude of 1000 feet above sea level. Suppose also that during the competitive firing a wind blows from a direction of 10:30 o’clock relative to the line of sight from the firing point to the target, and that this wind has a horizontal speed of 15 mph. Suppose also that there is an updraft along the firing range estimated at 1.5 mph.
The total wind speed for this example is then: Total wind speed = Square root [ 15.02 + 1.52] = 15.075 mph
Resolving this wind into three components for the purpose of analysis gives the following: Headwind component = 10.60 mph (from target toward shooter) Crosswind component = 10.60 mph (left to right) Vertical wind component = 1.5 mph (upward) The data in Table 3.2-1 then show the effects of these three wind components, first with each component acting alone, then with two horizontal components acting together, and finally with all three components acting together. The first row in the table shows the effect of a headwind with a speed of 10.6 mph acting alone. The increased drag on the bullet caused by the headwind would make the bullet strike the 600 yard target just 0.62 inch lower than it
Table 3.2-1 Wind Deflections at 600 yards Range Distance for 308 Winchester with Sierra’s .308 dia 175 grain MatchKing Bullet Loaded to 2550 fps Caused by a 15 mph Wind Blowing from 10:30 o’clock and with a Small Vertical Speed
Wind Direction and Speed Bullet Deflection at 600 yards Headwind Crosswind Vertical Wind Horizontal Vertical 10.6 mph 0.0 mph 0.0 mph 10.6 mph 10.6 mph
0.0 mph 10.6 mph (L to R) 0.0 mph 10.6 mph (L to R) 10.6 mph (L to R)
0.0 mph 0.0 mph 1.5 mph (upward) 0.0 mph 1.5 mph (upward)
0.0 in 32.76 in (R) 0.0 in 33.03 in (R) 33.03 in (R)
- 0.62 in (more drop) 0.0 in + 4.64 in (less drop) - 0.62 in (more drop) + 4.05 in (less drop)
would if there were no wind at all, and there would be no horizontal deflection. The second row in the table shows the effect of a crosswind blowing from the shooter’s left to right with a speed of 10.6 mph acting alone. In this case, the bullet would turn to follow the wind, and at 600 yards, it would be deflected nearly 33 inches to the right. A small vertical deflection also would occur, caused by the spin stabilization of the bullet, but Infinity computes a value of 0.0 inches for this small effect for reasons explained later in Section 4. The third row in the table shows the effect of a vertical updraft with a speed of 1.5 mph acting alone. The bullet would turn upward to follow the wind, resulting in a vertical deflection of the bullet on the target of 4.64 inches. A small horizontal deflection also would occur, caused by the spin stabilization of the bullet, but Infinity again calculates a value of 0.0 inches for this small effect for reasons explained in Section 4. It is interesting to note that the sensitivity of the vertical deflection caused by a vertical wind is the same as the sensitivity of the horizontal deflection caused by a crosswind. That is, referring to the second and third rows in Table 3.2-1, 32.76 inches divided by 10.6 mph gives a sensitivity of 3.09 inches horizontal deflection per mph of crosswind. The vertical deflection 4.64 inches divided by the vertical wind speed of 1.5 mph also gives 3.09 inches of vertical deflection per mph of vertical wind speed. This specific sensitivity number applies only to this example bullet fired at this example velocity, but in general the sensitivity to crosswinds and vertical winds is very large for all bullets. The deflections caused by headwinds (or tailwinds), however, are much less sensitive to wind speed, as the example in Table 3.2-1 shows. Furthermore, the vertical deflections caused by headwinds or tailwinds are not linearly related to wind speed. That is, it cannot be said that the vertical deflection caused by a 10 mph headwind is ten times
more than the deflection caused by a 1.0 mph headwind. This same statement is true for tail-winds. Returning to Table 3.2-1, the fourth and fifth rows show the effects of the wind components acting together. If a headwind of 10.6 mph acts with a crosswind of 10.6 mph (a horizontal wind of 15.0 mph blowing from the 10:30 o’clock direction), comparing the fourth row to the second row and then the first row shows that the crossrange deflection grows from 32.76 to 33.03 inches. The vertical deflection remains the same, compared to the effects of the wind components acting separately. The reason that the crossrange deflection grows is that the time of flight of the bullet is slightly longer when the headwind acts on the bullet, and this longer time of flight increases the effect of the crosswind. The same increase in the time of flight, of course, occurs when the headwind acts alone, and so the vertical deflection (0.62 inch) does not change. When all three components of wind act together in this example, the last row in Table 3.2-1 shows that the downward deflection caused by the head-wind component just reduces the upward deflection caused by the vertical wind. Again, there is an interaction among the wind components that changes the time of flight to the target, and so the effects of the wind components acting separately cannot be simply added (or subtracted) to exactly get the effects of the wind components acting simultaneously. Table 3.2-2 has been prepared for a 44 Magnum handgun cartridge with Sierra’s .4295" dia 240 grain Jacketed Hollow Cavity Sports Master bullet loaded to 1300 fps muzzle velocity. Suppose that the handgun has been sighted in at 100 yards under still air conditions (no wind), and the firing range also is located at an altitude of 1000 feet above sea level. Suppose also that during a target shooting session on a different day, a wind blows from a direction of 4:30 o’clock relative to the line of sight from the firing point to the target, and that this wind has a horizontal speed of 15 mph. Suppose that there also is an updraft along the firing range estimated at 1.5 mph.
The total wind speed for this example is then: Total wind speed = Square root [ 15.02 + 1.52] = 15.075 mph Table 3.2-2 Wind Deflections at 100 yards Range Distance for 44 Magnum with Sierra’s .4295" diameter 240 grain Jacketed Hollow Cavity Bullet Loaded to 1300 fps Caused by a 15 mph Wind Blowing from 4:30 o’clock and with a Small Vertical Speed
Wind Direction and Speed Tailwind Crosswind Vertical Wind
Bullet Deflection at 100 yards Horizontal Vertical
+ 0.08 in (less drop)
10.6 mph (R to L)
4.62 in (L)
1.5 mph (upward)
+ 0.66 in (less drop)
10.6 mph (R to L
4.48 in (L)
+ 0.08 in (less drop)
10.6 mph (R to L)
1.5 mph (upward)
4.48 in (L)
+ 0.71 in (less drop)
As before, resolving the total wind into its three components for the purpose of analysis gives the following: Tailwind component = 10.60 mph (from shooter toward target) Crosswind component = 10.60 mph (right to left) Vertical wind component = 1.5 mph (upward) In this handgun example, the horizontal wind has been reversed from the previous rifle example, but the vertical component of the wind is still an updraft of 1.5 mph. The data in Table 3.2-2 show the effects of these three wind components, first with each component acting alone, then with two horizontal components acting together, and finally with all three components acting together. The first row in the table shows the effect of a 10.6 mph tailwind acting alone. The decreased drag on the bullet caused by the tailwind would make the bullet strike the 100yard target 0.08 inch higher than it would if there were no wind at all, and there would be no horizontal deflection. The second row in the table shows the effect of a 10.6 mph crosswind blowing from the shooter’s right to left acting alone. The bullet would turn to the left to follow the wind, and at 100 yards it would be deflected 4.62 inches to the left. A very small vertical deflection also would occur, but Infinity does not compute this deflection, as noted above in the rifle example. The third row in the table shows the effect of a 1.5 mph vertical wind acting alone. The bullet would turn upward to follow the wind, resulting in a vertical deflection of the bullet on the target of 0.66 inch. A small horizontal deflection also would occur, but Infinity again calculates a value of 0.0 inches for this small effect as in the case of the rifle example. Again, we note that the sensitivity of the vertical deflection caused by a vertical wind is the same as the sensitivity of the horizontal deflection caused by a crosswind. In this specific example the sensitivity is 0.44 inch per mph of wind speed. This specific sensitivity number applies only to this example bullet fired at this example velocity, but in general the sensitivity to crosswinds and vertical winds is very large for all bullets, handgun as well as rifle. The fourth and fifth rows in Table 3.2-2 show the effects of the wind components acting together. If a tailwind of 10.6 mph acts with a crosswind of 10.6 mph (a horizontal wind of 15.0 mph blowing from the 4:30 o’clock direction), comparing the fourth row to the second row and then the first row shows that the crossrange deflection decreases from 4.62 to 4.48 inches. The vertical deflection remains the same, compared to the effects of the wind components acting separately. The reason that the crossrange deflection decreases is that the time of flight of the bullet is slightly shorter with the tailwind acting on the bullet, and this shorter time of flight decreases the effect of the crosswind. When all three components of wind act together in this example, the last row in Table 3.2-2 shows that the upward deflection caused by the tailwind component slightly increases the upward deflection caused by the vertical wind. Again, there is an interaction among the wind components that changes the time of flight to the target, and so the effects of the wind components acting separately cannot be simply added (or subtracted) to exactly equal the effects of the wind components acting simultaneously. To summarize this subsection: A wind from any direction can be resolved into at most three components, a horizontal headwind (or tailwind) component blowing along the line of sight between the shooter and the target, a cross-wind component blowing in a horizontal direction across the shooter’s line of sight to the target, and a vertical wind component blowing upward or downward across the shooter’s line of sight to the target. Headwinds or tailwinds generally have a quite small effect on bullet trajectories, unless the wind is very strong and the range is very long. Crosswinds and vertical winds, however, have serious effects on bullet trajectories. The effect of each component wind
can be analyzed separately, and this approach gives insight into wind effects. However, to get accurate calculations of the wind’s effects from any direction, all three components must be analyzed simultaneously, because the wind effects interact, primarily by changing the time of flight of the bullet to the target. Infinity can be used to calculate the effect of any wind component, or to calculate the effects of all components acting simultaneously. There is a common misconception among shooters that a wind ―blows‖ a bullet off its course as it travels downrange. It is very important to realize that a wind does not ―blow‖ a spin-stabilized bullet off its course. Rather, because of its spin stabilization a bullet turns to follow the wind if the wind direction is perpendicular to the line of sight between the firing point and the target. This will be described in greater detail in Section 4. In the case of a headwind or tailwind, the moving air simply changes the drag on a bullet, because drag depends on the speed of the bullet relative to the air and not the ground. A headwind will increase the drag a small amount, in turn increasing the time of flight and causing the bullet to shoot low. A tailwind will decrease the drag a small amount, in turn decreasing the time of flight and causing the bullet to shoot high.
