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Chapter 8: MOMENTUM Homework-Updated 10/26/11 Momentum (p=mv) is an important concept because it’s another quantity that is conserved in the universe. We can relate the change in momentum of a body to what is called the impulse (I=∫Fdt) as follows. As a result of Newton’s 3rd law, whenever two bodies interact each one applies an equal and opposite force and impulse to the other (FAB =-FBA). Newton’s 2nd law (F=ma) can then be rewritten as ∫[Fdt]=p (“impulse equals change in momentum”). Therefore, during an interaction momentum is transferred from one body to the other in equal amounts and the total amount of momentum of a “closed system” doesn’t change. The Conservation of Momentum Principle is most useful in analyzing collisions and explosions. In general, when the interactive forces between two (or more) bodies are much larger than any other force acting on the bodies, those other forces are negligible and the sum total of the momenta of those bodies is conserved during the interaction. The total momenta just before an interaction can be compared to the total momenta just after the interaction in order to solve problems. Finally, since momentum is a vector quantity, the principles apply independently to the x, y, and z-directions and you have one equation for every dimension. 1. Consider two gliders (A=0.2 kg and B=0.3 kg) that collide on an air track. During the collision the gliders apply an average force of 6 N for 0.05 sec to each other. We will look at the situation just before the collision takes place, during the collision, and just after the collision. Information about the initial velocities are given in the chart below. Use dynamics chart. and kinematics principles to fill in the rest of theCollision 5 m/s 4 m/s forces=6N A B ΩZ Velocity just before Average Change in velocity Velocity just after collision acceleration during during collision collision collision A 5 m/s B 4 m/s A+B Now fill in the following chart about the momentum and impulses in this collision: Glider Momentum just Average Change in momentum Momentum just before collision impulse during during collision after collision collision A B A+B By comparing these two charts you may begin to appreciate the usefulness of the momentum-impulse concept in analyzing collisions. Fnet a) Are there any quantities in the first chart that are “conserved”? b) Are there any quantities in the second chart that are “conserved”? 640 N Area=Imp=∆p 2. Consider a child with a mass of 40 kg who is playing on a trampoline. During one bounce the child hits the trampoline with a vertical speed of 1.0 s 2.0 s 3.0 s 10 m/s. A graph of the net, impulsive force is shown. Assume that the trampoline applied only a vertical force on the child. a) Using the graph estimate the change in the momentum of the child. b) What was the average force that the trampoline applied to the child? Don’t forget that the force applied by the trampoline is not the only force on the child; you must also consider the weight of the child to get the net force. c) Determine the velocity of the child after the bounce. Is this a positive or negative velocity? d) If the child misses the trampoline and instead falls on the ground, what is the change in his momentum? Why is this more dangerous than bouncing off the trampoline? e) On the graph above sketch a curve representing the interaction with the ground. What would the two curves have in common? f) Momentum is supposedly a conserved quantity, yet here the momentum of the child has changed, what other object is also changing momentum? How much does the momentum of that other object change? Why do we ignore this part of the problem? 3. A 200-g puck sliding on ice strikes a barrier at an angle of 53o and it bounces off at an angle of 45o. The speed of the puck before the bounce was 15 m/s and after the bounce it is 12 m/s. The time of contact during the bounce is 0.05 sec. a) Write these velocities in rectangular (x&y) components relative to the barrier. b) Determine the x and y components of the impulsive force. Which one can be identified as a normal force and which can be identified as a friction force? 53o 45o c) What was the direction of the impulsive force with respect to the barrier? d) How much energy was lost to heat in the bounce? e) If the impulsive force had the same normal component but the friction component were zero, what would be the final velocity of the puck? 4. On a horizontal plane, mass m moves uniformly in a circle of radius R in a time T. Determine: a) The average impulse on the mass over: (i) one quarter a revolution, (ii) one half a revolution, (iii) a full revolution. b) Draw a vector diagrams illustrating the relationship between momenta and impulse in (a). c) The average impulsive force on the mass during each interval in (a). Don’t forget to give directions. d) We know that in circular motion centripetal forces cannot do work (Fc is perpendicular to ∆s). Can they apply impulses? Explain. 