The Ballistic Coeficient Explai.

ROFESSORMIKE INVESTIGATES...

THE BALLISTIC COEFFICIENTEXPLAINED

provides coefficient MikeWr,ight of ballistic Professor explanation a technical

andhowit affectsourpellets shootersthe idea of a fo ' bmost allistic | c o e f f i c i e n t ' ( B Ci)s b o t h f a m i l i a ra n d b a f fl i n g a t t h e s a m et i m e , M a n y o f u s r e g u l a r l y use softwarepackageswhich use t h e b a l l i s t i cc o e f f i c i e n to f o u r c h o s e np e l l e tt o c a l c u l a t e trajectory,energyretentionand o t h e r v a r i a b l e sa t v a r i o u sr a n g e s , b u t h o w m a n yo f u s r e a l l y u n d e r s t a n dw h a t t h e B C o f a p e l l e t a c t u a l l yi s ? I n e s s e n c et,h e b a l l i s t i cc o e f f i c i e n to f a p e l l e t i s a m e a s u r eo f h o w w e l l i t r e s i s t sa i r d r a g a n d r e t a i n sv e l o c i t ya s i t travelsdownrange:the largerthe c o e f f i c i e n t t, h e l o w e rt h e v e l o c i t y l o s s .T h e b a s i cs c i e n c eo f h o w a i r d r a g a f f e c t sp e l l e tv e l o c i t yi s q u i t e straightforward,but it has been obscuredby the long historyof b u l l e tb a l l i s t i c sf,o r w h i c h t h e c o n c e p to f t h e b a l l i s t i cc o e f fi c i e n t

This is a very longway from the a i r g u np e l l e t ,w h i c h h a s m u c h s m a l l e rd i m e n s i o n as n d w e i g h si n at aroundone 500th or"soof the weightof the standardbullet. lt is not surprisingthereforethat the BCs of the pelletswe use are very, very much lowerthan 1.0. The valueof BC for most available p e l l e t sf a l l sw i t h i n a s p a no f v a l u e s r a n g i n gf r o m a b o u t0 . 0 1 t o 0 . 0 4 , d e p e n d i n go n t h e c a l i b r e w , eight and shapeof the pelletconcerned. In practice,the actual BC of the p e l l e ti n f l i g h t a l s od e p e n d so n t h e rifling patternof the gun's barrel and the mechanismof oower delivery;springor PCP. In manywaysit is unfortunate that the ballisticcoefficientidea, which has subsequently been refinedand redefinedby a varietyof researchers, was evercarriedover

Figure2: VelocityLoss dueto Air Drag . : : * 4SS r.t {t u E

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dragfactorslike straightforward everyotherbranchof engineering Butthe notionof the science. is nowfirmly ballisticcoefficient and in the literature embedded andwe of airgunballistics software whatit really needto understand Thebestwayto geta handle means. on whatthe BCis all aboutis to takea lookat thewayin whichair dragaffectsa pelletin flight.

**ffiffi ffiffie ffiffiffiffiffimffi-wffiffi ffiffitutuHffiwKffi ffiffiffiffiffiHffiffiffiffiww Kffieffiffiffiffiffiffiffi ffiffitutuffiw ffiffiffiffiwwffitutuKwffiffiffiffiffiwffi &ffiffiffiffi&ffi-&ffiffiffiffiw&Hffiffi wffituffimffiww-w a s o r i g i n a l l yd e v i s e d . I n t h e 1 8 6 0 ' s ,a B r i t i s h clergyman,the wonderfullynamed FrancisBashforth,was conducting velocitylossexperimentson artillery p r o j e c t i l e us s i n ga b a l l i s t i c p e n d u l u mO . w i n gt o t h e d i f f i c u l t y velocities of measuringdown-range for a whole rangeof bullets,he 'standard conceivedthe idea'ofa b f r o m w h i c h bullet' the allistic performance of all other bullets c o u l d b e c a l c u l a t e db, y s c a l i n gu p or down,without recourseto further he proposed tests,Consequently, that the ballisticcoefficient(or, morespecifically,its reciprocalthe 'dragfactor')be basedon a standardcylindricalprojectilewhich w a s o n e i n c h i n d i a m e t e ra n d w e i g h e do n e p o u n d .T h e B C o f t h i s s t a n d a r db u l l e tw a s d e f i n e da s 1 . 0 .