3.3 Effects of Shooting Uphill or Downhill When a gun is sighted in on a level or nearly level range and then is fired either uphill or downhill, the gun will always shoot high. This effect is well known among shooters, particularly hunters, but how high the gun will shoot is a subject of considerable controversy in the shooting literature. In fact, at the present time some literature has information that is simply erroneous. In this subsection, we will try to explain the physical situation carefully so that it can be understood clearly, and then provide some examples using Infinity to perform precise calculations. Throughout this subsection the terms ―bullet drop‖ and ―bullet path‖ will be used frequently, so we will review the definitions of those terms before we begin to explain the physical situation. One may refer back to Figure 3.0-1 concerning these definitions. Bullet drop is always measured in a vertical direction regardless of the elevation angle of the trajectory. At any range distance measured along either a level range or a slant range, drop is then the vertical distance between the extended bore line and the point where the bullet passes. Drop is expressed as a negative number, denoting that the bullet falls away from the extended bore line as the bullet travels. Bullet path, on the other hand, is always measured perpendicular to the shooter’s line of sight through the sights on the gun. Thus, it would be where the shooter would ―see‖ the bullet pass at any instant of time while looking through the gun sights, if that were possible. At the gun’s muzzle, the bullet path is negative because the bullet starts out below the line of sight of the shooter. Somewhere near the muzzle, the bullet will follow a path that rises and crosses the line of sight, then travel above the line of sight until the target is reached. The bullet path is then positive throughout this portion of the trajectory. The bullet will arc over and cross the line of sight at the zero range. So, the bullet path is zero at the zero range, and then becomes negative at distances greater than the zero range. The explanation of the physical situation for uphill/downhill shooting begins with a simple observational fact — that bullet drop at any given range from the muzzle is almost independent of firing elevation angle. What this means is that if the drop of a bullet trajectory at, say, 150 yards is measured when the gun is fired on a level range, then the drop at a slant range distance of 150 yards will be almost the same value when the gun barrel is elevated at +45 degrees, - 15 degrees, - 60 degrees, or any other positive or
negative elevation angle. It is very important to remember that we use ―start range‖ because that is the range that the bullet must actually travel to reach the target. This is true for all range distances practical for small arms fire. To illustrate this point, Table 3.3-1 has been prepared for a group of five cartridges, three for rifles and two for handguns. The table shows drop numbers at a specific range distance for each cartridge, as a function of the bore elevation angle of the gun at the firing event. These drop numbers have been computed with Infinity. These trajectories have been computed for a firing point altitude of 2500 feet above sea level. The selected cartridges in Table 3.3-1 illustrate typical behavior of drop at a specific (and relatively long) range distance versus the bore elevation or depression angle. The 338 Winchester Magnum cartridge exhibits the worst case in the table. At a range distance of 600 yards, there is only about 0.5 inch difference in drop value between a level trajectory and a trajectory elevated 60 degrees or depressed 60 degrees. This is because the major driving cause of bullet drop is gravity acting over the bullet’s time of flight. There are two other smaller effects on drop as the bullet travels. When a bullet is traveling upward on an elevated trajectory, there is a component of gravity that adds to the drag deceleration of the bullet, but the bullet is traveling into less dense atmosphere that reduces the aerodynamic drag. So, these small effects tend to offset one another. The opposite small effects occur when the bullet is traveling downward along a depressed trajectory. This result is true in general. At practical range distances for small arms fire the change in vertical drop with firing elevation or depression angle is very small, even for very steep angles. However, the bullet path can change dramatically, particularly at steep angles. Figure 3.3-1 shows how this happens. Ordinarily, a shooter will sight his gun in on a target range that is level or nearly level. Figure 3.3-1 (a) shows this situation. When sighting in, the shooter adjusts his sights so that the line of sight intersects the trajectory at the range (Ro in the figure), which is the range where he wants his gun zeroed in. Ro is called the zero range for level fire. The vertical distance between the line of departure (extended bore line) of the bullet and the point where the bullet passes is the drop (do). This symbol is used to denote the drop at the range where the gun is zeroed in. Note that the angle between the bullet’s line of departure (extended bore line) and the line of sight is very small. This angle is greatly exaggerated in Figure 3.3-1 for purposes of illustration. Even for very longrange target shooting (1000 yards or more), the angle A is much less than 1.0 degree, and it is typically less than 10 minutes of arc for sporting rifles and handguns.
Table 3.3-1 Bullet Drop at a Specific Range Distance versus Bore Elevation Angle for a Selection of Cartridges Cartridge and Load Range Distance
22 Hornet, Sierra’s 200 yds 45 gr. Hornet bullet, 2700 fps Mzl Vel
0 deg (level) 20 45 - 20 - 45 0 deg (level) 20 45 60 - 20
- 13.39 in - 13.38 - 13.36 - 13.40 - 13.41 - 39.98 in - 39.94 - 39.90 - 39.89 - 40.01
270 Winchester 400 yds Sierra’s 140 gr. SBT GameKing, 2900 fps Mzl Vel
338 Winchester 600 yds Magnum, Sierra’s 250 gr. SBT GameKing, 2700 fps Mzl Vel 44 Magnum, Sierra’s 150 yds 240 gr. JHC bullet, 1300 fps Mzl Vel
38 S&W Special, 100 yds Sierra’s 125 gr. JSP bullet, 1100 fps Mzl Vel
- 45 - 60 0 deg. (level) 45 60 - 45 - 60 0 deg (level) 20 45 - 20 - 45 0 deg (level) 20 45 - 20 - 45
- 40.05 - 40.06 - 109.05 in - 108.66 - 108.57 - 109.44 - 109.53 - 28.34 in - 28.33 - 28.32 - 28.36 - 28.37 - 16.04 in - 16.03 - 16.03 - 16.04 - 16.05
Now consider the situation where the shooter fires his gun uphill at a steep angle, as shown in Figure 3.3-1 (b), with no changes in the sights. Since the true bullet drop changes very little, at a slant range distance Ro from the muzzle the bullet has a vertical drop nearly equal to do, as shown in the figure. However, the line of sight at slant range distance Ro still is located a distance do in a perpendicular direction away from the line of departure. Because of the firing elevation angle, the bullet trajectory no longer intersects the line of sight at the slant range Ro. In fact, the bullet passes well above the line of sight at that point, as Figure 3.3-1 (b) shows. In other words, the bullet
shoots high from the shooter’s viewpoint as he or she aims the gun, and at steep angles it may shoot high by a considerable amount at longer ranges. Figure 3.3-1 (c) depicts the situation when the shooter fires the gun downhill. Again the vertical drop at the slant range distance Ro changes a very small amount from the value do for level fire, but the line of sight and line of departure are still separated by the perpendicular distance do at that range point. Compared to the case of level fire, the bullet again shoots high from the shooter’s viewpoint as he or she aims the gun. Furthermore, if the gun is fired uphill at some elevation angle, and then fired downhill at an
equivalent depression angle, the two bullets will shoot high by nearly the same amount at the same slant range distances. A careful look at Figure 3.3-1 (a) or (b) shows us that the amount by which the bullet shoots high at the slant range distance Ro is equal (approximately) to the perpendicular distance do from the line of sight to the extended bore line minus the projection of the drop do on that same perpendicular line. From plane trigonometry, the distance by which the bullet shoots high at Ro is:
Amount by which the bullet shoots high = do [1.0 – cosine A] where A is the elevation angle (or depression angle). Now, if you have forgotten or never studied trigonometry in school, don’t worry. The Infinity program will make exact calculations for you, and two examples of these calculations will be shown below. First though, let us point out that this explanation of the physics of uphill or downhill shooting has been given specifically for a slant range distance equal to the zero range distance for level fire, and this has been done just for convenience. The sketches are easier to draw and to understand for that situation. The result, however, applies for all slant range distances. At any range distance from the muzzle, the amount by which the bullet will shoot high at any elevation or depression angle A is very nearly equal to the drop for level fire at that range distance multiplied by the quantity [1.0 – cosine A]. Two examples for uphill or downhill shooting have been prepared using Infinity, and they are shown in Tables 3.3-2 and 3.3-3. The first example is for a 7 mm Remington Magnum, a flat-shooting rifle cartridge. The second example is for a 44 Remington Magnum handgun cartridge that has a trajectory with much more arc. It is presumed that both the rifle and the handgun have telescope sights and are sighted in at an altitude of 2500 feet. Then, they are fired uphill or downhill while at the same altitude. The tables show the reference bullet path for level fire together with the changes in bullet path depending on the elevation angle and slant range distance. When reviewing Tables 3.3-2 and 3.3-3, keep in mind that a depression angle is a negative elevation angle. Two conclusions are evident from these examples. First, shooting uphill or downhill can have a strong effect on the trajectory of any bullet, always causing the bullet to shoot high relative to the bullet path for level fire. This effect grows larger as the slant range distance grows longer and the elevation angle grows steeper. The second conclusion is that a bullet always shoots slightly higher when it is fired downhill than when it is fired uphill at the same angle. The reason for this, as explained above, is that when the bullet travels upward, there is a component of gravity acting as drag on the bullet that increases the drop slightly. When the bullet travels downward, on the other hand, there is a component of gravity acting as drag on the bullet that decreases the drop slightly.
Table 3.3-2 Example of Bullet Path Changes for a Rifle Bullet Fired Uphill or Downhill Cartridge: 7 mm Remington Magnum with Sierra’s 140 grain Spitzer Boat Tail bullet at 3000 fps muzzle velocity Zero range: 300 yds for level fire Shooting environment: 2500 ft altitude with standard atmospheric conditions
Elevation Angle (deg.)
Bullet Path (in)
Slant Range Distance (yds.) 200 300 400 500
+ 15 - 15 + 30 - 30 + 45 - 45
Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in)
0.07 0.07 0.27 0.27 0.59 0.59
0.29 0.29 1.13 1.15 2.49 2.50
0.68 0.70 2.68 2.73 5.89 5.94
28.06 2.08 2.19 8.31 8.50 18.27 18.48
1.26 1.32 5.03 5.13 11.05 11.17
Table 3.3-3 Example of Bullet Path Changes for a Handgun Bullet Fired Uphill or Downhill Cartridge: 44 Remington Magnum with Sierra’s 240 grain Jacketed Hollow Cavity bullet at 1300 fps muzzle velocity Zero range: 100 yds for level fire Shooting environment: 2500 ft altitude with standard atmospheric conditions Elevation Angle (deg.)
Slant Range Distance (yds.) 50 100 150 200
0 + 15 - 15 + 30 - 30 + 45 - 45
Bullet Path (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in) Bullet Path Change (in)
2.40 0.09 0.10 0.37 0.37 0.80 0.81
0.0 0.38 0.42 1.55 1.61 3.42 3.50
- 9.88 0.89 1.04 3.67 3.92 8.16 8.44
- 28.35 1.63 2.01 6.84 7.49 15.28 16.03
3.4 Trajectory Considerations for Sighting in a Gun The terms ―sighting in,‖ ―zeroing in,‖ or ―zeroing‖ a gun all mean the same thing: adjusting the sights on a gun so that it shoots to point of aim at a selected range distance. Sighting in a gun always takes place on a target range, which may be an informal range in the countryside or an established range in some convenient location. The procedure for sighting in is familiar to almost all shooters. However, three questions arise frequently, and Sierra’s Infinity program can be used to answer them all.
