Textbook Reference: Tipler 6th edition: Chapter 8, Section 3 and Section 1 (in that order) Glencoe Physics: Chapter 9 Section 1 and Section 2

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1 AP Physics Momentum and its Conservation Introduction: Newton originally wrote for the second law of motion, "The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed." Notice there is no mention of mass or acceleration but change in motion. What Newton called motion we now call MOMENTUM. The quantity known as momentum is one of the cornerstones of physics. It is a concept that explains much of the behavior of matter. The idea of momentum was first introduced by Rene Descartes in Newton included this concept in his laws of motion. If the universe is to go on forever then the stuff of which it is made cannot disappear. The idea that the total amount of material in the universe does not change is really a very old idea. The Roman poet Lucretius restated (in the first century B.C.) a belief held in Greece as early as the fifth century B.C.: The amount of matter in the universe is constant. Years before Newton was born the conservation of mass was an accepted scientific principle. Looking at moving things in the world around us easily leads to the conclusion that everything set in motion eventually stops--that every clock, every machine eventually runs down. It would appear that the amount of motion in the universe is decreasing and that the universe, like any machine, must be running down. To many philosophers of the1600's the idea of a universe that was running down was incompatible with the idea of the perfection of God; surely He would not construct such an imperfect mechanism. It was felt that some definition of "motion" could be found which would permit the statement that "the quantity of motion in the universe is constant". It has been shown experimentally that the momentum of a closed system is conserved. The law of conservation of momentum is equal in importance to the law of conservation of energy, which you will study later. MOMENTUM of a particle is a dynamic quantity defined as the product of the mass and the velocity of a particle. The symbol for momentum is p and:. Only an external force can change the momentum of a body. The change in momentum caused by an external force depends upon the amount of force and the time interval over which the force acts. The product of the force and the time interval over which the force acts is called the IMPULSE. The symbol for impulse is J and:. An impulse applied to a body produces a change in momentum. Both impulse and momentum are vector quantities and:. Performance Objectives: Upon completion of the readings and activities in this unit and when asked to respond either orally or on a written test, you will: define momentum and impulse. Know that impulse and momentum are vector quantities. recognize the universality of momentum. Be able to calculate changes in momentum. show an understanding of the relationship between the third law of motion and momentum changes by solving problems and explaining situations. state the law of conservation of momentum. Recognize its implications. Apply the law to the solution of problems. explain the general nature of the law of conservation of momentum. Apply vectors to the solution of momentum problems. Textbook Reference: Tipler 6th edition: Chapter 8, Section 3 and Section 1 (in that order) Glencoe Physics: Chapter 9 Section 1 and Section 2 " I have concluded that this question of impulsive forces is very obscure, and I think that, up to the present, none of those who have treated this subject have been able to clear up its dark corners which lie almost beyond the reach of human imagination." --Galileo Galilei "To understand motion is to understand nature." -- Leonardo Da Vinci

2 Questions and Problems 1. Compute the momentum of a golf ball that has a mass of 60.0 g and is moving with a velocity of 70.0 m/s. 4.2 kg-m/s 2. A force of 50.0 N is applied to a hockey puck for 2.0 s. Calculate the magnitude of the impulse N-s 3. Assume that the puck in Problem 2 has a mass of 0.50 kg and is at rest before the impulse acts upon it. With what speed does it move across the ice after the 2.0 second period? m/s 4. A force of 6.0 N acts on an object for 10.0 seconds. The mass of the object is 3.0 kg. a) What is the object's change in momentum? 60 N-s b) What is its change in velocity? 20 m/s 5. In terms of impulse and momentum, why are padded dashboards safer in automobiles? 6. In terms of impulse and momentum, why are nylon ropes, which stretch considerably under stress, favored by mountain climbers? 7. Why is a punch more forceful with a bare fist then with a boxing glove? 8. Railroad cars are loosely coupled so that there is a noticeable time delay from the time the first and last car is moved from rest by the locomotive. Discuss the advisability of this loose coupling and slack between cars from the point of view of impulse and momentum. 