Figure l: AirDrag

&ilpffiraR theflightof a Figure1 illustrates pelletwhichhasbeenfiredfroma Wind barrel. aligned horizontally effectapart,the pelletis subjected to two externalforces:a constant anda verticalforcedueto gravitY forcedueto air drag.The varying of theseforces combination the familiardownwardoroduces of the pelletshown curvingtrajectory in thefigure. howthe ln orderto understand gravityanddragforcescombine to we produce trajectory, a pafticular needto knowhowair dragvaries flight. duringthe pellets' with Thedragforceincreases on theffi velocityin a waythat depends

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natureof the air flow overthe pellet surface.Forthe subsonicregime, with air flow velocities substantially belowMach 1 (i.e,for velocitiesup to about 950 fVsecond)that applies to virtuallyall airgunpellets,the drag force increaseswith the squareof the velocityof the pellet.Sincethe deceleration of the pellet,or the 'air drag', is proportional to this force, the followingsimpleformulaapplies:

it will loseanother18%, arrivingát the 20 yard mark with 403 ftlsec, At 30 yards,the velocitywill be a further 18% lessat 337 ft/sec and s o o n , i n s i m i l a rf a s h i o n a, s t h e pellet movesdown range. Figure2 showshow air drag robsthe pelletof velocityas it movesdown range.

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This patternof constantpercentage lossovera given intervalgoverns Changein Velocity_ Vetocityz Air Drag= BallisticRatnge many phenomenain the worldaround Tine of Pellet us. Forexample,it is the waythat The constantin this for:mula, the radioactivity decaysand is the basis 'Ballistic Rangeof the Pellet',is very of the carbondatingmethodfor' closelyrelatedto the ballistic ancientartefacts.lt is alsothe pattern coefficient,BC, as we shall see. for ring separationin a tree trunk and In practicalshootingwe are, of showshowthe valueof money course,more interestedin the wav declinesduringa periodof constant that pelletvelocityvarieswith inflation.lt is calledan exponential distanceratherthan time and the decaycurveand the formulafor it air dragformula is better recastin alwayscontainsthe same number thoseterms: Change in Vetocity Distance _ Ballistic Velocity Ranqe of Pellet lA0 x Distance = 0r o/o.Reduction in Velocity BallisticRanoe of Pellet'

Notethat,for a givendistance, the percentage reduction in velocity dueto air dragis constant. Thisis pointto graspabout the essential velocity retention andit is the key factthatdetermines theshapeof the velocity/ distance curve.To illustrate this point,let'slookat an example for a verypoorpellet. Consider a pelletwith a muzzle velocity of 600 fVsec,whichis sloweddownby air dragto492 ft/secat 10 yardsrangei.e.the pellethaslost18 % of its velocity in 10 yards.Overthe next10 yards

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number 3.1416,w hi chi s l abel l ed as pi ( ) because it comesup so frequently. Using'e',,the velocity formulais: MuzzleVelocity VelocitY at Range'd'=--¡------, elBail¡stlc?ilge

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This is all verywell, but we needto knowwhat the BallisticRangeof a particularpelletis beforewe can do anycalculations. By puttingthe velocityformulainto logarithmicform, we find that the followingformula emergesfor the BallisticRange: Ballistic Range=

d ,tn(ffi¡,)

Where'ln' denotesthe natural logarithmof the velocityratio Youcan find the 'ex' and 'ln' buttonson all scientificcalculators, includingthe one that I purchased lastweekfor a poundfrom a well-

Wecannowusethisvalueto estimate thevelocity at another range,say35 yards: = vetocity at3s rards frl=

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Thisvelocity valueis markedup , Figure2, whereit canbeseenthat linesup precisely withthe35 yard range. TheBallistic Range formul provides uswitha simplemethodo converting twomeasured velocitie takenat twodifferentrangesintoa measurement of thevalueof the Ballistic Range.,But howdoesthis constant, the Ballistic Range tie up withthecommonly used'' Ballistic Coeff icient'?