3.4.1 Sighting in on a Short Target Range A shooter sometimes is faced with the following problem. He or she would like to sight in a gun for a zero range of, for example, 250 yards, but sighting in must be done on a shorter target range, say, 100 yards. The question is where should a group of shots be centered at 100 yards so that the gun will be zeroed in at 250 yards? The answer to this question is straightforward with Infinity. A trajectory is calculated using a zero range of 250 yards and using the altitude and atmospheric conditions at the target range. Then, the
bullet path is read from the calculated trajectory for a range distance of 100 yards. This is the point where the group should be centered on a paper target located at 100 yards to assure that the gun is sighted in at 250 yards. Of course, any other pair of zero range and target range distances can be used. It is frequently necessary, for example, to use a target range distance of 25 yards to sight in a handgun at a zero range of 100 yards.
3.4.2 Determining Zero Range from Firing Test Results This is the opposite question to the one immediately above, and it also occurs frequently. In this situation, a shooter knows that the center point of groups fired from his gun at a local target range are located a certain amount above the point of aim at a measured range distance. The question then is what is the zero range of the gun? Infinity can answer this question. We will use a specific example to illustrate how this question is answered. Suppose a shooter has a rifle chambered for the 25-06 Remington cartridge and uses Federal factory 25-06 Remington ammunition with the 117 grain Spitzer Boat Tail GameKing bullet. The shooter’s local target range is 610 ft above sea level, and he or she shoots on a pleasant day when the local atmospheric conditions are 30.05 inches of Hg barometric pressure, 75°F, and 60 percent relative humidity. The center of the groups fired is 3.25 inches high at 100 yards. In the ―Normal Trajectory‖ Operations list ofInfinity, we first go to the ―Load Bullet‖ library and select Federal Cartridges. From the list of Federal cartridges, we select the 257 caliber and then select the 25-06 cartridge with the 117 grain Spitzer Boat Tail GameKing bullet, placing that bullet in the active bullet list on the monitor. Then, we prepare to calculate a reference trajectory for this cartridge under the conditions at the shooter’s local target range. In the ―Trajectory Parameters‖ list, the muzzle velocity is 2990 fps (specified by Federal), the maximum range is set at 600 yards, the range increment for the print list is set at 50 yards, the zero range (for this reference trajectory calculation) is set at 200 yards, the elevation angle is set at 0 degrees (level fire), and the sight height is set to 1.5 inches for a telescope sight with a 40 mm objective lens. After these numbers are entered, we accept all the trajectory parameter values listed in the sidebar on the monitor. In the ―Environment Parameters‖ list, the barometric pressure is set to 30.05 inches of Hg, the temperature is entered as 75°F, the altitude is entered as 610 feet, and the relative humidity is entered as 60 percent. The wind velocity is entered as 0 mph, since there is little or no wind on this day at the target range. All the values of the environment parameters in the sidebar on the monitor are then accepted, and a reference trajectory is then calculated. In the Operations list of Infinity, we select the ―Calculate Zero‖ Operation, and enter the measured range of 100 yards and 3.25 inches above the line of sight. After accepting these values and pushing the ―Calculate‖ button, the monitor shows that the zero range is 274 yards for this particular situation. Of course, this is just an example. The same procedure can be repeated for any set of shooting conditions or firing test results.
3.4.3 Sighting in for a Change in Shooting Location Both hunters and target shooters often face the problem of preparing for a hunt or target competition at a location much different in altitude and shooting conditions from their local shooting range where they sight in their guns. Again, an example will help with the explanation of this situation. Suppose a hunter with a 30-06 Springfield is preparing to hunt deer and elk in a western location where the altitude is around 7500 feet above sea level, but he or she lives and uses a target range at an altitude of about 500 feet. The shooter wants the rifle to be zeroed in at 250 yards at the hunting location. The question is, where should the gun be sighted in at the local target range to make this happen? Infinity can be used to answer this question. The first step is to calculate a trajectory with a 250 yard zero range for the shooting conditions at the hunting location. Suppose the hunter handloads the Sierra .308" dia 180 grain Spitzer Boat Tail GameKing bullet in the 30-06 to a muzzle velocity of 2500 fps. We enter this muzzle velocity, a 250 yard zero range, and other appropriate trajectory parameters in the ―Trajectory Parameter‖ list inInfinity. Note that we can enter a 1.0 mph crosswind (from the 3:00 o’clock or 9:00 o’clock direction) just to obtain a crosswind sensitivity for the trajectory at the hunting location. In the ―Environment Parameters‖ list, we will use the standard atmospheric conditions because the actual weather conditions in the hunting location cannot be predicted in advance, and we set the altitude at 7500 feet. Then, we calculate the trajectory. On the top toolbar on the monitor screen, we then select ―Trajectory Variations.‖ From the dropdown menu, we select ―Environment Parameters‖ and make the following changes appropriate for the local target range. The barometric pressure is changed to 30.05 inches of Hg, the temperature to 75°F, the altitude to 500 feet, and the humidity to 65 percent, which are the local conditions for the day that the gun is sighted in. We then calculate the variations in the trajectory. Infinity tells us that if the gun centers groups 0.52 inch low at 250 yards at the local shooting range, the gun will be sighted in for a 250 zero range at the hunting location. Of course, this is a specific example, but this procedure can be used for any other set of shooting conditions.
3.5 Point Blank Range Point blank range is a concept that is very important to hunters and silhouette target competitors. The point blank range of any gun is the range distance out to which a shooter can hold right on his game or target and be assured of a hit within the vital zone of the animal or target. In other words, the shooter does not have to hold high or low to correct for the bullet trajectory. This eases the problem that many hunters have of estimating the distance to a game animal. As long as the range to a game animal is not farther than the point blank range of the gun, the hunter can aim at the center of the vital zone on the animal and be assured of a hit. For silhouette competitors, the point blank range idea can reduce the number of sight adjustments necessary for the targets at longer ranges. The vital zone of an animal is a zone within which a bullet will put the animal down, either killing it instantly or disabling it so that it can be quickly dispatched with another shot. For a silhouette target, the vital zone is an area on the steel animal profile within which a hit will tumble the target. Of course, the size of the vital zone depends on the size of the animal. Only the vertical dimension of the vital zone is important for calculation of the point blank range, because the bullet trajectory arcs upward and downward in a vertical plane. In the horizontal direction, the shooter must aim and place his bullet within the horizontal edges of the vital zone, but on a large animal the vital zone may be considerably wider than it is high.
For animals like white tail deer, the vertical dimension of the vital zone is about 10 inches, from a point on the shoulder of the animal down to the level of its heart. For the larger mule deer, the height of the vital zone might be 11 or 12 inches and about 14 inches for elk. For small varmints, such as prairie dogs, ground squirrels and ground hogs, a vital zone height of 5 inches is appropriate. For rifle silhouette targets, a vital zone height of 6 inches seems appropriate. These estimates, of course, can be changed based on the shooter’s judgment. Figure 3.5-1 illustrates the concept of the point blank range. The vital zone region extends from the gun muzzle to the game animal or silhouette target. The shooter’s line of sight goes down the center of the vital zone region, indicating that the shooter aims at the center of the vital zone on the animal or silhouette. The figure shows the special situation in which the point blank range is maximized. The bullet trajectory rises above the line of sight until it just touches the upper edge of the vital zone. Then the trajectory falls through the zero range and on through the lower part of the vital zone region. The maximum point blank range is then the range distance at which the trajectory crosses the lower edge of the vital zone. If the animal or target is positioned anywhere within this point blank range, a well-aimed bullet will put it down. Point blank range is maximized by choosing the correct zero range for the gun, that is, the zero range that causes the trajectory to rise just to the upper edge of the vital zone region. If a shorter zero range is chosen, the trajectory will not rise as far as the upper edge of the vital zone region, and the trajectory will then cross the lower edge of the vital zone region at a point blank range that is shorter than the maximum possible. It is clearly desirable to maximize the point blank range for game in all hunting situations, and there are advantages in the silhouette games as well.
Figure 3.5-1 Illustrating the Point Blank Range of a Gun
Infinity will calculate point blank range for either of two situations. The first is when the shooter has already zeroed in his or her gun for a specific zero range. This situation is the selection labeled ―PBR – Given Zero‖ in the drop-down menu under ―Operations‖ in the topmost Infinity toolbar. For this situation, Infinity first determines whether the specified zero range is less than or greater than the zero range that maximizes point blank range for the cartridge, load and environment parameters inputted by the shooter. If the zero range is less than that which maximizes the point blank range, a message on the monitor screen will inform the shooter, that this condition is true, and also tell the user what the reduced (submaximal) point blank range is. On the other hand, if the specified zero range is greater than that
which maximizes point blank range, a message on the monitor screen will inform the user, that this condition is true, and then tell him the maximum point blank range for the vital zone height that was chosen and what zero range to use to obtain the maximum point blank range. The second situation is the selection labeled ―PBR – Maximum PBR‖ in the ―Operations‖ dropdown menu. In this situation the user wants to know the maximum point blank range for his or her cartridge, load and shooting conditions, and what zero range to use to obtain that maximum range. In this case, Infinity will output those parameters on the monitor screen and also list a trajectory calculated for the maximum point blank range condition. This gives the shooter the necessary information to sight in the gun so that the maximum point blank range is obtained. Before using Infinity for either of these point blank range calculation modes, recall that it is necessary to first calculate a reference trajectory for the cartridge, load and shooting conditions at the hunting location or the target range location. This is done using the ―Trajectory‖ selection in the drop-down menu under the ―Operations‖ selection in the topmostInfinity toolbar.
Maximum point blank ranges are surprisingly long for both rifles and handguns. Of course, the high-velocity, flat-shooting magnum calibers have a decided advantage over lower-velocity cartridges with bullets having lower ballistic coefficients. But, the maximum point blank ranges of even ballistically inferior bullets are quite long. To cite a few examples, against deer-size animals
(vital zone height 10 inches), calculated for an altitude of 1000 feet and standard atmospheric conditions: Cartridge Bullet
300 Win. Mag. 180 gr SBT GameKing 308 Win. 165 gr SBT GameKing 30-30 Win. 150 gr FN Pro-Hunter 45-70 Gov’t 300 gr HP/FN Pro-Hntr 44 Rem. Mag. 240 gr JHC Sports Mstr. 357 Magnum 170 gr JHC Sports Mstr.