9. You push a body with a force of 3.0 N for 0.50 s. What impulse do you give the body? 1.5 N-s 10. How great is the impulse that gives an 8.00 kg mass a change in velocity of 4.00 m/s? 32 N-s 11. A 3.0 kg body has been accelerated by a constant force of 12 N from 10.0 m/s to 18 m/s. a) What impulse was given to the body? 24 N-s b) How long was the force acting? 2 sec A rope trails behind it. a) A reasonable estimate of the largest forces you could apply to stop the car by pulling on the rope is 250 newtons. How long would it take you to bring the car to rest? (Recall: the acceleration and the force must be in the direction opposite to the velocity in order to slow down the freight car.) b) Ten meters from the point where you start pulling, another freight car is standing. Will there be a crash? 60 s 9 m 14. Suppose you throw a ball against a wall and catch it on the rebound. How many impulses were applied to the ball? Which impulse was the greatest? 15. Would a head-on collision between two cars be more damaging to the occupants if the cars stick together or if the cars rebound upon impact? 16. A boxer tires more readily when he misses his opponent than when he hits him. Why is this so? 17. A car weighing 15,680 N and moving at 20 m/s is acted upon by a force until it is brought to a halt. a) What is the car's mass? b) What is its initial momentum? c) What change in the car's momentum does the force bring about? d) How long does the braking force act on the car to bring it to a halt? 1600 kg N-s N-s 18. A 1500-kg car moving with a speed of 15 m/s collides with a utility pole and is brought to rest in 0.30s. Find the average force exerted on the car during the collision N 19. A 3.0 kg particle has a velocity of (3i - 4j) m/s. Find its x and y components of momentum and the magnitude of its total momentum. 9N-s 12 N-s 15 N-s 20. A 3.0 kg ball with an initial velocity of (12i + 5.0j) m/s collides with a wall and rebounds with a velocity of (-12i + 5.0j) m/s. a) What is the impulse exerted on the ball by the wall? b) If the ball was in contact with the wall for 3.0 ms, what is the magnitude and direction of the average net force on the ball? 72i N-s 24,000i N 12. A constant force applied to a 2.0 kg object at rest moves it 4.0 m in 2.0 seconds. What impulse was applied? 8 N-s 13. A railroad freight car with a mass of 5.0 x 10 4 kg is rolling along a level track at 0.30 m/s (.7 mph).

3 21. A 3.0 kg ball with an initial velocity of 13 m/s, 23 north of east collides with a vertical wall and rebounds with a velocity of 13 m/s, 23 north of west. a) What is the impulse exerted on the ball by the wall? b) If the ball was in contact with the wall for 3.0 ms, what is the magnitude and direction of the average net force on the ball? 72i N-s 24,000i N 22. A golf ball strikes a hard, smooth floor at an angle of 30.0º wrt the vertical and, as the drawing below shows, rebounds at the same angle. The mass of the ball is kg, and its speed is 45 m/s just before striking the floor. What is the magnitude of the impulse applied to the golf ball by the floor? 3.7 N-s Find an expression for the force acting on it as a function of time. 26. A particle has a momentum given by p(t) = (4.00t ) i in SI units. What is the net external force acting on the particle at t = 2.00 s? 48 N i If force is a function of time, then we need a forcetime graph or the function describing the relationship in order to determine the impulse applied to the body. We now use the more general form of the impulse formula: This means if a force time graph is given, the area under the curve will yield the impulse. If you are given the force as a function of time, you must integrate the force function to determine the impulse. 23. A garden hose is held parallel to the ground as shown in the figure below. What force is necessary to hold the nozzle stationary if the discharge rate of the water is kg/s with a speed of 25.0 m/s? 15 N 27. An estimated force-time curve for a baseball struck by a bat is shown in the figure below. From this curve, determine (a) the impulse delivered to the ball, (b) the average force exerted on the ball, and (c) the change in momentum of the ball. J = 13.5 N-s = Δp F avg = 9,000 N 24. An interaction between a golf club and the ball in Problem 1 gave the ball its momentum. a) If the ball was at rest on the tee before the interaction, what was the impulse on the ball by the club? b) If the impact between the golf club and the ball lasted for 2.0 x 10-4 seconds, what was the rate of change of momentum? 2.1 x 10 4 N Up to this point we were working with a constant force or an average force. In reality the force involved in an interaction is rarely constant, rather the force varies with time. 25. The momentum of a vehicle traveling in a straight line is given in SI units by the equation 28. The force, F, acting on a 1.5-kg particle varies in time as shown in figure below. Find (a) the impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the average force exerted on the particle during the first five seconds. (d) Find the final velocity of the particle if its initial velocity were -3.0 m/s instead of zero. 8 N-s 5.3 m/s 1.6 N 2.3 m/s 29. The force, F, acting on a 2.00-kg particle varies in time as shown in figure below. Find (a) the

