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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 7**

**Multiple-Choice Problems**

1. A spiral spring is compressed so as to add U units of potential energy to it. When this spring is instead stretched two-thirds of the distance it was compressed, its remaining potential energy in the same units will be (see Section 2.5 for a review of proportional reasoning)

A. 2U/3 B. 4U/9 C. U/3 D. U/9

2. A block slides a distance d down a frictionless plane and then comes to a stop after sliding a distance s across a rough horizontal plane, as shown in the accompanying figure. What fraction of the distance s does the block slide before its speed is reduced to one-third of the maximum speed it had at the bottom of the ramp?

A. s/3 B. 2s/3 C. s/9 D. 8s/9

3. You slam on the brakes of your car in a panic and skid a distance d on a straight and level road. If you had been traveling twice as fast, what distance would the car have skidded under the same conditions?

A. 4d B. 2d C. √2d D. d/2

4. You wish to accelerate your car from rest at a constant acceleration. Assume that there is negligible air drag. To create a constant acceleration, the car’s engine must

A. maintain a constant power output.

B. develop ever-decreasing power.

C. develop ever-increasing power.

5. Consider two frictionless inclined planes with the same vertical height. Plane 1 makes an angle of 30° with the horizontal, and plane 2 makes an angle of 60° with the horizontal. Mass m1 is placed at the top of plane 1, and mass m2 is placed at the top of plane 2. Both masses are released at the same time. At the bottom, which mass is going faster?

A. m1

B. m2

C. Neither; they both have the same speed at the bottom.

6. A brick is dropped from the top of a building through the air (friction is present) to the ground below. How does the brick’s kinetic energy (K) just before striking the ground compare with the gravitational potential energy (Ugrav) at the top of the building? Set y = 0 at the ground level.

A. K is equal to Ugrav

B. K is greater than Ugrav

C. K is less than Ugrav

7. Which of the following statements about work is or are true? (More than one statement may be true.)

A. Negative net work done on an object always reduces the object’s kinetic energy.

B. If the work done on an object by a force is zero, then either the force or the displacement must have zero magnitude.

C. If a force acts downward, it does negative work.

D. The formula can be used only if the force is constant over the distance

8. A 10 kg stone and a 100 kg stone are released from rest at the same height above the ground. There is no appreciable air drag. Which of the following statements is or are true? (More than one statement may be true.)

A. Both stones have the same initial gravitational potential energy.

B. Both stones will have the same acceleration as they fall.

C. Both stones will have the same speed when they reach the ground.

D. Both stones will have the same kinetic energy when they reach the ground.

E. Both stones will reach the ground at the same time.

9. Two identical objects are pressed against two different springs so that each spring stores 50 J of potential energy. The objects are then released from rest. One spring is quite stiff (hard to compress), while the other one is quite flexible (easy to compress). Which of the following statements is or are true? (More than one statement may be true.)

A. Both objects will have the same maximum speed after being released.

B. The object pressed against the stiff spring will gain more kinetic energy than the other object.

C. Both springs are initially compressed by the same amount.

D. The stiff spring has a larger spring constant than the flexible spring.

E. The flexible spring must have been compressed more than the stiff spring.

10. For each of two objects with different masses, the gravitational potential energy is 100 J. They are released from rest and fall to the ground. Which of the following statements is or are true? (More than one statement may be true.)

A. Both objects are released from the same height.

B. Both objects will have the same kinetic energy when they reach the ground.

C. Both objects will have the same speed when they reach the ground.

D. Both objects will accelerate toward the ground at the same rate.

E. Both objects will reach the ground at the same time.

11. Two objects with different masses are launched vertically into the air by identical springs. The two springs are compressed by the same amount before launching. Which of the following statements is or are true? (More than one statement may be true.)

A. Both masses reach the same maximum height.

B. Both masses leave the springs with the same energy.

C. Both masses leave the springs with the same speed.

D. Both masses leave the springs with the same kinetic energy.

E. The lighter mass will gain more gravitational potential energy than the heavier mass.

12. Two objects with unequal masses are released from rest from the same height. They slide without friction down a slope and then encounter a rough horizontal region, as shown in the accompanying figure. The coefficient of kinetic friction in the rough region is the same for both masses. Which of the following statements is or are true? (More than one statement may be true.)

