HSC Physics Space Notes

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HSC Physics Summary ©

Ben 2010-present

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UNIT 1: Space a. b. c. d. e.

Definitions Earth’s Gravitational Field Factors of a Rocket Journey (Projectile Motion) Gravity in the Solar System Theories of Time & Space (Aether + Special Relativity)

a. Definitions

Weight (N)

Work (W)

Gravitational Field

The force (Newtons) that acts upon an object due to the presence of a gravitational field. The magnitude of the weight force depends on the strength of the field at a point and the mass of the object: FW = mg FW is weight in Newtons m is mass in Kilograms g is acceleration due to gravity in ms-2 Work is a measure of energy required to displace an object a specific distance. Work is given by the formula: W = Fs W is work in joules F is force in Newtons s is displacement in metres Region in which a mass experiences a force towards the centre of gravity – usually the centre of a large mass (e.g. planet) The gravitational force experienced by a mass at a point is given by Netwon’s Gravitational Force Equation:

Fg = where: Fg is the gravitational force in Newtons (N) G is the universal gravitational constant (6.67x10-11 Nm2kg-2) d is the distance between the centre of the two masses (m) mo & mp are the masses of the object and planet respectively (kg)

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The acceleration due to gravity (g) caused by a mass (planet) is given by:

g=G

Universal Gravitational Constant

Gravitational Potential Energy (Ep)

where: g is the acceleration due to gravity in ms-2 G is the universal gravitational constant (6.67x10-11 Nm2kg-2) d is the distance from the centre of the mass (planet) in metres Acceleration due to gravity at earth’s surface is 9.8ms-2 downwards A numerical constant existing in many of Newton’s equations. It equal to:

G = 6.67 x 10-11 ( units: Nm2kg-2 ) Potential energy possessed by a mass according to its position within a gravitational field. The work done on an object to raise it from the surface of a planet to a higher altitude is equivalent to the object’s Ep: Ep = mass x gravity x height = mgh Conversely, work is done by gravity to lower an object and reduce its Ep. On an Astronomical scale, Ep = 0 at an infinite distance away ( ) i.e. At any tangible distance, Ep < 0, as represented by the equation:

Ep = where: Ep is Gravitational Potential Energy in joules G is the universal gravitational constant (6.67x10-11 Nm2kg-2) d is the distance between the centre of the two masses (m) m1 & m2 are the masses of the object and planet respectively (kg) Any moving object that moves only under the sustained force of Projectile gravity. The velocity that must be attained by an object in order to escape the gravitational field of a planet. Escape velocity is determined by the -1 Escape Velocity mass and radius of the planet. Earth’s escape velocity is 11.2 kms

v2 =

G-Force

A ‘G-Force’ is a unit of force acting upon an Astronaut. Multiple GForces equate to multiples of the Astronauts regular Weight Force. (i.e. 2 G-Force = 2x Normal Weight) The G-force scale is an easily understood and communicated scale. The scale is applicable to all Astronauts, regardless of their mass. This is because the force they experience will be relative their personal weight.

g-force = Frame of time during which a rocket needs to be launched so that it Launch Window reaches its destination at the right time. Launch windows are largely based upon Earth’s rotation and Earth’s orbit around the sun.

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Uniform Circular Motion is undergone by objects travelling along a circular path. The circular path is caused by the object’s velocity, which Uniform attempts to keep it travelling straight, while an external centripetal Circular Motion force (such as gravity) directed towards the axis at a right angle causes it to follow a circular path for as long as the centripetal force acts.

=

Kepler‟s Law of This equation, derived from Newton’s Law of Universal Gravitation, Periods

Orbital Decay Atmospheric Drag Ionosphere Exosphere Re-Entry Point of Weightless ness

can be used to find the orbital period, T, of any orbiting mass around any planet. Orbital Decay refers to the orbital descent and eventual fall to earth experienced by satellites in LEO orbits. It is caused by atmospheric drag. Atmospheric friction causes a satellite to lose forward velocity, and hence causes it to lose altitude (according to Fc < Fg.) Friction slows the satellite, causing it to lose altitude where there is more friction which further slows the satellite… and so on!