4 impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the average force exerted on the particle during the first five seconds. (d) Find the final velocity of the particle if its initial velocity were -2.00m/s instead of zero. 12 N-s 6 m/s 2.4 N 4 m/s is in seconds and F is in newtons. If the velocity of the ball is3.0 m/s at t = 0, what is the magnitude of v x when t = 2seconds? 4 N-s 1.0 m/s 33. The only force acting on a 2.0 kg mass moving along the x-axis is given by F x (t) = 3t, where t is in seconds and F is in newtons. If the velocity of the ball is2.0 m/s at t = 0, at what value of t will v x be equal to 8.0m/s? 2.8 s 34. Calculate the impulse on a body when it experiences a force: acting over an interval from t = 0 to t = 0.30 s N-s 30. The forces shown in the force-time graph in figure below act on a 1.5-kg mass. Find (a) the impulse for the interval t = 0 to t = 3.0 s and (b) the impulse for the interval t = 0 to t = 5.0 s. (c) If the forces act on a 1.5-kg particle that is initially at rest, find the particle's speed at t = 3.0 s and at t = 5.0 s. 12 N-s 8 N-s 8 m/s 5.3 m/s 35. A 20.0-kg space probe fires an ion engine for 10.0 s during which time it experiences a force given by F(t) = (6.00 N/s 2 )t 2 i. If it was originally moving in the x-direction at 40.0 m/s, what is its velocity at t = 5.00 s? 52.5 m/s i 36. A balloon filled with helium has a mass of 20.0 g and is sailing due east at 0.10 m/s when it is caught in a gust of wind out of the west that lasts 0.50 s and exerts a force on the balloon of (0.025 N/s)t. What is the balloon's velocity at the end of 0.40 s? 0.2 m/s 37. A 1.00-kg cart on a large horizontal air track can move along its length without any appreciable friction. Given that the cart was at rest when, at t = 0, it began to experience a force F(t) = (15.0 N/s)t acting along the length of the track, find the speed of the cart at t = 6.00s. 270 m/s 31. The only force acting on a 2.0 kg object moving along the x-axis is shown in the graph below. If the velocity is +2.0 m/s at t = 0, what is the velocity at t =4.0 seconds? 4 m/s 32. The only force acting on a 2.0 kg mass moving along the x-axis is given by F x (t) = -4t + 2, where t 38. A 10.0 kg block of wood rests motionlessly on an air table at t = 0. It subsequently experiences a force in the x-direction that increases linearly from zero to 50.0N in 10.0 s. Given that there is no friction of any sort, what is the velocity of the block at t = 10.0 s? 25 m/s 39. A force acting along the z-axis on a vehicle is kn at t = 0, whereupon it decreases linearly to kn at t = s. (a) Make a graph of F(t) versus t, and from the geometry determine the area under the curve between t = 0 and t = s. (b) Write an equation for F(t), and use the calculus to determine the net impulse applied to the vehicle between t = 0 and t = s. (c)compare your two answers kn-s

5 40. A 1.00-kg cart which is initially at rest on a straight horizontal air track can move frictionlessly along the track's length. If it experiences a force specified by the expression: F(t) = (4.00 N/s)t (5.00 N/s 2 )t 2, a) write an equation for v(t). b) If the force acts for 1.00 s, what is the maximum speed attained by the cart and at what time does that occur? 0.8 sec 0.43 m/s 41. A kg asteroid is being acted on by a "tractor beam" from an alien spaceship. At t = 0 the asteroid has velocity components of v x = 24.0 m/s, v y = 15.0 m/s, and v z = m/s. If the force exerted on the asteroid has components of F x = 2.00t kn, F y = ( t) kn, and F z = 0, find the asteroid's velocity components at t = 5.00 s. v = (274, 90, -60) m/s The Law of Conservation of Linear Momentum states that the total momentum of an isolated system cannot change. In a collision, one object gains the momentum lost by the other. In an explosion the sum of the momenta of all the particles after the explosion it equal to the original momentum of the intact object. Total momentum before collision = Total momentum after collision 42. A plastic ball of mass g moves with a velocity of 30.0 cm/s. This plastic ball collides with a second plastic ball of mass g which is moving along the same line with a velocity of 10.0 cm/s. After the collision, the velocity of the g mass is 26 cm/s along the same line. What is the velocity of the g mass? 22 cm/s 43. A ball of mass 10.0 g moving with a velocity of 20.0 cm/s collides with a second ball of mass 20.0 g moving along the same line with a velocity of 10.0 cm/s. After the collision, the first ball is still moving in its original direction, but it has velocity of only 8.0 cm/s. Determine the velocity of the second ball after the collision. 