A. Both masses start out with the same gravitational potential energy.

B. Both objects have the same speed when they reach the base of the slope. Both masses have the same kinetic energy at the bottom of the slope.

D. Both masses travel the same distance on the rough horizontal surface before stopping.

E. Both masses will generate the same amount of thermal energy due to friction on the rough surface.

13. Spring #1 has a force constant of k, and spring #2 has a force constant of 2k. Both springs are attached to the ceiling, identical weights are hooked to their ends, and the weights are allowed to stretch the springs. The ratio of the energy stored by spring #1 to that stored by spring #2 is

A. 1:1 B. 1:2 C. 2:1 D. 1:√2 E. √2:1

14. Two balls having different masses reach the same height when shot into the air from the ground. If there is no air drag, which of the following statements must be true? (More than one statement may be true.)

A. Both balls left the ground with the same speed.

B. Both balls left the ground with the same kinetic energy.

C. Both balls will have the same gravitational potential energy at the highest point.

D. The heavier ball must have left the ground with a greater speed than the lighter ball.

E. Both balls have no acceleration at their highest point.

15. The stone in the accompanying figure can be carried from the bottom to the top of a cliff by various paths. Which path requires more work?

A. AC

B. ABC

C. The work is the same for both paths.

Problems

1. A fisherman reels in 12.0 m of line while landing a fish, using a constant forward pull of 25.0 N. How much work does the tension in the line do on the fish?

2. A tennis player hits a 58.0 g tennis ball so that it goes straight up and reaches a maximum height of 6.17 m. How much work does gravity do on the ball on the way up? On the way down?

3. A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of 15° with the forward direction of motion. If the tension in the rope is 180 N, how much work does the rope do on the skier during a forward displacement of 300.0 m?

4. A constant horizontal pull of 8.50 N drags a box along a horizontal floor through a distance of 17.4 m. (a) How much work does the pull do on the box? (b) Suppose that the same pull is exerted at an angle above the horizontal. If this pull now does 65.0 J of work on the box while pulling it through the same distance, what angle does the force make with the horizontal?

5. You push your physics book 1.50 m along a horizontal tabletop with a horizontal push of 2.40 N while the opposing force of friction is 0.600 N. How much work does each of the following forces do on the book? (a) your 2.40 N push, (b) the friction force, (c) the normal force from the table, and (d) gravity? (e) What is the net work done on the book?

6. A 128.0 N carton is pulled up a frictionless baggage ramp inclined at 30.0o above the horizontal by a rope exerting a 72.0 N pull parallel to the ramp’s surface. If the carton travels 5.20 m along the surface of the ramp, calculate the work done on it by (a) the rope, (b) gravity, and (c) the normal force of the ramp. (d) What is the net work done on the carton? (e) Suppose that the rope is angled at 50.0o above the horizontal, instead of being parallel to the ramp’s surface. How much work does the rope do on the carton in this case?

7. A factory worker moves a 30.0 kg crate a distance of 4.5 m along a level floor at constant velocity by pushing horizontally on it. The coefficient of kinetic friction between the crate and the floor is 0.25. (a) What magnitude of force must the worker apply? (b) How much work is done on the crate by the worker’s push? (c) How much work is done on the crate by friction? (d) How much work is done by the normal force? By gravity? (e) What is the net work done on the crate?

8. An 8.00 kg package in a mail-sorting room slides 2.00 m down a chute that is inclined at below the horizontal. The coefficient of kinetic friction between the package and the chute’s surface is 0.40. Calculate the work done on the package by (a) friction (b) gravity, and (c) the normal force. (d) What is the net work done on the package?

9. Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.8 × 106 N, one 14o west of north and the other 14o east of north, as they pull the tanker 0.75 km toward the north. What is the total work they do on the supertanker?