TttmffimHtistüe ffioeffisfrmmt arld ffimtlistüc ffiangm In orderto understand the torturo definition of the present:day

**wmm ffiffiewffiwffiffiK $vxffiffiffitu wffi ffiffiffiffiffiffiq"ffffiffi

tr&ffiffiffiWffiffi.Yffiffifuffiffiffifu ffitrWffi&hfuffi $'&fuTffiffi ffiffiWffif ffiMHWffiffi ffiWey-ffiffi &ffiffiKwpgffiAwwffiruTffiwffiffi wffi ffiKMPfuHffiY ffiXEYffiWffiKK-ffi ffiffiMffifuWffi'(2.7I83 to four decimalplaces) knowndiscountchain.These irrespective of the field of application. functionsare alsobuilt into a variety The equationfor the velocitydecline of computerspreadsheet packages. of our pelletwith distanceis: Let'stake a lookat how these formulaeapplyto our example.We = Velocity at Range'd' will first find the valueof the Ballistic #ffi Range.We knowthat the muzzle Because thenumber 2.7783' velocityof 600 fVsechas fallen to occursoverand overagainin all 403 fVsec at20 yards,so the forms of dynamicanalysis,it is a BallisticRangewill be givenby: giventhe 'shorthand'label'e', where 'e' = = Slyar¿s Battistic nanle is shorthandfor 'exoonential ffi constant'.This is just likethe

Ballistic Coefficient, l th:inkit's helpfulto tr,ace a littlebit of the historybehindit. A l thougthhe not ionof a ballis coefficient in its present form,as measure of bulletf lightef f icienc originated withthe Reverend Bashforth, it wasProfessor Peter Guthri Tait e ,wor king in Cam br id de a n dE d i n b u r gw hh , op u b l i s h et h firstaccurate trajectory formulaei t h e 1 8 6 0 s i,n c l u d i ntgh e i d e ao f B al l i sti R c a nge. Som e20 year s l ater,the K ruppCom pany in Germany, madethe firstaccurate measurements on the effectof air dragon bullettrajectory in the 1881.Thi swasdoneby t estf ir in largeflat-based, blunt-nosed bulletsmanufactured to a standa desi gn.Fo llowing t he Kr upp experi mentms,ilit ar yengineerins (Ma yevski) R ussi a and lt aly( Siac workedto derivea mathematica modelto predictbullettrajectory apparently unaware that Profess Tatehadalreadybrokenthe back of the problem(therewasno Googlein thosedaysl).The Mayevski modell wassubsequen takenup by Colonel Jam esI ngall of the UnitedStatesArmy,who

TECHNICALAIRGUN

a t t e m p t e dt o s i m p l i f yM a y e v s k i ' s resultsfor use by non= m a t h e m a t i c aal r m y p e r s o n n e l .t w a s I n g a l l sw h o r a t i o n a l i s etdh e b a l l i s t i cc o e f f i c i e nat s a m e a s u r eo f t h e b a l l i s t i ce f fi c i e n c yo f a n y p r o j e c t i l er,e l a t i v et o a s t a n d a r d b u l l e ta n d p u t t h e B C i n t h e f o r m still usedtoday.The resultswere p u b l i s h e di n w h a t a r e s t i l l k n o w n a s ' l n g a l l sT a b l e s ' . T h e i d e ao f s c a l i n gt h e b a l l i s t i c e f f i c i e n c yo f a s t a n d a r db u l l e tt o determinethe performánceóf a n o t h e rc h o s e np r o j e c t i l e( i . e . a n a i r g u n p e l l e t )r e q u i r e st h a t t h e physicsof the problembe taken f u l l y i n t o a c c o u n t .I h a v ep r e v i o u s l y w r i t t e na d e t a i l e da r t i c l eo n t h i s i n A i r g u nS p o r t ,a s u m m a r yo f w h i c h is as follows: The air flowingoverthe pellet createsa dynamicair pressure, which dependson the squareof the velocityof the pelletand the density of the air.This is depicted l