5. Consider the following one-dimensional, inelastic collision between two cars. One car, with a mass of 2 kg and a speed of 5 m/s, collides with another car with a mass of 3 kg moving with a speed of 2 m/s in the same direction. After the collision the 2-kg car has a speed of the -1 m/s. Even though energy is not conserved during this collision, momentum is conserved if we restrict the analysis to a comparison of the momenta of the cars just before and just after the collision. a) What conditions are required for the momentum conservation principle to be valid? Why do we restrict our analysis to the moments just before and just after the collision? b) Determine the velocity of the 3-kg car just after the collision. See chart below. c) If the collision had been perfectly inelastic, what would be the final speed of the carts? The cart would “stick” together and move with speed v=[total p/ total m]=16/5=3.2 m/s d) It is instructive to look at a collision from different “frames of reference”. The information in this problem was given with respect to the ground frame of reference. Now you’ll redo the same problem from two other different frames of reference: (1) a frame of reference moving with the initial velocity of the 2-kg car (5 m/s), and (2) the center of mass frame of reference (here the center of mass speed is 3.2 m/s). In each case determine the initial and final speeds of the cars relative to each frame. To do this, imagine yourself looking at each of the colliding cars from another car that is moving at the speed of the frame. Remember that the relative velocity formula for velocities is: vA/B + vB/C = vA/C. In this equation “A” represents the car being looked at, “B” represents the moving frame, and “C” represents the ground. The following chart may help you summarize your results. Answers in chart. Frame of Velocity just before Momentum just before Velocity just after Momentum just after Reference collision (m/s) collision (kg-m/s) collision (m/s) collision (kg-m/s) 2-kg 3-kg 2-kg 3-kg 2-kg 3-kg 2-kg 3-kg Ground 5 m/s 2 m/s -1 m/s 2-kg car 0 Center of 1.8 m/s Mass e) Does the Momentum Conservation Principle apply in any inertial frame of reference? 6. A child is playing on a swing. He swings down from an angle of 53o. The mass of the child is 40 kg and the seat of the swing has a mass of 5 kg. The length of the swing is 5 m. At the bottom of the swing the boy quickly picks up his 5-kg dog from rest and continues to swing up with it on his lap. 53o a) How far will the child and dog swing up before coming to rest? b) Instead of picking up the dog, the child slips off the swing at the bottom with zero speed relative to the swing. How far will the swing rise all by itself? c) In a third version of this problem, the child slips of the swing and the swing rises to an angle of 37o all by itself. What was the speed of the child relative to the ground when he slipped off? …relative to the swing? d) In which of part of this problem is energy conserved? …is momentum conserved? 7. In this problem we compare the momentum and kinetic energy of bodies in motion. a) Show that in general K=p2/2m. b) If a 0.04 kg marble and a 0.16 kg ball have the same kinetic energy, which has more momentum? Determine the momentum ratio of the two objects. c) If 700 N man and 500 N woman had the same momentum, which has more kinetic energy? Determine the kinetic energy ratio of the two objects. 8. Two cars are headed for a collision with equal speeds v. Cart A, however, has a larger mass M and cart B a smaller mass m. The two cars stick together after the collision so this is a perfectly inelastic collision. v v a) If M=2m, show that the combined speed of the cars is v/3. A B b) Determine the speed of the center of mass of the two cars before the collision and compare to the speed in (a). Does the comparison make sense? c) Determine the change in the velocity of each car, the change in the kinetic energy of each car, and the change in momentum of each car. d) Which car exerted the larger force on the other car? Which car is likely to sustain more damage? Justify your answers. e) Determine the speed of the cars after the collision in the extreme case of M being much larger than m (m/M0). This would be like a car colliding with a wall. 9. Determine the center of mass of the following objects or system of objects. System Needed info Chosen origin X center of mass Y center of mass M1 M2 Two unequal Origin added M2 D/(M1 + M2 ) 0 point masses to illustration 0 D Uniformly dense Origin added L/4 L/4 L bent rod to illustration 0 L a Uniformly a Origin added 2.64a 0.786a a b b dense surface: to illustration 0 b=2a Uniformly Origin added 5/6R 0 R R circular surface to illustration 0 Hole with hole Triangular Origin added 2a/3 b/3 b uniform to illustration Requires calculus. Done in class. 0 a surface Non-uniform Origin added (10/18)oL 0 0 L linear density: to illustration Requires calculus. = o(1+x/L) o=constant 10. A toy cannon mounted on top of a cart fires a ball at an angle of 37o to the horizontal. l The initial speed of the ball is 20 m/s. The mass of the ball is 0.5 kg and the mass of the cannon and cart combined is 2 kg. a) Determine the recoil velocity of the cart due to the cannon firing. b) How do you account for the change in the y-component of the momentum of the ball? c) The cart is attached to a spring of constant k=400 N/m that is initially relaxed. What will be the maximum stretch of the spring? 