16 cm/sec in original direction 44. A ball of mass 3.0 kg, moving at 2.0 m/s eastward, strikes head-on a ball of mass 1.0 kg that is moving at 2.0 m/s westward. The balls stay together after the impact. a) What is the magnitude and direction of the velocity of the combined mass after collision? 1.0 m/s east 45. A car of mass kg travels at 20 m/s. The car collides with a stationary truck of mass kg. The two vehicles interlock as a result of the collision. What is the velocity of the car-truck system. 6.7 m/s in original direction 46. A bullet of mass 50.0 g strikes a wooden block of mass 5.0 kg. The bullet becomes embedded in the block. The block with the bullet in it then flies off at 10.0 m/s. What was the original velocity of the bullet? 1010 m/s 47. A billiard ball of unknown mass travels at 40.0 cm/s. The ball overtakes a second billiard ball of mass g which is traveling along the same line at 15 cm/s. After the collision, the first ball moves in the same line at 20.0 cm/s. The second ball is moving at 42 cm/s. What is the mass of the first ball? g 48. A g ball traveling horizontally in a straight groove at 200 cm/s strikes a g ball that was originally at rest. After the collision the first ball rebounds at 120 cm/s, while the other is sent forward. Calculate the velocity of the second ball cm/s 49. A 40.0 kg projectile leaves a kg launcher with a velocity of m/s forward. What is the recoil velocity (speed and direction) of the launcher? 16 m/s in opposite direction 50. A cesium nucleus emits a beta particle of mass 9.11x kg with a velocity of 1.50 x 10 7 m/s. The resulting barium nucleus has a mass of 2.20 x kg. What is the recoil speed of the barium nucleus? 62 m/s 51. A proton (mass = 1.67 x kg) with a speed of1.00 x 10 7 m/s collides with a motionless helium nucleus, and the proton bounces back with a speed of 6.00 x 10 6 m/s. The helium nucleus moves forward with a speed of 4.00 x 10 6 m/s after the bombardment. a) Can you compute the mass of the helium nucleus? If so, what is it? b) Can you compute the force that acted during the collision? If so, what is it? c) If you answered "no" to either (a) or (b), discuss why you gave this answer x kg

6 52. Would you care to fire a gun that has a bullet ten times as massive as the gun? Explain. 53. When an apple falls to the ground and strikes the earth without rebound, what "becomes" of the momentum of the apple? 54. In terms of conservation of momentum explain why it is not a good idea to jump from a rowboat to a dock that seems just within jumping distance, if the boat is not tied to the dock. 55. Is it possible for a rocket to attain a speed greater than the velocity with which the exhaust gases leave it? Explain. 56. An open freight car is coasting along on a smooth track when rain starts to fall thereby increasing the mass of the car. Does the velocity of the car remain constant? Explain. 57. If you throw a ball horizontally while standing on roller skates, you will roll backward with a momentum that matches that of the ball. Will you roll backward if you go through the motions of throwing the ball but instead hold onto it? Explain. 58. When traveling in your car at highway speed the momentum of a bug suddenly changes as it splatters into your windshield. Compared to the change of momentum of the bug, by how much does the momentum of your car change? 59. If a Mack truck and a new Volkswagen Beetle have ahead-on collision, which vehicle will experience the greater force of impact? The greater impulse? The greater change in its momentum? The greater acceleration? 60. Suppose there are three astronauts outside a space ship, and two of them decide to play catch with the third one. All the astronauts weigh the same on earth and are equally strong. The first astronaut throws the second one toward the third one and the game begins. Describe the motion of the astronauts as the game proceeds. How long will the game last? 62. A rifle with a weight of 30.0 N fires a 5.0-g bullet with a speed of 300 m/s. (a) Find the recoil speed of the rifle. (b) If a 700-N man holds the rifle firmly against his shoulder, find the recoil speed of man and rifle m/s 0.02 m/s kg astronaut is on a space walk away from the shuttle when her tether line breaks! She is able to throw her 10.0-kg oxygen tank away from the shuttle with a velocity of 12.0 m/s to propel herself back to the shuttle.(see the figure below.) Assuming that she starts from rest (relative to the shuttle), determine the maximum distance she can be from the craft when the line break sand still return within 60.0 s (the amount of time she can hold her breath). 120 m 64. An 8.00-g bullet is fired into a 250 g block that is initially at rest at the edge of a 1.00-m high table as shown in the figure below. The bullet remains in the block, and after the impact the block lands 2.00 m from the bottom of the table. Determine the initial speed of the bullet m/s 65. A 0.11-kg tin can is resting on top of a 1.7-mhigh fencepost. A kg bullet is fired horizontally at the can. It strikes the can with a speed of m/s, passes through it, and emerges with a speed of 720 m/s. When the can hits the ground, how far is it from the fencepost? Disregard friction while the can is in contact with thepost m 61. High-speed stroboscopic photographs show that the head of a 200-g golf club is traveling at 55 m/s just before it strikes a 46-g golf ball at rest on a tee. After the collision, the club head travels (in the same direction) at 40 m/s. Find the speed of the golf ball just after impact m/s