2800 fps 2650 2200 1550 1300 1050
345 yds 319 247 179 151 131
292 yds 271 211 152 127 110
Examples of maximum point blank ranges against small varmint-size animals (vital zone height 5 inches) calculated for an altitude of 1000 feet and standard atmospheric conditions are the following: Cartridge Bullet
243 Win. 55 gr. BlitzKing 22-250 Rem. 55 gr. Blitz 223 Rem. 55 gr. Blitz 22 Hornet 45 gr. Hornet
3800 fps 3600 3000 2650
311 yds 301 257 208
269 yds 259 221 180
These examples clearly show that the maximum point blank ranges of cartridges are surprisingly long for animals with relatively small vital zone heights. This targeting technique has a great deal of utility.
3.6 Maximum Horizontal Range of a Gun The maximum distance that a gun will shoot in some direction and the barrel elevation angle necessary for the bullet to reach that maximum distance are questions that arise often when an outdoor shooting range is being designed, particularly in an urban or suburban area. The maximum ranges can be more than a mile for some handgun bullets and more than 4.5 miles for some rifle bullets. It is necessary quite often to place barriers forward and above the firing lines to block bullets accidentally discharged from elevated guns from traveling far downrange to threaten inhabitants, homes, or business establishments. Infinity will calculate the maximum range distance for any cartridge, either horizontally or along any reference slope with either a positive or negative inclination angle, together with the bore elevation angle of the gun necessary for the bullet to reach that maximum range. A reference slope is an upward or downward slope along which the maximum range must be calculated. This feature is incorporated in Infinity because shooting ranges are often located on a hillside or in a valley, and the maximum range in an upward or downward direction is needed. The maximum range of any particular cartridge of course varies with shooting conditions, especially with altitude of the firing point and with the inclination angle of the reference slope and also with atmospheric conditions at the firing location. The maximum range computation capability in Infinity is reached by selecting the ―Maximum Range‖ entry in the ―Operations‖ dropdown menu. For the cartridge, load and shooting conditions of interest, a reference trajectory need not be calculated. However, those conditions must be entered in the ―Trajectory Parameters‖ and Environment Parameters‖ lists in the sidebar that appears on the monitor screen in the ―Trajectory‖ operation. After those conditions are entered, the user can proceed directly to the ―Maximum Range‖ operation. Key parameters for the ―Maximum Range‖ operation appear in a sidebar in this mode and can be changed to examine the effects of varying any of these parameters.
The examples calculated below show the surprising maximum range of just a single handgun cartridge, the 45 ACP loaded with Sierra’s 230 grain Full Metal Jacket Match bullet to a muzzle velocity of 850 fps under several variations of firing conditions. All these examples are calculated for standard atmospheric conditions, of course adjusted for altitude automatically within Infinity. Firing Pt. Altitude 500 ft. 500 500 5000 5000 5000
Bore Elevation Angle
0 deg. + 15 - 15 0 + 15 - 15
2096.3 yds. 1851.0 2450.4 2265.7 1992.7 2663.0
+ 33 deg + 42 + 23 + 33 + 43 + 24
It is evident that the maximum horizontal range of this cartridge is well over a mile (1760 yards) and that it varies significantly with the altitude of the firing point. It is also quite apparent that the reference slope angle has a large effect on the maximum range.
IMPORTANT: Note also that the bore elevation angle is with respect to the local level at the firing point, not with respect to the reference slope. A common misconception among many shooters is that the bore elevation angle that maximizes the horizontal range is 45 degrees. It may be seen that for this 45 ACP cartridge that angle is 33 degrees. A 45 degree bore elevation angle would maximize the range if the gun were fired in a vacuum. Air drag, however, changes the physics of the trajectory dramatically. It turns out that the bore elevation angle for maximum horizontal range is around 30 degrees for all small arms bullets fired on the surface of the earth.
3.7 Maximum Height of Fire of a Gun The maximum height, or maximum altitude, that a bullet will reach if fired straight up is also a concern in the design of outdoor shooting ranges. If a bullet can rise to an altitude at which it could threaten aircraft over the shooting range, or upon returning to earth threaten local residents or property, then reinforced roof barriers must be placed above the firing lines to prevent accidental discharge of bullets in an upward direction. The maximum height that a bullet can reach depends on shooting conditions, especially on the altitude of the firing point and also on atmospheric conditions at the firing location. Infinity can be used to closely estimate the maximum height that can be reached by a bullet from any cartridge. The maximum height computation capability in Infinity is reached by selecting the ―Vertical Fire‖ entry in the ―Operations‖ dropdown menu. For the cartridge, load and shooting conditions of interest, a reference trajectory need not be calculated. However, those conditions must be entered in the ―Trajectory Parameters‖ and ―Environment Parameters‖ lists in the sidebar that appears on the monitor screen in the ―Trajectory‖ operation. After those conditions are entered, the user can proceed directly to the ―Vertical Fire‖ operation. Key parameters for the ―Vertical Fire‖ operation appear in a sidebar in this mode and can be changed to examine the effects of varying any of these parameters. When a vertical fire trajectory is calculated, the monitor screen displays a message that states the maximum height the bullet will reach above sea level and above the firing point, taking account of the firing point altitude. The message also states the flight time of the bullet to reach the maximum height. Bullets can reach great heights if fired straight up or nearly straight up. This is especially true for rifles firing magnum cartridges. But bullets from rifles firing standard cartridges or from handguns are capable of reaching significant heights. For example; Fired from a point 500 feet above sea level with standard atmospheric conditions;
o A .300 Winchester Magnum cartridge with the 180 grain SBT GameKing bullet at 2800 fps muzzle velocity will reach a maximum height of 11,993 feet (2.27 miles) above the firing point, which is a maximum altitude of 12,493 feet above sea level. o A 30-30 loaded with the 150 grain FN Pro-Hunter bullet at 2200 fps fired from the same location will reach a maximum height of 7232 feet (1.37 miles) above the firing point, or 7732 feet above sea level. o A 45 ACP loaded with the 230 grain FMJ Match bullet at 850 fps fired from the same location will reach a maximum height of 4198 feet (0.8 mile) above the firing point, or 4698 feet above sea level. If the firing point altitude is changed to 5000 feet;
o The 300 Winchester Magnum bullet will reach a maximum height of 12,707 feet (2.407 miles) above the firing point, a maximum altitude of 17,707 feet (3.35 miles). o The 30-30 bullet will reach a maximum height of 7637 feet (1.446 miles), an altitude of 12,637 feet (2.393 miles).
o And the 45 ACP bullet will reach a maximum height of 4465 feet (0.846 miles), an altitude of 9465 feet (1.793 miles). These numbers show that a bullet will travel upward considerably farther when air drag decreases (because the air is less dense). The flight path altitudes of aircraft are referenced to sea level, of course, and so the maximum altitudes of the bullets become very important for aircraft safety considerations.
4.0 Six Degree of Freedom Effects on Bullet Flight As described in the opening paragraphs of Section 3.0, a flying bullet has six degrees of dynamic freedom (6 DOF), three translational degrees of freedom and three rotational degrees of freedom. All sporting bullets, except round balls from smoothbore black powder guns, are spin-stabilized during flight. For a flying bullet that is well stabilized (gyroscopically stabilized), trajectories calculated with a three degree of freedom (3 DOF) model of bullet flight are almost exactly correct. The 3 DOF model treats only the three translational degrees of freedom of the bullet (e.g., bullet position in downrange, vertical and crossrange coordinates), producing a trajectory based on a simplified bullet model that is a point mass with a ballistic coefficient. The three degrees of freedom of rotational motions of the bullet cause only very small variations in the trajectory calculated from the 3 DOF model. Fundamentally, this is because the spinning bullet is so well stabilized that the rotational motions, other than the spinning motion, are very tiny.
However, there are at least four, possibly five, effects caused by rotational motions of the bullet that are small but observable in small arms trajectories. These are: (1)
The small rotation downward (aerodynamic pitch) of the nose of a bullet as it flies along an arced trajectory, so that the longitudinal axis of the bullet stays almost exactly parallel to the velocity vector throughout the trajectory. This motion is caused by a very small aerodynamic sideforce on the bullet resulting from a yaw angle known as the ―yaw of repose.‖ This angle is true aerodynamic yaw. The nose of the bullet is pointed very slightly to the right of the trajectory plane for a bullet of right hand (RH) spin, or very slightly to the left of the trajectory plane for a bullet of left hand (LH) spin. A small crossrange deflection of the bullet, to the right for RH spin or to the left for LH spin. This crossrange deflection is caused by the tiny aerodynamic sideforce on the bullet resulting from the yaw of repose. The bullet turning horizontally to the right or left to follow a crosswind, or turning upward or downward to follow a vertical wind. This turning motion causes a large crossrange deflection of the bullet to follow a crosswind, or a large vertical deflection of the bullet to follow a vertical wind. These bullet deflections were described in Section 3.2. A small vertical deflection (upward or downward) of the bullet together with the large crossrange deflection, resulting from a crosswind. This small vertical deflection is caused by a tiny aerodynamic lift force, or negative lift force, on the bullet, which is necessary to make the bullet turn to follow the crosswind. A small horizontal deflection of the bullet (right or left) together with the large vertical deflection,
resulting from a vertical wind. This small horizontal deflection is caused by a tiny aerodynamic sideforce on the bullet, which is necessary to make the bullet turn upward or downward to follow the vertical wind.
These effects seem very strange. For example, it does not seem correct that a small horizontal sideforce would cause a bullet to rotate downward to keep the bullet longitudinal axis tangent to the arc of the trajectory as the bullet flies, although we can easily imagine that such a sideforce would deflect the bullet horizontally as it flies. These effects truly do happen, but in general they are observable only at longer ranges of 300 yards or more. This is for two reasons. First, as described in Section 2.4, a bullet exits the muzzle with some ballistic yaw, generally an angle on the order of one degree. This initial yaw causes the bullet to precess, or cone about the velocity vector. As the bullet flies, this coning motion damps out or damps to some minimum value over the first 200 yards or so. This motion is, of course, a 6 DOF effect, and it is initially much larger than the small effects that we will describe here. The second reason is that the small effects grow with range distance (or flight time). They are overwhelmed by the coning motion at short ranges, but they become observable at longer ranges when the coning motion damps out. To explain the causes of the small 6 DOF effects, we need to delve into a branch of physics, specifically into the dynamics of rigid, spin-stabilized bodies. Because spin-stabilized bullets are very simple rigid bodies, we can do this without using advanced mathematics. Readers who are familiar with rigid body dynamics will be able to understand the following explanation with no trouble. Readers who have never studied this branch of physics will need to accept a number of statements on faith, but they will be able to follow the logic of the explanation and understand the causes of the effects. It is important to understand that we cannot quantify the effects, that is, we will not be able to calculate how far a bullet will deflect as a result of the rotational motions. The deflections depend on dynamic properties of bullets that simply are not known and cannot be measured with testing facilities available to commercial bullet manufacturers. These properties have been measured for a few bullets in military testing facilities, and the results of such tests verify that the 6 DOF effects on bullet trajectories are indeed small variations on the 3 DOF trajectories for bullets used in small arms.