11. Two blocks of unequal masses (m and 5m) are compressing a spring of constant k between them. When the masses are released, the spring pushes the masses apart over a frictionless surface as it returns to equilibrium. m2 m1 a) After the spring is decompressed determine the following ratios: Velocity Momentum Kin. Energy Impulse Work Power v1/v2 p1/p2 K1/K2 I1/I2 W1/W2 P1/P2 Ratio -5/1 -1/1 5/1 -1/1 5/1 5/1 b) What fraction of the energy stored in the spring went to the smaller mass? c) When a rifle is fired the change in the momentum of the bullet is the same as the change in the momentum of the rifle. Explain why it is the bullet that is the more deadly part. 12. A block of mass M moving with speed v hits a surface at an angle . The block lands on a flat side and doesn’t bounce but it slides a distance D on the horizontal surface slowed by friction during a time interval “t” until it stops. The coefficient of kinetic friction here is k. a) Determine the value of the speed v using momentum and impulse principles. v b) What fraction of the original kinetic energy was dissipated as heat by the sliding? µk c) What happened to the rest of the original K that was not dissipated by the sliding D, t friction? vx 13. A projectile is fired with speed vo at an angle ø with respect to the horizontal. Assume no air resistance. The projectile returns to the same level and it has a range R and altitude H. vo ø a) In going from the ground to the maximum height, determine the change in momentum in the x-direction and in the y-direction. b) In going from ground back to the ground to the, determine the change in momentum in the x-direction and in the y-direction. c) What is the impulsive force in the case of a projectile? Draw a graph of F vs. t for the projectile. 2vx d) Now suppose that, at the maximum height H, the projectile breaks into two equal pieces. One piece ends up with no velocity and moves directly downward H taking a time “t” to hit the ground. How much time will it take the other piece to hit the ground? How far apart will the two pieces be on the ground? ∆x 14. In lab we will experiment with a ballistic pendulum. This is a massive bob hanging from a light, pivoting rod that is set to catch a rapidly moving projectile. After catching the projectile the rod rises to a maximum height “h=L-Lcos”. This can be used to determine the initial speed of the projectile. Assume that the mass of the bob is M, that the mass of the projectile is m, and that the length of the rod is L, but has negligible mass. The initial speed of the projectile is vo. a) In what part of this problem is momentum conserved? In what part is energy conserved? b) Determine the maximum swing on the pendulum m in terms of the other given quantities. M c) If the bob fails to catch the projectile and the projectile bounces vo back, does the pendulum swing higher or lower? Justify your answer. d) What minimum speed would the projectile need to have initially to make the rod swing all the way to the top? Note: The bob doesn’t need a minimum speed at the top if connected to a rigid rod. e) The rod in the apparatus is replaced with a string. Is the speed determined in (d) for the projectile enough to make the bob swing all the way to the top? Explain. How much speed would you need at the bottom to allow the pendulum on a string to reach the top? 15. In a perfectly elastic collision both momentum and energy are conserved. Consider two carts of mass mA and mB with initial velocities vA and vB and final velocities v´A and v´B. a) Write the momentum conservation principle for this collision and the kinetic energy conservation principle. b) Derive the very useful relative velocity formula for this collision: (vA - vB)= -(v´A– v´B). c) Explain the significance of the relationship. Why is the difference (vA - vB) called the relative velocity between the two carts? The values vA,vB,v´A,v´B are measured with respect to which frame of reference? d) Describe the collision as seen by an observer in one of the carts. e) Describe the collision as seen by an observer moving with the center of mass. 16. Consider an elastic collision between two carts. The collision is elastic because an ideal spring of constant 425 N/m is attached to one of the carts. Cart A has a mass of 3 kg and cart B has a mass of 1 kg. Initially cart A is moving with a velocity of 1 m/s toward cart B that is at rest. The graph below plots the positions of both carts vs. time from the beginning to the end of the collision: 0.20 Position (m) Car B Collision ends 0.10 Collision starts Car A 0.0 0.04 0.08 0.12 0.16 0.20 time (sec) Use the information in the graph to fill in the relevant values in the chart below: (Note: This graph may not print out properly so your answers may not agree too closely with mine!) Before Collision Maximum Compression After Collision cart A B A B A B velocity momentum Kinetic energy a) Verify that the relative velocity before the collision is the same (but negative) as the relative velocity after the collision. Also explain why the carts have equal velocities when spring is most compressed. b) Verify that momentum is the same before, during, and after the collision. c) Verify that kinetic energy is the same before and after, but not during the collision. What happens to the “missing kinetic energy” during the collision? d) Determine the maximum compression of the spring. e) Sketch a velocity vs. time graph for both carts throughout the interaction. 