7 Collisions in Two Dimensions 66. Ball A of mass 2.0 kg is moving at a velocity of 5.0m/s. Ball A collides with stationary ball B, also of mass 2.0 kg. After the collision, ball A moves off in a direction 30 to the left of its original direction. Ball B moves off in a direction 60 to the right of ball A's original direction. a) Draw a vector diagram to find the momentum of ball A and of ball B after the collision. b) What is the speed of each ball after collision? 4.33 m/s 2.5 m/s 67. Ball A of mass 5.0 kg moves at a speed of 40.0 m/s. It collides with a second stationary ball B, also of mass5.0 kg. After the collision, ball A moves off in the direction 45 to the left of its original direction. Ball B moves off in the direction 45 to the right of ball A's original direction. a) Draw a vector diagram to determine the momentum of ball A and of ball B after the collision. b) What is the speed of each ball after the collision? 141 kg-m/s 68. Object A of mass 6.0 kg moves at a speed of 3.0m/s. It collides with object B also of mass 6.0 kg. After the collision, object A moves off in the direction 50 to the left of its original direction. Object B moves off in a direction 40 to the right of object A's original direction. a) Draw a vector diagram to determine the momentum of object A and of object B kg-m/s kg-m/s b) What is the speed of each object after the collision? 1.93 m/s 2.30 m/s 69. A billiard ball of mass 0.50 kg moves at a speed of 10.0 m/s. It collides with a second stationary billiard ball, also of mass 0.50 kg. After the collision, the first ball moves off in a direction 60 left of its original direction. The second ball moves off in a direction 30 to the right of the first ball's original direction. a) Draw a vector diagram to determine the momentum of each billiard ball. b) What is the speed of each ball after the collision? 2.5 kg-m/s 5 m/s 4.33 kg-m/s 8.7 m/s 70. A star of mass 2.0 x kg moving with a velocity of 2.0 x 10 4 m/s collides with a second star of mass 5.0 x10 30 kg moving with a velocity of 3.0 x 10 4 m/s in a direction at right angles to the first star. If the two join together, what is their common velocity? 2.2 x 10 4 m/s 71. A space probe of mass 7600 kg is traveling through space at 120 m/s. Houston Control determines that a change in course of 30 is necessary and by electronic communication instructs the probe to fire rockets perpendicular to its direction of motion. If the escaping gas leaves the craft's rockets at an average speed of 3200 m/s, what is the mass of the gas Houston Control should allow to be expelled? 164 kg 72. The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of kg and is moving along the x axis with a velocity of +5.5 m/s. It makes a collision with puck B, which has a mass of kg and is initially at rest. After the collision, the two pucks fly apart with the angles shown in the drawing,. Find the final speed of (a) Puck A and (b)puck B. 3.4 m/s 2.58 m/s 73. A mine car, whose mass is 440 kg, rolls at a speed of 0.50 m/s on a horizontal track, as the shown in the drawing below. A 150-kg chunk of coal has a speed of 0.80 m/s when it leaves the chute. Determine the velocity of the car/coal system after the coal has come to rest in the car m/s 74. By accident, a large plate is dropped and breaks into three pieces. The pieces fly apart parallel to the floor. As the plate falls, its momentum has only a vertical component, and no component parallel to the floor. After the collision, the component of the total momentum parallel to the floor must remain zero, since the net external force acting on the plate has no component parallel to the floor. Using the

8 data shown in the drawing below, find the masses of pieces 1 and 2. m 1 = m 2 = 1.00 kg 75. A fireworks rocket is moving at a speed of 50.0 m/s. The rocket suddenly breaks into two pieces of equal mass and they fly off with velocities v 1 and v 2 as shown in the drawing. What is the magnitude of (a) v 1 and (b)v 2? 86.5 m/s 50.0 m/s Relative Velocity. The velocities given in the preceding problems were relative to the earth. However, in problems where two or more things are separating, you may find some problems where the velocity of one body is given relative to the other body. These problems won t be as hard to solve if you remember that the velocity substituted into the conservation of momentum is the velocity relative to the earth. Given the velocity of m 1 relative to m 2, and you want to find the velocity of m 1 relative to the earth, use the following relationship. v 1 relative to m2 v 2 = v 1 relative to earth