and is alwayscalculatedin lb-inch units. The factor; 'i 'is the Form Factorof the pellet,whichdetermine5 its effectivefrontalareaand the amount of drag. U s i n gB C a n d c o m p a r i n gt h e f o r m u l a ef o r a i d r a g i t s h o u l db e a p p a r e ntth a t t h e B a l l i s t i c Coefficientis simply a scaled v e r s i o no f t h e B a l l i s t i cR a n g e .l f w e factor in the correctunits, an appropriatefigure for air density and the correctbasefor 'i' it works out that: Range oarto - Ballistic Battistic coefficient 8000 So for our examplepellet,the BC is 50/8000 = 0.0063. (l saidthat it was a poorpellet!)Let'ssummarise the key facts so far:

ffi At subsonicspeeds,ai dragvaries with the squareof pelletspeed. ffi Velocityreductionwith increasing in Figure3. diagrammatically distanceis suchthat a constant percentage This pressuremultipliedby the of velocityis lost for effectivefrontalareaof the pelletgives each equal incrementof range. the dragforceon the pellet,which, ffi The velocityversusdistancecurve is an exponentialdecay. whendividedby the pelletmass, the deceleration air drag; determines ffi The rate of decaydependson the B a l l i s t i cR a n g e( B R )o f t h e p e l l e t ; Pettet Effective Area Vetocitv, .. the higherthe BR, the betterthe x atroensnY AIrDrag= prttrt z: uuu pellet retainsits velocity. The term in squarebracketsis ffi The BR can be found from ' effectivelythe reciprocalof the measuring the pelletvelocityat which is usually ballisticcoefficient, two rangepoints. given,as: ffi The BatlisticCoefficientis the BallisticRangein yardsdividedby Pettet Mass 8000, Dlr-

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Figure3: VelocltyLsss dueto Air Drag

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25 yards.Theyareveryeasyto use. Figure5 is a diagramof the process areset up a Twochronos involved. issue, articlein this In his companion distance apart. Ily measured carefu Jim Tylerrecountsthe trialsand ' d' is t hesepar at ion in Thedi stance the ballistic of,measuring vicissitudes yardsbetween the centresof the coefficientfor a particularpellet.You The sensorsof the two instruments. to mightthink that it is unnecessary chrono(redin the diagram) nearest go to the trouble,becausethe BC'sof V1.The, givesa velocity reading a vast rangeof pelletshavealready chronoreadsV2.To further(green) beenpublishedon the internetand fi ndthevel ocitryat io,Vl is divided are availableat the click of a button. 'd' is 15, , 20 point, byV 2. l f thedist ance but that Youmight indeedargue or 25 yards,thevalueof BCcan you wouldbe wrong.The ballistic then be readoff the appropriate coefficientof a pelletdependson its the Vl suppose preciseweightand dimensional curve.Forexample, was750 ftlsecandthe reading tolerancesand its exactshapeafter it 20 reading on thesecondchrono, has beenfired from a barrel.So,with yardsaway,V2,was647 ft/sec.The the bestwill in the world,the ratiois 750/647= 1.16. publishedfigurescan only be regarded velocity ( 0. 017)can c ef f icient Thebal l i stico withinplusor as an approximation off the 20 yard directly get be read now To an accurate minusabout 15%. , curvein Figure4. figure,you reallyneedto measureit as Figure5 shows, pelleVrifle combination. Alternatively, for your'orrrrn wecanusetheformulato getthe The most direct methodfor sameresult: rneasuring the ballisticcoefficientis : to fire a oellet overtwo 2o =o. ol7 B c= which are separated

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T h i s a r t i c l e , , a l o nwgi t h J i m : T y l e r ' sc' o m p a n i o np i e c e ,s h o u l d Velocityfron NearestChrono p r o v i d ea r e a s o n a b l ien t r o d u c t i o n VelocityRatio= Velocityfrom FurthestChrono t o t h e b a l l i s t i cc o e f f i c i e nat n d , e x p l a i nw h a t ' sb' e h i n dt h e l i t t l e = BallisticCoefficient 800x In(VelocityRatio) n u m b e rw e t y p e i n t o o u r b a l l i s t , i c Figure4 showsa veryconvenient p r o g r a m m e sB. u t , a s J i m ' s a r t i c l e whichdo the mathsfor warns,there are morethan a few setof curves youfor threetypicalchronograPh w r i n k l e st o b e i r o n e do u t y e t , Watchthis space! ffi 15, 20 and distances; separation

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