17. For the following 2-D collisions or explosions draw the appropriate vector diagram relating the momentum vectors before and after the event. Also write the momentum conservation relationships and give an expression for the velocity of the center of mass. Example Exploding object: Inelastic Collision Elastic Collision Before After Before After Before After Illustration v 3m vA 2m v0 v=0 m v√3 m v 5m 2m m v 2v v’ m 2m 2m vB Vector diagram Momentum Conservation Expressions Center of Mass Velocity 18. Atomic particles collide all the time. Consider a helium nucleus which is also called an alpha particle (4He) striking a gold nucleus (197Au) head on. In nuclear notation, the superscript denotes the atomic mass of the particle in what are called atomic mass units (1 “amu” is about the mass of a proton). Assume the collision is perfectly elastic and use “amu” as the mass unit. He vo Au a) Determine expressions for the final velocities of the two nuclei in terms of the initial speed vo of the alpha particle, assuming that they move in the same direction after the head-on collision. b) What fraction of the alpha particle’s energy is transferred to the gold nucleus? c) If the collision is not head-on the particles move on in different directions. Assume the alpha particle is deflected at right angles to its initial direction and determine the final speed of the nuclei after the collision and the direction of motion of the gold nucleus. (You can simplify this problem a lot if you assume that the speed of the smaller particle changes only negligibly. The answers are very close to the correct ones). 18A. Show that if two equal masses collide elastically the angle between their paths after the collision will always be 90o regardless of the actual initial and final velocities of the masses. 19. Two stars start from rest very far apart (think infinity) and are attracted together by gravity. Assume that the stars masses are M1 and M2 and that no other mass will interfere with their D interaction. Consider the starts when they are a distance D apart. a) Write the energy conservation formula that applies here and explain why it is valid here. b) Write the momentum conservation formula that applies here and explain why it is valid here. c) Determine the velocities of each star when they are a distance D apart. d) Determine the center of mass of the system and its velocity. e) Couldn’t you simply use the acceleration of each mass to determine the speeds? Explain. 20. A bullet of mass m moving with velocity vo crashes into a block of mass M at rest on a frictionless horizontal surface. The block is connected to a spring of constant k. Assume that the time of the bullet/block interaction is very short compared to the time it takes the spring to respond. In each of the following cases determine maximum compression of the spring. To get some numerical values make m= 20 g, M=0.5 kg, vo = 300 m/s, and k= 400 N/m. a) In the first case assume that the bullet sticks to the block. b) In the second case assume that the bullet bounces back m M directly with half its original speed. c) In the third case the bullet crashes through the block emerging on the other side with half its original speed but still moving in the same direction. d) Which has the bigger impact on the large mass, the sticky interaction or the bouncy interaction? Explain. 21. In another version of the problem above, assume that the block is not connected to the spring but it is sitting at the edge of a table that is H high. When the bullet hits the block it flies off the table and it lands a distance X from the bottom of the table. In each case below, determine the initial velocity of the bullet in the terms of the given quantities. m M a) In the first case assume that the bullet sticks to the block. b) In the second case assume that the bullet bounces back directly with half its original speed. c) In the third case the bullet crashes through the block emerging on the other side with half its original speed but still moving in the same direction. d) Which makes the block move farther, the sticky bullet or the bouncy bullet? Explain. 22. A child stands on a cart initially at rest. The cart and child together have mass of 60 kg. Also on the cart are three large blocks 20 kg each. The child throws each block horizontally away from the cart with a speed of 2 m/s relative to the cart always. Vblock/cart =2 m/s a) Determine the speed of the cart and child, relative to the ground, after each block is tossed out by the child. b) If the child could manage to throw all the blocks at once, what would be the recoil velocity of the cart and child? Explain why this answer is not the same as the final velocity in (a). c) Assume that, instead of individual blocks, there is a tank carrying 60-kg of water. The child uses a hose to eject the water horizontally away from the tank at a rate of 1 kg/sec. The water leaves the hose with a speed of 2 m/s relative to the child. Determine the final speed of the cart and child after all the water is ejected. d) Does the rate at which the water is ejected in (c) affect the final speed of the cart? Explain. 