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This topic contains notes, examples and exercises which deal with objects that are moving horizontally with constant speed and vertically downward with the acceleration due to gravity.

This topic contains notes, examples and exercises which deal with objects that are moving horizontally with constant speed and vertically downward with the acceleration due to gravity.

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- 1. PROJECTILE MOTION constant acceleration in a straight line; free fall under gravity, projectile motion; Relative motion, change in velocity, velocity vector components, circular motion (constant speed with one force only providing centripetal force). 1. Recall that in the absence of friction a falling object will have a constant acceleration of 10 ms-2 and that this value is referred to as “gravitational acceleration, g” 2. Use F = mg to show that Nkg-1 is an equivalent unit to ms-2. 3. Explain the effect that air friction has on the acceleration of a falling object (resulting in free fall) 4. Define the term projectile. 5. Describe projectile motion in terms of its uniform horizontal motion and it accelerated vertical motion. 6. Use equations of motion to calculate time, distance, velocity and acceleration.
- 2. FREE FALL - IGNORING FRICTION If the object is falling slow: • We consider these objects to have a constant acceleration of 9.81 ms-2. This means that the acceleration does not change. The object speeds up by 9.81 ms-1 every second. • This is not strictly true but is very close to being true because air friction at slow speeds is insignificant. We can now perform straightforward calculations. • This acceleration is often rounded to 10 ms-2 to allow calculations to be performed easily. This value is called the “gravitational acceleration” of the object and is given the symbol, “g”. g = 10 ms-2 THE OBJECT ACCELERATES THROUGHOUT ITS FALL The reason the acceleration not changing is because the force of gravity on a given object does not change. It is the force on an object that causes the object to accelerate. This is supported mathematically by the equation, F = mg or ... doesn’t change g = F so g cannot change because a is a product of m two physical quantities that do not change. doesn’t change
- 3. FREE FALL - IN THE PRESENCE OF FRICTION If the object is falling fast: Air friction is too large to ignore. Here we consider the object to have two forces acting on it. Air friction and gravity. A. When the object is initially falling slowly, air friction is much smaller than the force of gravity on the object. B. As the object speeds up the air friction increases. C. The object will eventually reach a speed at which the air friction balances the force of gravity on the object. The object cannot fall any faster than at this speed (unless it changes shape). This speed is called terminal velocity. This situation is illustrated by the following example: B A C Vector arithmetic with notes
- 4. WHAT IS A PROJECTILE? Brainstorm Examples of Projectiles “Why do you think that a ball dropping vertically (in free fall) is not an example of projectile motion?” ______________________________________________________ ______________________________________________________
- 5. WHAT IS A PROJECTILE? • A projectile is an object that is both moving horizontally and falling vertically at the same time. • The path of any projectile travelling in a vacuum is a parabola Brainstorm Examples of Projectiles “Why do you think that a ball dropping vertically (in free fall) is not an example of projectile motion?” ______________________________________________________ ______________________________________________________
- 6. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 7. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 8. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 9. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 10. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 11. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 12. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 13. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 14. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vx vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 15. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vx vx vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 16. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vx vx vy vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 17. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy vx vx vy vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 18. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy vx vx vy = 0 vy vx vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 19. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy vx vx vy = 0 vy vx vy vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 20. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy vx vx vy = 0 vy v vx vy vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 21. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy v vx vx vy = 0 vy v vx vy vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 22. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy v v vx vx vy = 0 vy v vx vy vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 23. PROJECTILES - HORIZONTAL AND VERTICAL MOTION • A projectile is an object that has constant horizontal velocity and constant vertical acceleration due to gravity. In other words it is moving horizontally at the same time it is in free fall. The projectiles velocity is the sum of these two velocities. • The path of a projectile is parabolic and symmetrical • Air resistance is considered to be negligible • The force of gravity is the only force acting on the object. vy v v vx vx vy = 0 vy v vx vy v vx v = the size of the velocity of the projectile To determine the velocity: vx vx = the horizontal component of the vy projectile’s velocity add the horizontal and v vy = the vertical component of the vertical components projectile's velocity
- 24. SYMMETRICAL FLIGHT PATH • The path of a projectile is parabolic and symmetrical Symmetrical Path It is often convenient to work with half the flight path of a projectile. Remember When the projectile is at the maximum height the vertical velocity is zero. At t : vy = 0 2
- 25. SYMMETRICAL FLIGHT PATH • The path of a projectile is parabolic and symmetrical Symmetrical Path It is often convenient to work with half the flight path of a projectile. d Remember When the projectile is at the maximum height the vertical velocity is zero. At t : vy = 0 2
- 26. SYMMETRICAL FLIGHT PATH • The path of a projectile is parabolic and symmetrical Symmetrical Path It is often convenient to work with half the flight path of a projectile. t 2 d Remember When the projectile is at the maximum height the vertical velocity is zero. At t : vy = 0 2
- 27. SYMMETRICAL FLIGHT PATH • The path of a projectile is parabolic and symmetrical Symmetrical Path It is often convenient to work with half the flight path of a projectile. t 2 t d Remember When the projectile is at the maximum height the vertical velocity is zero. At t : vy = 0 2