Thermosphere. Upper layers of Earth’s atmosphere (80km – 640km) Outermost sphere of Earth’s Atmosphere extending 9600km. Return of a spacecraft into Earth’s atmosphere and subsequent descent to Earth An object between the moon and earth will experience a point of weightlessness where the gravitational attraction due to gravity from both the Earth and moon will be equal and opposite. a.k.a. Gravity Assist Effect. Method used by astronauts to “slingshot” a

Slingshot Effect spacecraft around a planet, exploiting its gravitational field to accelerate the craft.

a.k.a Geosynchronous orbit. An orbit in which a satellite travels with the earth’s atmosphere, remaining at the same point in the sky relative to earth’s surface. (e.g. Foxtel, Communications, GPS.) Satelites in LEO are usually between 250-1000km above sea level, and Low Earth Orbit never higher than 1500km. They have shorter periods (1-5 hours) and (LEO) their position in respect to earth is constantly changing. (e.g. satellite imaging, weather forecasting, spying.) The art of making scientific discoveries accidentally. Many major breakthroughs in science have been ‘stumbled upon’ in this manner. Serendipity (e.g. Michelson & Morely) Electromagnetic Self-propagating waves of varying wavelengths that travel at the speed of light (c). They do not need a medium through which to travel and Radiation (EMR) include all the radiation on the electromagnetic spectrum. (e.g. x-rays) A comparison between a quantity to a selected standard and expressing Measurement the measured quality as a factor of that standard. (e.g. 2 x std. metre) All measured quantities are relative quantities. SI unit / Standard of length. The distance travelled by light in a vacuum Metre Geostationary Orbit

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in the fraction

Simultaneity Limiting Velocity

of a second. (i.e. defined in terms of time)

Events will occur at different times in different frames of reference based upon the observer’s velocity. Both perspectives are correct. Observers in relative motion will disagree on the simultaneity of events separated in space. No object can travel faster than the speed of light (c = 3x108 ms-1)

Scalar or Vector Quantity

SI Units

Distance / Displacement / Radius Time / Period (T) Speed / Velocity Impulse / Momentum Work Done / Gravitational Potential Energy Momentum

metres (m) seconds metres-per-second (ms-1) Newton-seconds (Ft) Joules Kilogram-metres-per-second (Kgms-1)

b. Earth’s Gravitational Field DOT POINTS 1.1.1 – 1.1.3 

Mass (the amount of matter of which an object consists) does not change with location  Weight (the force acting upon a mass due to gravity) changes according to gravity  Gravitational Fields are regions in which a mass experiences a force towards the centre of gravity – usually the centre of a large mass (e.g. planet) the gravitational force of such a field at a point is given by Newton’s Gravitational Force Equation:

Fg = This equation is derived from the below equation, from which g (a) is replaced by 

(F = ma)

The acceleration due to gravity (g) at a point caused by a large mass is given by:

g=G This formula can be used to ascertain the acceleration due to gravity (g) on the surface of any given planet

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PRACTICAL: Perform an investigation to determine a value for acceleration due to gravity using pendulum motion and identify reasons for possible deviations from the correct value of 9.8ms-2

AIM: Determine acceleration due to gravity using a pendulum and compare experimental results to published results. The relationships between the period (T) of a simple pendulum is related to its length (l) and acceleration due to gravity (g) is shown by:

T=2 EQUIPMENT: Retort stand, bosshead and clamp, roll of string, masses, stopwatch METHOD: 1. 2. 3. 4. 5. 6. 7.

Set up a retort stand and clamp on the edge of a desk and tie a length of string to it Tie a 200g mass 1 metre down the string (cut off excess string) Release the masses from 20o deviation from vertical Using a stopwatch, time how long it takes for the pendulum to complete 10 full periods Record this time in a results table with the corresponding length of string Perform a total of three times for each length of string Shorten the length of string by 10cm after each set of three trials and repeat steps 4-7 until results are obtained for a string length of 50cm.

EXPERIMENTAL ERRORS:  The trial for each different length of string could be repeated several more times to allow for greater accuracy  Observing the time taken for 20 Periods to pass instead of 10 will reduce the error involved with the reaction time of the person with the stopwatch  A light gate could be used to gather more precise measurements of the period of the swing FACTORS AFFECTING THE VALUE OF g ON EARTH:  Due to Earth’s spin, there is a slight bulge at the equator and flattening at the poles. Because the force of a gravitation field (Fg) acting upon an object is directly proportional to

, an object’s elevation affects the force it experiences.

 The density and chemical composition of the Earth’s crust between an object and the origin of force influences the magnitude of the force the object experiences this is because there is more mass per volume, which equates to greater force.

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How is a change in gravitational potential energy related to work done?

 The Gravitational Potential energy possessed by an object is determined by its mass and its distance from the centre of a gravitational field.  The work done (Force x Displacement) on an object to move it away from the centre of gravity is equivalent to the Ep possessed by that object: Work Done = Gravitational Potential Energy W = Ep This method of deriving Ep applies exclusively in terrestrial situations within Earth’s atmosphere Define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field.