23. Consider a rocket in outer space away from any gravitational attraction. It is ejecting fuel at a rate of 5 kg/s with a speed of 200 m/s relative to the rocket. Assume that 40% of the original 40-kgrocket mass is fuel. Vexhasut/rockt a) Determine the thrust of the rocket. b) Determine the initial acceleration and the final acceleration of the rocket. c) Determine the final speed of the rocket and the time that it took to attain that speed. d) How would you determine the distance traveled by the rocket? (You don’t need to get the final answer) e) Show that you can derive the basic rocket equations from the momentum conservation principle. f) Redo parts (b) and (c) above assuming that the rocket is actually lifting off from the surface of the earth so that the weight of the rocket must be taken into consideration. Assume that “g” is constant during the burning of the fuel. g) If you run out of fuel to eject in outer space is there another way to move your rocket? Explain. 24. A raft of length L and mass M is a distance d from the dock. At the end closer to the dock stands a person of mass m. For the purpose of getting some numerical values assume that M=200 kg, m= 50 kg, L= 10 m, d= 4 m. Dock d a) As the person walks the center of mass of the system doesn’t change. Explain why. b) The person walks to the other end of the raft. How far did the raft move relative to the dock? How far did the person move relative to the dock? c) If the person accelerated until it reaches the end of the raft with a speed 6 m/s relative to the raft, what would be the speed of the raft relative to the water? What would be the speed of the person relative to the water? 25. A canoe with a mass of 200 kg is moving along with a speed of 12 m/s. When it passes by a second canoe with a mass of 300 kg, the second canoe transfer 50 kg of its mass to the first canoe directly across (perpendicular to) the motion (lowering its mass to 250 kg). Assume that the canoes cannot move sideways, so all changes in velocity occur parallel to their motions. Determine the final velocities of both canoes in the following cases. a) Explain why the motion of the second canoe is not affected by the 12 m/s mass transfer, but the motion of the first canoe is affected by it. 200 kg b) In the first case the second canoe is at rest when it transfers the 50 kg 50 kg mass. 250 kg c) In the second case the second canoe is moving with the same velocity as the first canoe when it transfers the 50 kg mass. d) In the third case the second canoe is moving with a speed of 10 m/s in the same direction as the first canoe when it transfers the 50 kg mass. e) In the fourth case the second canoe is moving with the same speed as the first canoe but in the opposite direction when it transfers the 50 kg mass. 26. A tennis ball moving at 18 m/s strikes the 45o hatchback of a car moving away at 12 m/s. Both speed are given with respect to the ground. The ball rebound elastically from the car. Assume that the car is much more massive than the ball, which is reasonable. Car frame of a) Determine the speed of the ball relative to the car before the bounce. reference b) If the ball bounces elastically, how fast and in what direction will it be moving with respect to the car after the bounce? c) What is the velocity of the ball with respect to the ground after the bounce? Where did the lost kinetic energy of the ball go? d) Explain why in this problem it was safe to assume that the speed of the car is not noticeably affected by the ball striking it. Challenge problems: 27. A frame with a mass of 0.15 kg stretches a spring a distance of 0.50 m when it is hung from it. A lump of putty with a mass of 0.20 kg is dropped from rest onto the frame from a height of 30 cm. 30 cm a) Find the spring constant. b) In which part of this problem is momentum conserved?… in energy conserved? c) Determine the maximum stretch of the spring. d) Determine the stretch of the spring when the system comes to a final state of equilibrium. 28. Two identical billiard balls are initially at rest when they are struck symmetrically by a third identical ball moving with velocity vo in the x-direction. The collision is perfectly elastic. a) Explain why the struck balls must end up with the same speed vo and also be deflected by 300 to the horizontal. b) Draw an appropriate vector diagram relating the initial and final momenta of the masses. c) Find the velocities of all three balls after the collision in terms of vo. Determine the impulses on each ball during the collision. 29. A small mass m is moving with velocity vo on a horizontal frictionless surface. The small makes contact with a curved shaped wedge of mass M initially at rest which it free to move on the frictionless surface. As the two masses interact, the small mass rises on the curved side of the wedge to a height h and the wedge accelerates to the right. Finally the mass slides down and returns the way it came but with one-third its original speed and the wedge moves on with a constant speed. vo vo/3 h Before Middle After a) In which part of this problem is momentum conserved?...energy conserved? b) When the mass reaches the height h what is the relative velocity between the mass and the wedge? c) Determine the speed of the mass and wedge together when the mass reached the height h. d) Determine the height h in terms of the other quantities. e) Determine the final velocity of the wedge after its interaction with the mass. f) Determine the ratio m/M.