10. A tow truck pulls a car 5.00 km along a horizontal roadway using a cable having a tension of 850 N. (a) How much work does the cable do on the car if it pulls horizontally? If it pulls at 35.0° above the horizontal? (b) How much work does the cable do on the tow truck in both cases of part (a)? (c) How much work does gravity do on the car in part (a)?

11. A boxed 10.0 kg computer monitor is dragged by friction 5.50 m up along the moving surface of a conveyor belt inclined at an angle of 36.9° above the horizontal. If the monitor’s speed is a constant 2.10 cm/s how much work is done on the monitor by (a) friction, (b) gravity, and (c) the normal force of the conveyor belt?

12. It takes 4.186 J of energy to raise the temperature of 1.0 g of water by 1.0 °C (a) How fast would a 2.0 g cricket have to jump to have that much kinetic energy? (b) How fast would a 4.0 g cricket have to jump to have the same amount of kinetic energy?

13. A bullet is fired into a large stationary absorber and comes to rest. Temperature measurements of the absorber show that the bullet lost 1960 J of kinetic energy, and high-speed photos of the bullet show that it was moving at 965 m/s just as it struck the absorber. What is the mass of the bullet?

14. Animal energy. Adult cheetahs, the fastest of the great cats, have a mass of about 70 kg and have been clocked at up to 72 mph (32 m/s) (a) How many joules of kinetic energy does such a swift cheetah have? (b) By what factor would its kinetic energy change if its speed were doubled?

15. A racing dog is initially running at 10.0 m/s, but is slowing down. (a) How fast is the dog moving when its kinetic energy has been reduced by half? (b) By what fraction has its kinetic energy been reduced when its speed has been reduced by half?

16. If a running house cat has 10.0 J of kinetic energy at speed v (a) At what speed (in terms of v) will she have 20.0 J of kinetic energy? (b) What would her kinetic energy be if she ran half as fast as the speed in part (a)?

17. A 0.145 kg baseball leaves a pitcher’s hand at a speed of 32.0 m/s. If air drag is negligible, how much work has the pitcher done on the ball by throwing it?

18. A 1.50 kg book is sliding along a rough horizontal surface. At point A it is moving at 3.21 m/s and at point B it has slowed to 1.25 m/s (a) How much work was done on the book between A and B? (b) If –0.750 J of work is done on the book from B to C, how fast is it moving at point C? (c) How fast would it be moving at C if +0.750 J of work were done on it from B to C?

19. Stopping distance of a car. The driver of an 1800 kg car (including passengers) traveling at 23.0 m/s slams on the brakes, locking the wheels on the dry pavement. The coefficient of kinetic friction between rubber and dry concrete is typically 0.700. (a) Use the work–energy principle to calculate how far the car will travel before stopping. (b) How far would the car travel if it were going twice as fast? (c) What happened to the car’s original kinetic energy?

20. Meteor crater. About 50,000 years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Recent (2005) measurements estimate that this meteor had a mass of about 1.4 × 106 kg (around 150,000 tons) and hit the ground at 12 km/s (a) How much kinetic energy did this meteor deliver to the ground? (b) How does this energy compare to the energy produced in one day by a standard coal-fired power plant, which generates about 1 billion joules per second?

21. You throw a 20 N rock into the air from ground level and observe that, when it is 15.0 m high, it is traveling upward at 25.0 m/s. Use the work–energy principle to find (a) the rock’s speed just as it left the ground and (b) the maximum height the rock will reach.

22. A 0.420 kg soccer ball is initially moving at 2.00 m/s. A soccer player kicks the ball, exerting a constant 40.0 N force in the same direction as the ball’s motion. Over what distance must her foot be in contact with the ball to increase the ball’s speed to 6.00 m/s.

23. A 61 kg skier on level snow coasts 184 m to a stop from a speed of 12.0 m/s. (a) Use the work–energy principle to find the coefficient of kinetic friction between the skis and the snow. (b) Suppose a 75 kg skier with twice the starting speed coasted the same distance before stopping. Find the coefficient of kinetic friction between that skier’s skis and the snow.

24. A block of ice with mass 2.00 kg slides 0.750 m down an inclined plane that slopes downward at an angle of 36.9° below the horizontal. If the block of ice starts from rest, what is its final speed? You can ignore friction.