4.1 Basic Physical Concepts Vector – For our purposes, a vector is any physical quantity that has both a magnitude and a direction. Examples are a force, linear acceleration, linear velocity, linear momentum, a torque, angular acceleration, angular velocity, angular momentum, etc. We will discuss changes in vector quantities, and it is important to realize that a vector can change in magnitude, in direction, or both.
Translational motions – Translational motions are linear motions of a body (i.e., a bullet). These motions can occur in three directions, for example, in the downrange, vertical, and crossrange directions. The translational motions of a bullet are governed by Newton’s Laws, specifically Newton’s Second Law, which states that the rate of change of the linear momentum of a body with respect to time is equal to the force applied to the body.
When the mass of the body is constant, not changing with time, this relationship becomes the familiar mass times acceleration is equal to the force applied to the body. These are relationships among the vector quantities of force and momentum (or acceleration) of the bullet. When these vectors are resolved into components along the three translational directions of motion of the bullet, the results are called the equations of linear motion of the bullet.
Rotational motions – Rotational motions of a rigid body (a bullet) are caused by torques applied to the body. A force that does not act at or through the center of mass of a body produces a torque. The rotational motions of a bullet are governed by an angular vector relationship, which is the rate of change of the angular momentum of a body with respect to time is equal to the torque applied to the body. Normally, the center of mass of the body is considered the origin, and the principal axes of the body are used for calculating components of forces, torques, moments of inertia, and angular momentum. This choice of origin and axes greatly simplify the angular vector relationship, and the resulting equations are called Euler’s equations of angular motion of the body. This choice is used to analyze the angular motions of bullets. The implications of these laws of motion are clear. If a bullet changes its linear state of motion in flight (speeds up, slows down, moves up or down or sideways), the changes must be caused by forces applied to the bullet such as gravity, drag, lift, negative lift, or sideforces. If a bullet changes its angular orientation in flight (pitches downward to keep the velocity vector tangent to the trajectory, or turns to follow the wind), the angular changes must be caused by torques applied to the bullet. Torques are caused by forces. Except for gravity, all the forces that can act on a bullet are aerodynamic. Aerodynamic forces that do not act through the center of mass produce the torques that cause the angular orientation of the bullet to change. Fortunately, the drag force on a bullet, which is by far the largest force, acts through the center of mass when the bullet is well stabilized, and thus creates no torque on the bullet. If this were not true, the bullet would lose stabilization and tumble erratically in flight. Figure 4.1-1 illustrates basic characteristics of a bullet in flight. When there is no wind, the trajectory path lies in a vertical plane that contains the bullet velocity vector and the gravity vector. The bullet has a center of mass and a center of pressure, both of which are located on the longitudinal axis of the bullet. The center of mass is a point at which the gravitational force acts on the bullet, and the center of pressure can be thought of as a point at which the aerodynamic forces act on a bullet. As a bullet flies, the velocity vector is tangent to the trajectory path at all points, and the longitudinal axis is almost exactly tangent to the trajectory path at all points. A very tiny yaw angle (nose left or right of the trajectory plane) or pitch angle (nose up or down but in the trajectory plane) will exist to cause the bullet to turn. Figure 4.1-1 will be referred to in the following subsections to explain the rotational motions of a bullet in flight. This figure will be redrawn from other points of view to illustrate the forces, torques, and angular momentum of the bullet.
4.2 Yaw of Repose and Resulting Crossrange Deflection Referring to Figure 4.1-1, as the bullet flies, the principal aerodynamic force on the bullet acts directly opposite to the velocity vector. The projection of this force along the longitudinal axis is the drag force
on the bullet. The drag force acts through both the center of pressure and the center of mass, and so it does not create a torque on the bullet. With a tiny yaw angle (i.e., the yaw of repose) a very small component of the aerodynamic force acts horizontally and sideward on the bullet. This small sideforce, acting at the
Figure 4.1-1 Bullet Flight Characteristics
center of pressure, creates a torque on the bullet equal to the force multiplied by the moment arm, that is, the distance between the center of mass and the center of pressure. The direction of this torque is downward for a right hand spinning bullet, or upward for a left-hand spinning bullet. To visualize this situation refer to Figure 4.2-1, which is Figure 4.1-1 viewed from directly above the bullet. Note that there is no wind acting in Figure 4.2-1. The principal trajectory parameters are displayed with the correct relationships for a bullet with a right hand spin. The velocity vector V is in the trajectory plane and tangent to the trajectory path. The aerodynamic force Faero is directed opposite to the velocity vector. The bullet has a spin angular momentum H that is directed along the longitudinal axis and forward for a right-hand spin. The yaw of repose is the small angle between the H vector and the V vector. This angle causes a small component of the aerodynamic force, called Fside, to act on the side of the bullet, and which can be thought of as acting at the center of pressure of the bullet. This sideforce creates a torque vector M on the bullet. The torque vector M is the vector cross product of the moment arm r, which extends from the center of mass to the center of pressure, and the sideforce Fside. The direction of the torque vector M is downward, that is, perpendicular to the plane containing r and Fside. The torque vector does not point exactly vertically downward, because of the inclination angle of the trajectory, but it is exactly perpendicular to the plane of r and Fside. Now consider the angular motion of the bullet, which is governed by the equations of angular motion. A key parameter in these equations is the angular momentum of the bullet, which consists of two components. The first component is the spin angular momentum H shown in Figure 4.2-1, which is large in order to guarantee stabilization of the bullet. Because the bullet rotates downward in the pitch direction as it flies, a second component of
angular momentum is directed horizontally in the direction opposite to Fside. This component is so small that it can be considered negligible compared to the spin angular momentum. The magnitude of the spin angular momentum of a bullet is nearly constant as the bullet flies. It changes very slowly because the rotational frictional force and torque acting on the bullet are small. Consequently, the change in the vector angular momentum of the bullet as it flies is very nearly limited to a change in direction of the spin angular momentum vector H, with no change in the magnitude of that quantity. Under this condition, the equations of angular motion tell us that the angular momentum vector H rotates toward the torque vector M applied to the bullet. Consequently, the spin angular momentum vector H, which is always along the central axis of the bullet, rotates downward toward the torque vector M caused by the sideforce Fside. So, in essence the sideforce causes the bullet to rotate downward in the pitch direction to keep the axis of the bullet almost exactly tangent to the trajectory curve as the bullet flies along the trajectory arc. Of course, as the axis of the bullet and the vector H rotate downward, the vector M also rotates at exactly the same rate, so that H always remains perpendicular to M. This entire situation is almost a steady state motion; everything changes very slowly as the bullet flies. The yaw of repose angle, the spin angular momentum magnitude, the torque magnitude, and the sideforce are nearly, but not quite, constant as the bullet flies from muzzle to target. Because the sideforce Fside acts throughout the flight of the bullet, a horizontal (crossrange) deflection of the bullet will result. This deflection is generally small, but it can be noticed, especially by long-range target shooters. This is because the deflection increases as time of flight to the target grows longer. Usually the observation comes about as follows. A rifle is sighted in at point of aim, say, at 200 yards. Then the range to the target is changed to 400 yards. The shooter makes an elevation correction to the rifle sights for the longer range, and a sighting shot (or group) is fired. The shooter notices that the shot (or group) is deflected to the right (for a RH twist barrel) by a few inches, but there is no crosswind to account for this deflection. The shooter can apply a windage correction for the 400-yard range, and everything goes well at that range distance. Then, if the range is changed to 600 yards, the shooter has the same experience. A satisfactory sight elevation correction can be made, but shots will be deflected a few inches to the right, necessitating a windage correction even in the absence of a crosswind. The sideforce arising from the yaw of repose is the cause of this unexpected crossrange deflection of bullets. [Here we assume that the crosshairs in the telescope
(and the adjustment axes) are aligned precisely vertically and horizontally, so that the sight adjustments are precisely vertical and horizontal.] Crossrange deflections occur also for bullets with left-hand spin, but the deflections are toward the left rather than toward the right. For a bullet with lefthand spin, the spin angular momentum vector is directed out of the tail of the bullet. To cause the bullet to rotate downward in pitch, an upward vertical torque is necessary so that the angular momentum vector will rotate upward. This in turn requires a sideforce directed from right to left across the trajectory plane, and this can result only from a yaw of repose angle to the left of the trajectory plane (a small, nose-left angle of the bullet as it flies). Consequently, the sideforce is directed to the left, and the bullet deflects in the crossrange direction to the left as it flies downrange.
4.3 Turning of a Bullet to Follow a Crosswind and Resulting Deflections The point was made in Section 3.2 that a crosswind does not ―blow‖ a bullet off course. Rather the bullet turns in the cross-range direction to follow the crosswind. This is a horizontal rotation of the bullet, and, if the bullet is to rotate horizontally, there must be a horizontal torque applied to the bullet. A horizontal torque requires a small vertical force to be applied to the bullet’s center of pressure, and this in turn requires a very small angle of attack of the bullet relative to the velocity vector. For a bullet with right-hand spin, this angle of attack must be positive for a crosswind blowing from left to right across the trajectory plane and negative for a crosswind blowing from right to left. This situation is reversed for a bullet with left-hand spin. Figure 4.3-1 illustrates the deflection of a bullet trajectory by a crosswind. Notice that the bullet has right-hand spin, the spin angular momentum vector H is directed out the nose of the bullet, the crosswind is blowing from the left to right as the bullet flies, and the bullet trajectory curves to follow the wind. The bullet velocity vector V is exactly tangent to the trajectory, but the nose of the bullet and the spin angular momentum vector H are tilted vertically upward by a small, positive angle of attack. As explained in the preceding subsection, the principal component of the aerodynamic force (the drag force on the bullet, not shown in Figure 4.3-1) acts through both the center of pressure and center of mass, causing no torque on the bullet. However, the small angle of attack causes a small aerodynamic lift force Flift to be applied to the bullet at the center of pressure. The torque vector M, which is the vector cross product of the moment arm r and the lift force Flift, is then directed horizontally to the right of the bullet as it flies. As explained in the preceding subsection, the equations of rotational motion of the bullet cause the spin angular momentum vector H to rotate toward the torque vector M, which causes the bullet to turn to the right as it flies. This effect causes the crossrange deflection (also called crosswind drift) of the bullet, which can be large if the cross-wind is strong. There can also be an observable vertical deflection of the bullet, in addition to the crossrange deflection. This vertical deflection is upward for the
Figure 4.3-1 Trajectory Deflection by a Crosswind (drawn for a bullet with right hand spin and a crosswind blowing from left to right situation pictured in Figure 4.3-1. It results from the upward force Flift acting on the bullet throughout its flight. Generally, this vertical deflection is small compared to the crossrange deflection, but it can be observed, particularly in long-range target shooting. If the crosswind blows in the opposite direction, that is, from the right to the left as the bullet flies, the bullet must turn to the left to follow the wind. This necessitates a torque vector directed horizontally and to the left as the bullet flies. Such a torque can be generated by a negative lift force (directed downward as the bullet flies), and this can happen with a small, negative angle of attack. The final result is a crossrange deflection of the bullet to the left, and a vertical deflection of the bullet downward. If a bullet has a left-hand spin, resulting from a barrel with a left-hand twist, the spin angular momentum vector is directed out the tail of the bullet. The torque vector directions that cause the bullet to follow the crosswind then must be opposite to those for a bullet with right-hand spin. This means that the angles of attack must be opposite, with the result that the vertical deflections are also opposite in direction. These effects are summarized in the table below. The twist direction in the shooter’s gun barrel — right-hand (RH) or left-hand (LH) twist — determines whether the bullet has RH spin or LH spin. The crosswind direction is determined as the shooter looks at the target; the crosswind can blow from the shooter’s left (L) to right (R) direction, or from the shooter’s right to left direction. The crossrange deflection of the bullet will always be in the direction of the crosswind. The vertical deflection will depend on the direction of spin of the bullet.