On an Astronomical scale, a separate trail of logic applies when finding Ep – Newton’s Law of Universal Gravitation is the basis for this logic:  The gravitational attraction forces existing between two objects decreases with d2  Therefore, Fg and Ep only reach 0 when the object is an infinite distance away  BUT! The Ep possessed by an object increases equivalent to the work done to move the object away from the centre of gravity and towards infinity  Hence, an object gains Ep as it gets closer towards infinity (where Ep = 0)  Therefore, the value of an object’s Ep at any tangible location has to be less than that which it possesses at infinity (i.e. at any tangible distance, Ep < 0)  Therefore, Ep has a negative value, as represented by the equation:

Ep = As the object travels from infinity to earth, Ep decreases until it reaches the surface, where Ep = 0

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c. Factors of a Rocket Journey (Projectile Motion) Describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical components.

TWO types of Projectile Motion:  Oblique Motion: o Launched at an angle of elevation o Initial velocity (u or v0) can be illustrated using a vector diagram [right] o Horizontal Velocity remains constant (Ux = Vx) o At the apex, Vy = 0 and ½ t o initial = final  Horizontal Motion: o Launched horizontally (usually from a height) o No vertical velocity at launch (Uy = 0) o Height is vertical displacement (Sy) o Horizontal Velocity remains constant (Ux = Vx)

U

Uy

Ux

Describe Galileo’s analysis of projectile motion

1. Projectiles follow a perfect parabolic path 2. Trajectory be split into two components: vertical and horizontal 3. Horizontal velocity remains constant ( Ux = Vx ) It isn’t influenced by a sustained force 4. Vertical velocity is uniformly accelerated downwards at 9.8ms-2 due to gravity 5. Projectiles are subject only to their own inertia and the sustained force of gravity http://static.newworldencyclopedia.org/thumb/7/73/Newton_Cannon.svg/220px

Solve Projectile Motion Problems using horizontal and vertical components in combination with Newton’s equations of motion

Newton’s Equations of Motion:  v = u + at  v2 = u2 + 2as  s = ut + ½ at2

Horizontal Ux = ucos Vx = ux (a = 0) Vx2 = ux2 Sx = Uxt (a = 0)

Vertical Uy = usin Vy = uy + 9.8t V2 = uy2 + 2aysy Sy = uyt + ½at2

Explain escape velocity in terms of the gravitational constant and the mass & radius of the planet

To escape Earth’s gravitational pull, a projectile fired from the surface of Earth needs to be given kinetic energy equal to its gravitational potential energy:

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EK = ½m1v2

Ep =

EK > Ep ½ m1v2 > v2 > Escape velocity increases with the planet’s mass and decreases with distance from the centre of gravity. Escape velocity is independent of the projectile’s mass.

Outline Newton’s concept of escape velocity

Newton theorised his principle based on a hypothetical scenario in which a projectile is fired from an impossibly high vantage point as such as speed (8000ms-1) that it never lands due to the balancing factors of the Earth’s curvature and gravity. (i.e. the object enters orbit when fired fast enough.) He thus reasoned that if an object were to be fired faster than this theoretical value (8kms-1), it could escape earth’s gravitational field.

Earth’s escape velocity is 11.2kms-1 Identify why the term ‘g-forces’ is used to explain the forces on an astronaut

  

G-forces are multiples of the normal weight force experienced on Earth The G-force scale is used to easily communicate the force acting upon an astronaut, expressing it in terms of what they normally experience. The G-force scale is applicable to every individual based on their personal, unique mass. This is because experienced forces are relative to their mass.

g-force = 1g = 9.8 ms-2 ; In a rocket accelerating upwards at 9.8ms-2, astronauts experience 2g = 19.6ms-2

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Perform a first-hand investigation to calculate initial and final velocities, range and time of flight of a projectile. i.e. Mega Marble LauncherTM Projectile Motion Class Assignment\Projectile Motion Assignment – Mega Marble Ludicrous Launcher with graph.docx

AIM: To determine and graph the relationship between the launch angle and range of an oblique projectile using the Mega Marble Launcher™ - and hence find the optimum launch angle that corresponds to the maximum range possible.

SAFETY:   

Ensure all personelle wear safety goggles at all times throughout the experiment Conduct experiment in isolated area secluded from students and other hazards Only fire projectile when the firing range is clear and all personnel are behind the line of fire

METHOD: 1. Set up the Mega Marble Launcher™ in a remote location, pointed in a direction with at least 50m of space and free from obstruction. 2. Arm the launcher, first setting the launch angle to 20o and placing a marble into the shaft. (Ensure all marbles launched in the experiment are of the same size and shape) 3. When the firing range is clear, launch the marble. 4. Measure the range with a measuring tape and retrieve the marble. Record this value for the range in a table like the one below with the corresponding launch angle: Launch Angle Range (m) 20o xx 5. Repeat the trial with the same launch angle a total of 5 times to ensure reliable results are collected. 6. Repeat steps 2-5 with a launch angle of 30o, 45o, then again for 60o, completing each trial a total of 5 times to ensure reliable results are collected. 7. Graph the results, with launch angle (o) on the horizontal axis and average range (m) on the vertical.