- 28. SYMMETRICAL FLIGHT PATH • The path of a projectile is parabolic and symmetrical Symmetrical Path It is often convenient to work with half the flight path of a projectile. t 2 t d d= the total distance travelled by a projectile t = the time taken to travel that distance t/2 = the time taken to reach maximum height Remember When the projectile is at the maximum height the vertical velocity is zero. At t : vy = 0 2
- 29. Forces The force due to gravity is the only force acting on the projectile. Not a projectile: An object in horizontal flight is not a projectile Fgrav because Fgrav is balanced by the lift force. An object that has not been launched but maintains a high horizontal speed will have air friction acting on it and therefore must have a thrust force to balance air friction. This will not be a projectile. Vertical motion calculations • Since the force of gravity is constant (10 N per kg of any object), the acceleration is also constant. (10 ms-2 for any object) • Kinematic equations must be used for analysing the vertical motion Horizontal motion calculations • A projectile progresses horizontally with constant speed. • Horizontal speed can be calculated in the usual way: Speed = distance travelled v=d time taken t Units of v: ms-1
- 30. PROCESS FOR PROBLEM-SOLVING [PIA] 1. Read the question and underline the relevant information 2. Draw a diagram of the situation 3. List the information that relates to the vertical motion and the horizontal motion. (Keeping it separate. Remember that vertical and horizontal motion are independent of each other) 4. Show up as a positive direction and down as negative. The signs that you use for the vertical information should reflect this. 5. Select the appropriate kinematic equation when calculating a vertical quantity. 6. Use v = d/t for calculation of a horizontal quantity. Example 1 Imagine a car is driven off a 250m cliff at 30 ms-1. How far from the base of the cliff will the car be when it lands? ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Diagram ________________________________________________
- 31. Example 2 A hockey ball is flicked with an initial velocity of 14 ms-1, 45o to the horizontal, as shown. A spectator 1.6 m tall is standing directly in the path of the ball, 20 m away. Will the ball hit the spectator? Diagram ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________
- 32. 12 PHYSICS PROJECTILES ASSIGNMENT Name ______________________ 1. A rock is dropped from the top of a bridge. It takes 4 seconds to reach the water below. (b) Explain why this is not an example of projectile motion. ________________________________________________________________ ________________________________________________________________ (c) What would need to be done to the rock to turn it into a projectile? ________________________________________________________________ (d) How far has the rock fallen? ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ (e) A second rock is now thrown horizontally at 6 ms-1 from the same bridge. ________________________________________________________________ (f) How long will it take for the rock to reach the water? ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ (g) Calculate the range of this second rock ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
- 33. 2. A 2 kg bowling ball is launched from ground level and follows the path shown below: At each of the labelled positions A, B, C and D, state the size and direction of: (a) the acceleration of the projectile (b) the net force on the projectile 3. When an aircraft was travelling in level flight at 200 ms-1, a nut fell off part of the landing gear. Assume air friction is negligible. (a) Sketch the paths of the aircraft and the nut. Sketch The nut covered a total horizontal distance of 3.0 km. (b) Find the total time it took to fall to the ground. ________________________________________________________________ ________________________________________________________________
- 34. (c) What was the altitude of the plane when the nut fell off? ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ 4. A ball is kicked with an initial velocity as shown below. The angle of inclination is 40o. 21 ms-1 40o (a) Calculate the horizontal component of the initial velocity. __________________ (b) Calculate the vertical component of the initial velocity. ____________________ (c) What is the instantaneous velocity of the ball at the top of its path? ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ (d) Find the height of the ball at the top of its path. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________
- 35. (e) Find the time taken for the ball to reach its maximum vertical displacement. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ (f) Calculate the range of the ball. ________________________________________________________________ ________________________________________________________________ If the ball was kicked at an angle of 60o to the horizontal, would it have travelled as far along the field? ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ 5. A potato was launched from the muzzle of a spud gun with an initial velocity of 30 ms-1 at an angle of 60o to the horizontal. Will it clear a 25 m tree that is 60 m away on the flight path? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 6. A cricket ball was hit with the following trajectory. Find the initial velocity of the ball. 25 m 100 m
- 36. 7. Johnny is competing in the javelin event of his school athletics competition. The javelin behaves like an ideal projectile. (a) Describe the shape of the path of the javelin. ________________________________________________________________ (b) Ignoring air resistance, draw arrow(s) on the drawing of the javelin below to show the force(s) acting on it when it is in the position shown. Name the forces. Joe now throws the javelin into the air at an angle of 40o above the horizontal at an initial velocity of 30 ms-1. Joe now throws the javelin into the air at an angle of 40° above the horizontal at an initial velocity of 30 m s–1
- 37. (c) Show that the horizontal component of the initial velocity of the javelin is 23 ms-1. ________________________________________________________________ ________________________________________________________________ (d) Calculate the range of the javelin under these conditions. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ [this is a 2004 NCEA exam question]
- 38. Starter 1 WARMUP QUESTIONS 1. A student stands against the rail of a bridge and drops a rock into the water below. His fellow classmates measure the time it takes for the rock to fall. They stop their watches the instant they see the rock make contact with the water. They calculate an average time of 5.4 s for the rock to fall. Calculate the distance through which the rock has fallen. 2. A second student, also on the bridge notes that from when the cars enter the bridge to when they exit the acceleration of each car is constant. The length of the bridge is 150 m and the time taken for that distance to be travelled is 10 s. For a car that exits the bridge at a speed of 27 ms-1, calculate the speed at which it enters.