Barrel Twist Crosswind Direction
RH L to R RH R to L LH L to R LH R to L
Right Left Right Left
Upward Downward Downward Upward
Note that in this description of deflections caused by crosswinds, the effect of the yaw of repose has not been considered. This approach has been used to simplify the explanation. There always will be a yaw of repose for a bullet flying an arced trajectory. However, since all these effects are small, they can be considered as approximately additive in an algebraic sense. That is, the horizontal deflection of a bullet caused by the yaw of repose either adds to or subtracts from the crossrange deflection caused by a crosswind.
4.4 Turning of a Bullet to Follow a Vertical Wind and Resulting Deflections A bullet must turn upward or downward to follow a vertical wind that blows upward or downward. This is a very similar situation to a bullet turning to follow a crosswind, as described in Section 4.3, except that the direction of the wind has changed from horizontal to vertical. Figure 4.4-1 has been drawn to illustrate the conditions for a bullet flying in the presence of a vertical wind. Figure 4.4-1 is drawn for a bullet with a right-hand spin and a vertical wind directed upward. Notice that the bullet velocity vector V is exactly tangent to the trajectory. To follow the vertical wind, the bullet must rotate vertically. If the bullet is to rotate vertically, there must be a vertical torque M applied to the bullet. A vertical torque requires a small horizontal force Fvwind to be applied to the center of pressure of the bullet and to be directed to the left as shown in Figure 4.4-1. This in turn requires the spin angular momentum vector H to be rotated to the left of the velocity vector V by a very small angle. For a bullet with right-hand spin and a vertical wind directed upward, this small angle must tilt H to the left of V, so that the aerodynamic force Fvwind is directed horizontally to the left. This causes an upward directed torque vector M. As explained previously, the equations of rotational motion of the bullet cause the spin angular momentum vector H to rotate toward the torque vector M, which causes the bullet to turn to follow the vertical wind upward as it flies. So, a vertical wind blowing upward causes an upward vertical deflection of the bullet relative to a trajectory with no wind. There also is a small horizontal deflection of the bullet, that is, a small crossrange deflection. This crossrange deflection is to the left for the situation pictured in Figure 4.4-1. It results from the horizontal force Fvwind acting on the bullet throughout its flight. Figure 4.4-1 Trajectory Deflection by a Crosswind (drawn for a bullet
If the vertical wind is directed downward for a bullet with a right-hand spin, the bullet must turn downward to follow the wind. This necessitates a torque vector that is directed downward, which in turn requires a horizontal force directed to the right of the trajectory plane. This requires the spin angular momentum vector and the nose of the bullet to be directed to the right of the velocity vector by a small angle. The resulting vertical deflection of the bullet will be downward relative to the trajectory with no wind, and the small cross-range deflection will be to the right. If a bullet has a left-hand spin, resulting from a barrel with a left-hand twist, the spin angular momentum
vector is directed out the tail of the bullet. The torque vector directions that cause the bullet to follow a vertical wind then must be opposite to those for a bullet with right-hand spin. This means that the horizontal forces must be opposite in direction, with the result that the horizontal deflections are also opposite in direction. These effects are summarized in the table below. The vertical wind direction is upward or downward, determined as the shooter looks at the target. The vertical deflection of the bullet will always be in the direction of the vertical wind. The crossrange deflection will depend on the direction of spin of the bullet as well as on the direction of the wind.
Barrel Twist Vertical Wind Direction
RH Upward RH Downward LH Upward LH Downward
Upward Downward Upward Downward
Left Right Right Left
Generally, the crossrange deflection caused by a vertical wind is small compared to the vertical deflection caused by the wind, and it is seldom ever observed because vertical wind velocities tend not to be large. The vertical bullet deflections caused by vertical winds, however, are frequently seen by hunters in hilly or mountainous terrain. In this description of deflections caused by vertical winds, the effects of the yaw of repose and crosswinds have not been considered for the purpose of simplifying the explanation. However, since all these effects are small, they can be considered as approximately additive in an algebraic sense. That is, the horizontal deflection of a bullet caused by a vertical wind adds to or subtracts from the crossrange deflections caused by the yaw of repose and/or a crosswind, and the vertical deflection caused by a vertical wind either adds to or subtracts from the vertical deflection caused by a crosswind.
5.0 Trajectory Tables If you have purchased this manual in book form, and you have called one of Sierra’s ballistic technicians for trajectory information, the trajectory tables you may receive from our technicians are custom prepared for you using Sierra’s Infinity software. The basic elements of a trajectory and the information contained in the tables are described below. We have included the most useful trajectory parameters (velocity, energy, drop, bullet path, wind drift and point blank range with the zero range to maximize it) in the Infinity tables. For each computed trajectory, they are tabulated for the user-selected muzzle velocity and shooting conditions. This discussion uses an example of a popular rifle bullet to discuss each element of the baseline table and explain the important terms.
Trajectory: The trajectory of a projectile, in this case a rifle bullet, is the actual path that the projectile follows after leaving the muzzle. It is very important that this term be understood before using the tables. The trajectory of the bullet is shaped by many factors: gravity, altitude (air pressure), temperature, humidity, muzzle velocity, wind conditions, and the ballistic coefficient of the bullet itself. Although all of these have been discussed in much more detail in previous sections, the major contributors will be redescribed here as they affect the example trajectory.
The figure above illustrates the parts of a trajectory as they are discussed here and used in the tables. As soon as the bullet leaves the muzzle, gravity causes it to begin to fall away from its line of departure. This creates the drop discussed later. The line of departure is an imaginary line through the longitudinal centerline (Bore Centerline) of the bore and is the line upon which the bullet is launched. The significance of this line will be discussed as each element of the tables is discussed. Rz in the figure is the Zero Range where the trajectory crosses the line of sight on its downward path, and R is the Range of the bullet from the muzzle at each point along the trajectory. Muzzle velocity, air resistance (ballistic coefficient), and gravity are the major contributors to trajectory shaping. Your Infinity tables are computed for the conditions defined in the ―Trajectory Parameters‖ and ―Environment Parameters‖ sidebars associated with the Infinity trajectory operations. The example we
will describe below uses level fire, which means that the line between the gun muzzle and the target is level, and the line of departure is nearly level. In Figure 5.0-1, the elevation angle of the line of departure is greatly exaggerated for clarity. In practice, the elevation angle is a small fraction of a degree. The elevation angle of this line is determined by the sight height and the zero range. (Since the bullet begins falling immediately after it leaves the bore, it must be launched with a slightly upward direction so that it will fall back to the line of sight at the zero range). Now, in order to illustrate the key elements in the Infinity tables, let’s discuss each element. The figure below shows the header for an Infinity trajectory table for the Sierra .257 inch diameter, 117 gr. Spitzer Boat Tail Bullet at a velocity that might be attained in a 25-06.