RESULTS & ANALYSIS: Results show range increasing with the launch angle to a maximum value achieved at 45o, (as was hypothesised) then decreasing for 60o. The closer to 45o the angle, the greater the range achieved. The relationship between the range and the launch angle is non-linear. The range of a projectile is not directly proportional to its launch angle; the relationship between these variables is more complex. An online source shows the relationship to be:

R = V2 x LIMITATIONS, ERRORS & IMPROVEMENTS: -

The muzzle velocity is only sufficient to achieve small ranges – experimental results are more “bunched up” and hence variations in range are harder to detect. There are only 4 launch angle settings (20o, 30o, 45o, 60o), making it difficult to conduct a comprehensive analysis.

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+ + +

The contour and texture of the test range (grass) caused the marble to roll or bounce upon landing. The crosswind proved to be a major factor that influenced the results Perform more trials for each launch angle to obtain reliable results Apply a greater consistent launching force to the marbles so that longer, more diverse (and hence comparable) ranges are reached. Use a firing range that is flat and sheltered to minimise wind resistance and/or projectile bouncing and rolling.

Analyse the changing acceleration of a rocket during launch in terms of Conservation of Momentum and the forces experienced by astronauts

Rocket launch, Momentum and Forces 

At launch, the downward momentum of exhaust gases provides equal upward impulse (F x t) to propel the rocket (Newtons 3rd law) : Momentum of Rocket

Momentum of Exhaust

Because change in momentum of an object is equal to the impulse of an applied force, so the impulse of the exhaust gases down will equal the impulse applied to the rocket upwards:

 



Pilots experience vision problems at 4g, and lose consciousness at 8g. 3g was once considered safe. Astronauts can survive up to 20g if:  They lying down (stops blood draining from head)  Facing opposite to the direction of force (stops eyes from popping out) The forces acting upon a rocket during its launch and flight include:  Weight Force (down)  Thrust (up; Thrust > Fw)  Reaction force (up while stationary; =0 when in flight)

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Discuss the effect of Earth’s orbital and rotational motion on rocket launches

Effect of Earth’s motion on Rocket Launch     

The earth spins counter-clockwise (when viewed from above the north pole) Rockets are launched eastward from the equator, where the rotational speed of the earth is greatest, and adds an extra 1700kmh-1 to their trajectory. The orbital speed of the Earth around the Sun can also be harnessed to attain greater velocity in respect to the solar system (used for Intra-Solar-System travel.) Less fuel needs to be spent to attain escape velocity and more storage mass (payload) can be carried if rotational speeds are harnessed. ‘Launch Windows’ are frames of time during which a rocket must be launched to arrive at its destination at the right time – taking full advantage of orbital speeds.

Analyse the forces involved in uniform circular motion for a range of objects, including orbiting satellites

Uniform Circular Motion is undergone by objects travelling along a circular path. The circular path is caused by the object’s velocity, which attempts to keep it travelling straight, while an artificial centripetal force directed towards the axis at a right angle causes it to follow a circular path for as long as the centripetal force acts. Centripetal Force / Acceleration:    

Fc Acts towards the centre of the Circular Locus Direction (and velocity) change continuously due FNET Velocity & Centripetal force keep object in motion Speed Remains constant

Fc =

ac =

In the context of a rocket orbiting earth, the force of gravity is considered the centripetal force as the rocket produces a right-angle velocity around earth. To stay in orbit, a satellite needs to maintain a speed in proportion with the earth’s gravity, its own mass (m) and its distance from Earth’s centre (r.) This is because the centripetal force (Fc = gravity, in this case) will remain constant, so the satellite must adjust its velocity to balance it and thus undergo uniform circular motion.

In Orbit: Fc = Fg where…

Fg =

Fg = Gravitational Force acting upon the object as given by Fg = R = radius of orbit centre-to-centre (in metres) v = Orbital speed of object [ms-1] m = mass of object in orbit (in kilograms)

=

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Compare qualitatively, Low Earth Orbit (LEO) and Geostationary Orbits

Feature

Low Earth Orbit Satellites

Geostationary Satellites

ALTITUDE PERIOD ORBIT POSITION

(