- 39. Starter 2 WARMUP QUESTION An aircraft is flying at a height of 500 m above the ground. Explain why the aircraft is not an example of projectile motion. __________________________________________________________________ __________________________________________________________________ While the aircraft is flying horizontally at a speed of 35 ms-1, a packet is dropped. Calculate the vertical speed of the packet when it reaches the ground. Using this vertical speed and knowing that the horizontal speed is 35 ms-1 draw a vector diagram that shows the velocity of the packet. ( Hint: the velocity of the packet is equal to the vertical velocity added to the horizontal velocity .... a vector calculation)
- 40. Starter 3 WARMUP QUESTION Falling 1. The diagram shows a strobe photo of a tennis ball’s motion. The ball is rolled along the top of a bench and drops off the edge of the bench at point A. The strobe flashes every 1/10 second against a background of 10 cm squares. (a) Calculate the ball’s horizontal component of velocity at position C? __________________________________________ __________________________________________ (b) What is the ball’s initial horizontal velocity at point A? __________________________________________ (c) How long does it take for the ball to go from position A to position F? __________________________________________ (d) What is the vertical distance travelled by the ball between A and F?__________________________________________ (e) What is the ball’s vertical component of velocity at position A?__________________________________________ (f) Use an equation of motion to calculate the ball’s acceleration? __________________________________________ __________________________________________
- 41. Starter 4 KERRY’S THROWING ARM Kerry has spent many years perfecting his cricket throw. One day whilst out on the field with the physics class he practises a throw. He is trying to throw the length of the rugby pitch which is 150 m long. A camera measures the speed and angle of his throw as 27.5 ms-1 and 40o respectively. Does Kerry achieve his goal?
- 42. Starter 5 HOLE IN ONE - TAUPO The hole in one challenge in Taupo involves standing at the lake front and swinging at the golf ball in an attempt to sink a hole in one on a platform that is floating some distance out in the lake. Adam is convinced that if he can achieve a ball speed of 31 ms-1 horizontally he will at least be able to land the ball on the platform. The situation is pictured below: Does Adam get the ball on the platform? 31 ms-1 10 m 45 m 3m
- 43. Starter 6 BATTER UP 150 m Joe De Maggio bats a baseball a distance of 150 m in 5 s. What is the altitude of the ball 2 s into the flight?
- 44. Starter 7 KERRY’S THROWING ARM Kerry has spent many years perfecting his cricket throw. One day whilst out on the field with the physics class he practises a throw. He is trying to throw the length of the rugby pitch which is 150 m long. A camera measures the speed and angle of his throw as 27.5 ms-1 and 40o respectively. Does Kerry achieve his aim?
- 45. Starter 8 PROJECTILES REVISION A student throws a cricket ball which lands 100 m from where it is launched. The rest of the physics class times the throw and averages the times that they measure. The average time is 5 s. (a) Sketch a diagram of the flight path of the cricket ball. Label your diagram with all the relevant information. (b) Calculate the horizontal component of the ball’s velocity. __________________________________________________________________ (c) Explain why this component of the ball’s velocity can be considered to be constant. __________________________________________________________________ (d) Calculate the vertical component of the ball’s velocity as it strikes the ground. __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ (e) Calculate the speed with which the ball strikes the ground. __________________________________________ __________________________________________ __________________________________________ __________________________________________ _____________________________________
- 46. Starter 9 Mass and Weight • Weight is a measure of the force due to _____________ on an object. • It is measured in _____________ , ___ • Mass is a measure of ________________________________________________ . • Near the earth’s surface the weight per kg on an object is _______ . • Weight and mass are related by the following formula: • g is called the __________ ____________ or _____________ ______________ . • As we travel away from the earth the force due to gravity _______________. In other words g is only a constant near the surface of the earth. It gets smaller as an object moves away from the earth’s surface. • The value of g depends on _______________________________________________ • Smaller planets will have ___________ values of g. For example on the moon, g = _________ • Gravitational attraction holds satellites in a ____________ ____________.

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