Trajectory for Sierra .257" dia. 117 gr. SPT at 2900 Feet per Second At an Elevation Angle of: 0 degrees Ballistic Coefficients of: 0.388 0.383
Velocity Boundaries (Feet per Second) of: 0.362 2500
1800 Wind Direction is: 3.0 o’clock and a Wind Velocity of 9.0 miles per hour Wind Components are (miles per hour): Down Range: 0.0 Cross Range: 9.0 Vertical: 0.0 Altitude: 0 Feet with a Standard Atmospheric Model. Temperature: 59°F Data Printed in English Units Figure 5.0-2 The header for a particular table documents the parameters used in computing the data table that follows after the header. The title, of course, is self-explanatory. The Elevation Angle defined as 0 degrees in the above example means that the baseline trajectory for this bullet was computed for level fire. The elevation angle permits computing a baseline trajectory for a shooting range that is not level. Although most shooting ranges are approximately level, some public and personal ranges are not. The next two lines of the header must be taken together in the discussion. The first of these two lines defines the five Ballistic Coefficients which are used to compute the trajectory. The next line defines the velocity boundaries for the velocity ranges within which these Ballistic Coefficients are used. In this example, the 0.388 coefficient is used for computations when the bullet velocity is above 2500 feet per second, 0.383 is used for calculations when the bullet velocity is below 2500 feet per second but above 1800 feet per second, and 0.362 is used when the bullet velocity is below 1800 feet per second. As we have discussed previously, we provide for five Ballistic Coefficient values for our computations because our testing over the last 32 years has proven that a single Ballistic Coefficient does not accurately model
the fit of the individual bullet drag to the G1 Drag Function. For this particular bullet we have found that three coefficients closely fit the bullet to the G1 Drag Function. The Ballistic Coefficients for Sierra bullets are all measured in controlled conditions in our underground test range. The next two lines should also be examined together. The Wind Direction line documents the wind direction and speed that were entered in the ―Environment Parameters‖ sidebar ofInfinity. The Wind Components line documents the actual downrange, crossrange and vertical wind components that have been resolved from the defined wind and used in the calculations. The next two lines define the atmospheric and temperature values that were used in the computations. The Altitude value is the altitude (or elevation) of the firing point above sea level, and the Temperature value is the temperature of the air at the firing point. The last line in the header documents the physical units used for the tabular material to follow. Let’s discuss the basic elements of the trajectory table and what they represent. The figure below highlights the basic table elements. The Range column documents the range distance for which the data in the remaining columns apply. Since the basic computations and data storage in Infinity are in one yard (or meter) increments, any integer printout interval value may be selected from 1 yard up to the maximum range. In figure 5.03 the printout interval has been chosen to be 50 yards, and this selection was entered in the ―Trajectory Parameters‖ side bar. Range
Bullet Path Wind Drift
Time of Flight (Seconds)
50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
2779.2 2661.6 2547.2 2434.6 2324.4 2216.9 2112.2 2010.4 1911.7 1816.0 1719.5 1626.1 1537.2 1453.4 1375.0 1302.6 1237.0 1178.7 1128.0 1084.8
2006.3 1840.1 1685.3 1539.6 1403.3 1276.6 1158.9 1049.9 949.2 856.6 768.0 686.8 613.8 548.7 491.1 440.8 397.5 360.9 330.5 305.7
1.44 1.38 1.32 1.26 1.21 1.15 1.10 1.04 0.99 0.94 0.89 0.84 0.80 0.76 0.71 0.68 0.64 0.61 0.59 0.56
-0.53 -2.19 -5.07 -9.3 -14.99 -22.29 -31.36 -42.38 -55.55 -71.1 -89.28 -110.41 -134.83 -162.94 -195.15 -231.97 -273.92 -321.54 -375.42 -436.14
0.67 1.71 1.53 0.0 -2.99 -7.59 -13.96 -22.28 -32.75 -45.6 -61.09 -79.53 -101.25 -126.65 -156.17 -190.29 -229.53 -274.46 -325.64 -383.66
0.18 0.72 1.65 3.0 4.79 7.07 9.86 13.19 17.12 21.68 26.93 32.95 39.79 47.5 56.12 65.69 76.22 87.72 100.15 113.46
0.052837693 0.107990399 0.165600200 0.225830888 0.288887624 0.354968217 0.424288238 0.497081750 0.573599874 0.654111878 0.738988797 0.828709231 0.923603742 1.023986006 1.130134794 1.242270846 1.360513639 1.484836153 1.615043409 1.750789074
Figure 5.0-3 The Velocity column is the speed with which the bullet is moving at any given point. In our trajectory table, it is the remaining velocity at the printout range. Remaining velocity is the velocity left after the bullet has been launched with the Muzzle Velocity and has then decelerated (slowed down) due to air resistance while traveling the distance to the printout range. The rate at which the bullet slows down is related to its ballistic coefficient(s) and the properties of the air (temperature, humidity, barometric pressure) over the path the bullet travels. The Energy column, as typically used in ballistics tables, is Kinetic Energy. Without getting involved in whether Kinetic Energy or Momentum provides the killing power and what the mechanisms are, we’ll discuss both. The Kinetic Energy of a bullet is based on the mass of the bullet and the velocity with which it is traveling. The mass is not identical to the bullet weight so the bullet weight must be converted to mass before we can use it.
The mathematical formula for the mass of a bullet is:
where: w = bullet weight in grains and g = 32.174 ft/sec2 (acceleration due to gravity) (The factor of 7000 converts bullet weight from grains to pounds) The Kinetic Energy of the bullet can then be found from the formula: where: m = mass of the bullet and: v = the remaining velocity
The Momentum column is included as an additional measure of effectiveness.
Momentum is calculated using the formula:
where: m = mass of the bullet and: v = the remaining velocity Drop, as it is used in ballistic computations, is really an intermediate parameter. That is, it is useful to calculate other parameters. The drop of the bullet is the vertical distance of the bullet referenced to the line of departure from the bore. The following figure illustrates drop when the line of departure is
horizontal. When the line of departure is not horizontal (tilted upward or downward), drop is still defined as the vertical distance between the bullet and the line of departure at any point in the trajectory. Drop is used in the computations of bullet path, but otherwise it has meaning primarily in a direct comparison of two bullets as to the shape of their trajectories.
Figure 5.0-4 The values in the Bullet Path column are the position of the bullet with respect to the shooter’s line of sight through the gunsights. For the average shooter, bullet path is one of the most useful parameters in the tables. Bullet path values are based upon the sight height (distance of the line of sight above the bore centerline at the gun) and the desired zero range. These two points determine the line of sight with respect to the gun and permit an accurate calculation, using the drop, of the bullet position with respect to the line of sight. The sight height above the bore centerline can be measured for your specific firearm and entered on the Infinity ―Trajectory Parameters‖ sidebar. We use 1.5 inches for the sight height for rifles and handguns with telescope sights and 0.8 inches for rifles with iron sights and most handguns as default values for sight height.
Figure 5.0-5 illustrates the bullet path. This rifle has a sight height of 1.5 inches and is zeroed at about 220 yards. The bullet path is 1.5 inches below the line of sight at the muzzle, 1.68 inches above the line of sight at 100 yards, 7.45 inches low at 300 yards, and 21.78 inches below the line of sight at 400 yards.
Wind Drift values are shown for the resolved value of crosswind. The deflection of the bullet due to crosswind is related to the crosswind velocity (or crosswind component of a wind from any direction) and the time of flight of the bullet. In this example, we used a 3 o’clock wind direction (crosswind only blowing from right to left across the shooter’s line of sight to the target) so there is little impact on the drop or bullet path. (There could be an impact at the longer ranges due to the increased time of flight over the slightly longer flight path.) If we use a quartering wind (1.30 o’clock for example) there will be an impact on the drop (increased drop), time of flight (also increased), and bullet path (bullet shooting lower) due to the headwind component. The Time of Flight column shows the flight time to the range distance in the range column. Now, let’s discuss Point Blank Range and the Infinity printout features related to it. The figure below shows a graphical view of Point Blank Range.
Point Blank Range (PBR) is that range distance out to which a shooter can always hold directly on his target, with no compensation for drop, and expect a hit within the vital zone. Maximum Point Blank Range (MPBR) refers to the maximum range for which the firearm can be zeroed such that the bullet will neither rise above the line of sight farther than one-half the vital zone height nor fall below the line of sight more than one-half the vital zone height. ―Vital Zone‖ refers to that area within which an animal may be hit and the hit is quickly fatal. In a deersized animal, that zone is approximately 10 inches high and centered on the heart/lung area. On a prairie dog or ground squirrel the vital zone is much smaller, typically 2 to 4 inches in height. The concept applies equally well to the metallic silhouette and other games. The Point Blank Range and the Maximum Point Blank Range values are printed as a result of running either the Point Blank Range or the Maximum Point Blank Range Operation in the Operations menu of Infinity. Both quantities represent the maximum range that you can hold directly on the target and hit within the vital zone. The difference is that, if your zero range is less than the zero range which maximizes the point blank range (MPBR zero), the bullet path at ranges closer than the zero range will not reach a value as high as one-half the vital zone height above the line of sight. So, your PBR is less than it could be for the game you are hunting. Running the Maximum Point Blank Range Operation in Infinity will tell
you where to zero your gun to maximize the point blank range, as the example below shows. Note that in the example below, the bullet path is computed for the Maximum Point Blank Range Zero even though the MPBR zero doesn’t lie at a printout range point. If you wish to maximize your point blank range, and you are sighting in on a 100-yard range, simply center your group about 4.02 inches higher than your aimpoint. (3.5 to 4.5 inches might be close enough for most hunters!). If you have a 200-yard range, you would center your group about 4.6 inches above your aimpoint. Then you can take full advantage of the 343 yard PBR of your rifle and load for deer.
Calculation of Maximum Point Blank Range for a Vital Zone of: 10 inches Maximum Point Blank Range is 343. Set Zero at 292 Trajectory for Sierra .257” dia. 117 gr. SPT at 2900 Feet per Second At an Elevation Angle of: 0 degrees Ballistic Coefficients of: 0.388 0.383 0.362 0.362 0.362 Velocity Boundaries (feet per second) of: 2500 1800 1800 1800 Wind Direction is: 3.0 o’clock and a Wind Velocity of: 9.0 miles per hour Wind Components are (feet per sec): Downrange: 0.0 Cross Range: 0.0 Vertical: 0.0 Altitude: 0 Feet with a Standard Atmospheric Model. Temperature: 59 F Data Printed in English Units
0 50 100 150 200 250 300 350 400 450
2900.0 2779.2 2661.6 2547.2 2434.6 2324.4 2216.9 2112.2 2010.4 1911.7
2184.5 2006.3 1840.1 1685.3 1539.6 1403.3 1276.6 1158.9 1049.9 949.2
1.51 1.44 1.38 1.32 1.26 1.21 1.15 1.10 1.04 0.99
0.0 -0.53 -2.19 -5.07 -9.3 -14.99 -22.29 -31.36 -42.38 -55.55
Bullet Path Wind Drift (inches) -1.5 1.82 4.02 4.99 4.62 2.78 -0.67 -5.88 -13.05 -22.36
(inches) 0.0 0.45 1.83 4.2 7.62 12.18 17.95 25.03 33.51 43.49
Time of Flight (Seconds) 0.000000000 0.052837693 0.107990399 0.165600200 0.225830888 0.288887624 0.354968217 0.424288238 0.497081750 0.573599874
6.0 Sierra’s Infinity Exterior Ballistics Software This section presents an overview description of Sierra’s Infinity software for personal computers. The description includes key features, modes of operation, and tables and graphs which the program provides. Features of Infinity Infinity incorporates a number of significant features. They include: • Provisions for determination of a reference trajectory for each of five bullets on a sight-in range which may either be level or not level. • Atmospheric corrections along the bullet trajectory for uphill or downhill shooting. • An extensive Database which contains the necessary information (Ballistic Coefficients and applicable velocity ranges) for the bullets of all leading manufacturers. The Exterior Ballistics Section of our previous manuals has given you a brief history of Ballistics, the significant factors that affect the flight of a bullet and how those factors are used in determining how the bullet ―flies.‖ We have used all of these factors and the most accurate mathematical methods available in developing Infinity and the information that it computes and prints. Our intent has been to develop software that computes information to be used by both the serious competitive shooter and the ―once-a-year‖ hunter. We hope we have succeeded in making this information truly useful to you. The Sierra Ballistics Program, Infinity, computes all essential elements of a small arms trajectory for any bullet that has a Ballistic Coefficient referenced to the ―G1‖ drag function and for any set of firing conditions. The program has eight computational modes of operation, and it performs these operations on any one of five selectable (active) bullets at any time. In the first operation, the Trajectory computation operation, it computes downrange, vertical, and crossrange positions and downrange, vertical, and crossrange velocities in slant range coordinates referenced to the extended bore line of the gun. That is, it performs the trajectory computations for level or non-level shooting and refers the data to the direction in which the shooter is pointing the bore of the gun. It also computes time of flight, energy, momentum and wind deflections. It does these computations for specifiable atmospheric conditions, altitudes and wind conditions. This operation permits the selection of 10 different tabular printout formats. There are ten individual tabular printouts that are available to the user in the normal Trajectory operation. The Uphill-Downhill and Trajectory Variations operations present two additional tabular printouts that are similar to the basic trajectory table but show the parameter changes resulting from the changed shooting conditions. The remaining operations provide unique textual or tabular material that relate to that particular operation. We recommend that you select your favorite bullet and run trajectories with each of the tabular modes to examine the data on each. The second operation in the Operations menu computes Point Blank Range for a game animal (or target such as a silhouette) for the case where a gun is already zeroed at a specific range. The third operation computes the Maximum Point Blank Range for a given game animal and the necessary zero range to use to achieve this maximum. The fourth operation, Uphill-Downhill, computes the bullet path difference for the case of zeroing in on a reference range and then shooting later at an elevated or depressed firing angle. This is an important situation for hunting.
The fifth operation, Calculate Zero, calculates the zero range for the case in which a gun shoots high by a measured amount on a target at a measured distance from the muzzle. This is an important situation for many hunters and target shooters. The sixth operation calculates the Maximum Range of a bullet along a given slope angle, which can be chosen as positive (uphill), zero (level fire), or negative (downhill). This operation also calculates the elevation angle of the muzzle (referenced to level) to reach the maximum range along the chosen slope. The seventh operation, Vertical Fire, utilizes a special case of the equations of motion to calculate the maximum altitude that a bullet can reach if fired vertically. The eighth operation, Trajectory Comparisons, is designed to answer a variety of ―what if‖ questions. It calculates variations from a reference (or baseline) bullet trajectory caused by variations in shooting conditions. This operation is very useful to determine sensitivities of trajectory parameters to changes in shooting conditions. It permits the user to determine the trajectory characteristics of varying bullet and environmental parameters without destroying the initial trajectory parameters. Although the program operates in normal English units to accommodate the G1 Drag function, it handles full metric input and output units, or mixed mode units for some of the current shooting games where ranges are in metric units and all other values are in English units. The Units mode is selectable on the Trajectory Parameters panel of the Trajectory operation. The program calculates all basic trajectories in 1-yard (or meter) increments to the specified Maximum Range specified on the Trajectory Parameters sidebar or to a maximum of 8000 yards (meters). The printout values, zero ranges and maximum ranges can be any multiple of one yard (or one meter). The printout ranges will be multiples of the Range Increment specified on the Trajectory Parameters panel with the exception of the Silhouette tables. All computed values for maximum range, point blank range, etc. are computed to one yard. In order to prevent computational overflow in the equations, we have included a computational limit when the bullet drop reaches a value which exceeds 9 feet within 3 feet of downrange travel. A special note on the table will be printed when this limit is exceeded, and the Maximum Range value in the Trajectory Parameters panel will be set to this number. The only operating restrictions we have placed on the user are ones that are necessary for proper mathematical and program function. It is incumbent upon the user to assure that his operational conditions are what he desires prior to computation. It is easy, for example, to run a trajectory at an altitude of 10000 feet and a non-standard temperature condition with a vertical wind of 20 miles per hour when what is really wanted is a trajectory at 1000 feet altitude with no winds. While we have tried to print all conditions on the outputs, the results can be misinterpreted if the user is lax or in a hurry when computing. It is easy, of course, to correct the input value and re-run the calculation. The basic means of navigating the software and the functions of each operation will be discussed briefly below in order to get you started. Any restrictions on your use should be self-explanatory with error messages requesting different input or by having the controls visible only when they are applicable.
Operating Infinity When you first start the program, a title displays the necessary credits while the initial information for the screens and computations loads. The time-consuming operation during this period is the loading
of the information computed during the last session. (On Initial start-up, we provide a pre-computed data set using the Sierra Bullets we love the most.) This data set contains the complete trajectory information, including the bullet trajectory and environmental parameters with which each trajectory was computed, for the five active bullets last used in the program. Thus, the user can resume operations where the last session ended. Once these data have been loaded, the initial screen appears with the menu and toolbars. The program initializes in the Trajectory operation showing the Trajectory Parameters panel on the right side of the monitor and with the Current Bullet from the last session highlighted in the Active Bullets window. The Menu bar operates just as the normal MS Windows or MS Office menu bar in that left clicking on a menu item drops down the sub-menu items that can be performed. An example of this can be found by left clicking on the Operations menu item to drop down the available operations (Trajectory Calculation, Point Blank Range Calculation with a Given Zero, Maximum Point Blank Range Calculation, Elevated Fire, Calculate Zero, Maximum Range Calculation, Vertical Fire and Trajectory Comparison). Right-clicking the mouse with the cursor located anywhere on the left panel of the screen and outside a defined window will bring up the same Operations in a panel on the screen. A similar function (Right-Click) has been included in the right panels to switch between the Trajectory Parameters panel and the Environment Parameters panel. The user may select any of these operations to perform. If the user has selected a new bullet or has not run a baseline trajectory on a new bullet and one is required for the commanded operation, the baseline trajectory will be run automatically prior to performing the commanded operation. The baseline trajectory will be run using the values specified in the Trajectory Parameters and Environment Parameters panels located on the right portion of the screen. Thus, the user should review these panels when a new bullet is loaded to assure they are consistent with the bullet. While selecting and loading a new bullet from the Load Bullet menu item should be self-explanatory, it should be noted that loading a new bullet replaces the currently selected (highlighted) bullet. With the exception of the parameters unique to the specific bullet (and a typical muzzle velocity) the new bullet will receive the same trajectory and environmental parameters that were present for the bullet being replaced. For example, if the bullet being replaced was run with a 1000 yard maximum range, the bullet replacing it will also have a 1000 yard maximum range. This may be undesirable for a bullet like the .458 diameter 300 gr. Flat Nose for the .45-70. Placing the cursor on any individual button on the toolbar just below the menu bar will bring up a label that defines the button’s functions. Reading from the left, the first button will permit editing a custom bullet. The second will print. The third is reserved for a Print Preview function. The fourth button will return to the chart (table) mode when available, and the fifth button will switch from the chart mode to graphics output when available. The sixth button will hide/show the data entry panels for those users with 640 x 480 screens to permit viewing of the entire output box. Note that a trajectory must be calculated prior to utilizing the graphics output mode. The next command buttons are only effective in the graphics mode to add/remove the trace label box, add/remove grids, add/remove labels on the graph, and zoom the graph. The Trajectory operation is designed to calculate the baseline trajectory of a selected bullet. It handles an elevated fire case where the shooting range is not level. It will handle up to +/- 65 degrees. The results are stored into the current bullet locations as the baseline trajectory for that bullet. You may change the information as defined on either the Trajectory Parameters panel or the Environment Parameters panel as you desire. However if you change anything, you must left-click the Accept Data control to make the change effective. You must then left-click the Calculate control to calculate the trajectory.
There are two operations associated with Maximum Point Blank Range. The first calculates the Maximum Point Blank Range of your weapon as you have zeroed it. That is, given that you choose a vital zone for your target or game animal and you have zeroed your gun in for a specific range, this operation will calculate whether your zero is such that the bullet will rise farther than one-half the vital zone height above your line of sight prior to reaching your specified zero range (point blank zero less than your zero range), and when it will be more than one-half the vital zone height below your line of sight at ranges farther than your zero. It will also determine if your zero range is less than the zero range for maximum point blank range. In either case, the maximum point blank range is determined for your gun as sighted. The second operation associated with Maximum Point Blank Range determines what the optimum zero range is to maximize the point blank range of your particular bullet given any reasonable vital zone height. We arbitrarily determined a limit of 36 inches for the vital zone height assuming that some hunters might be going after the few elephants left! The operation is performed on your selected bullet and its baseline trajectory. The Uphill-Downhill operation calculates the difference between your reference trajectory (which may have been computed for a non-level range) and the elevated (or depressed) firing angle. The printout will define the bullet path difference directly as a separate column in the tabular output. The remaining values in the table (remaining velocity, energy etc.) are based on the new trajectory at the new elevation angle so that the remaining differences at the elevation angle can be calculated from the reference trajectory output data. The reference trajectory is not destroyed so repetitive trajectories can be run without changing the reference trajectory. The Calculate Zero operation is designed to answer the question ―I’m sighted in ―x‖ inches high at ―Y‖ yards. What’s my zero range? The program permits measured bullet path height input to .01 inches (for those purists who believe that they can reliably determine the centroid of their group to .01 inches) and range increments of 1 yard out to 1000 yards. Only positive (above the line of sight) values are accommodated. The Maximum Range operation computes the maximum range of your selected bullet given the environmental conditions of your site (altitude, temperature, pressure and humidity), a reference slope angle, and the muzzle velocity of your bullet. The reference slope angle is designed to support determining whether the bullet will clear an object or not. The Maximum Range is determined with respect to this slope. A zero-degree value is level fire. The program outputs the maximum range along the reference slope and the bore elevation angle with respect to level (horizontal direction) necessary to achieve it. The Vertical Fire operation determines the maximum altitude that the selected bullet will reach given the muzzle velocity, firing point altitude, and environmental conditions defined by the user. This is a special algorithm for vertical fire. Since winds may vary in direction and magnitude at different altitudes (and almost always do!), there is no provision for wind. The program outputs maximum altitude in feet (or meters) above sea level, maximum height above the firing point, time of flight to the maximum point, and the environmental conditions at the firing point. The Trajectory Comparisons operation permits graphic comparison of up to five bullets. The values that can be compared graphically are remaining velocity, remaining energy, drop, bullet path and wind drift. Only trajectories with like computational units (English or Metric) can be compared. Since the graphs are plotted in 1-yard (or meter) increments, printout range increments, maximum ranges and zeros need not be the same. The zoom function clearly makes the value labels on each plot more readable, although for closely matched bullets it still requires some interpretation.
The Trajectory Variations menu item permits a ―what if‖ function. It is much the same as the UphillDownhill operation in that the reference trajectory is preserved for all calculations. The output table gives a specific bullet path difference column for the difference between the reference trajectory and the trajectory computed with the variations input data. The values in the body of the table are for the trajectory with the variations included. Note that there are four panels of information that can be varied to observe the effects on the bullet trajectory. Graphic output is available showing the velocities for both the reference and the variations trajectory. Those of us who hunt at altitudes and in weather conditions other than those at which we sight-in, use this operation to get an accurate feel for where the rifle shoots under real conditions.