25. To stretch a certain spring by 2.5 cm from its equilibrium position requires 8.0 J of work. (a) What is the force constant of this spring? (b) What was the maximum force required to stretch it by that distance?

26. A spring is 17.0 cm long when it is lying on a table. One end is then attached to a hook and the other end is pulled by a force that increases to 25.0 N, causing the spring to stretch to a length of 19.2 cm. (a) What is the force constant of this spring? (b) How much work was required to stretch the spring from 17.0 cm to 19.2 cm? (c) How long will the spring be if the 25 N force is replaced by a 50 N force?

27. A spring of force constant 300.0 N/m and unstretched length 0.240 m is stretched by two forces, pulling in opposite directions at opposite ends of the spring, that increase to 15.0 N. How long will the spring now be, and how much work was required to stretch it that distance?

28. An unstretched spring has a force constant of 1200 N/m. How large a force and how much work are required to stretch the spring: (a) by 1.0 m from its unstretched length, and (b) by 1.0 m beyond the length reached in part (a)?

29. The graph in the accompanying figure shows the magnitude of the force exerted by a given spring as a function of the distance x the spring is stretched. How much work is needed to stretch this spring: (a) a distance of 5.0 cm, starting with it unstretched, and (b) from x = 2.0 cm to x = 7.0 cm

30. A 575 N woman climbs a staircase that rises at 53o above the horizontal and is 4.75 m long. Her speed is a constant 45 cm/s (a) Is the given weight a reasonable one for an adult woman? (b) How much has the gravitational potential energy increased by her climbing the stairs? (c) How much work has gravity done on her as she climbed the stairs?

31. How high can we jump? The maximum height a typical human can jump from a crouched start is about 60 cm. By how much does the gravitational potential energy increase for a 72 kg person in such a jump? Where does this energy come from?

32. A 72.0-kg swimmer jumps into the old swimming hole from a diving board 3.25 m above the water. Use energy conservation to find his speed just he hits the water (a) if he just holds his nose and drops in, (b) if he bravely jumps straight up (but just beyond the board!) at 2.50 m/s and (c) if he manages to jump downward at 2.50 m/s.

33. A 2.50-kg mass is pushed against a horizontal spring of force constant on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

34. A force of magnitude 800.0 N stretches a certain spring by 0.200 m from its equilibrium position. (a) What is the force constant of this spring? (b) How much elastic potential energy

is stored in the spring when it is: (i) stretched 0.300 m from its equilibrium position and (ii) compressed by 0.300 m from its equilibrium position? (c) How much work was done in stretching the spring by the original 0.200 m?

35. Tendons are strong elastic fibers that attach muscles to bones. To a reasonable approximation, they obey Hooke’s law. In laboratory tests on a particular tendon, it was found that, when a 250 g object was hung from it, the tendon stretched 1.23 cm. (a) Find the force constant of this tendon in N/m (b) Because of its thickness, the maximum tension this tendon can support without rupturing is 138 N. By how much can the tendon stretch without rupturing, and how much energy is stored in it at that point?

36. A certain spring stores 10.0 J of potential energy when it is stretched by 2.00 cm from its equilibrium position. (a) How much potential energy would the spring store if it were stretched an additional 2.00 cm? (b) How much potential energy would it store if it were compressed by 2.00 cm from its equilibrium position? (c) How far from the equilibrium position would you have to stretch the string to store 20.0 J of potential energy? (d) What is the force constant of this spring?

37. In designing a machine part, you need a spring that is 8.50 cm long when no forces act on it and that will store 15.0 J of energy when it is compressed by 1.20 cm from its equilibrium position. (a) What should be the force constant of this spring? (b) Can the spring store 850 J by compression?

38. As part of your daily workout, you lie on your back and push with your feet against a platform attached to two stiff springs arranged side by side so that they are parallel to each other. When you push the platform, you compress the springs. You do 80.0 J of work when you compress the springs 0.200 m from their uncompressed length. (a) What magnitude of force must you apply to hold the platform in this position? (b) How much additional work must you do to move the platform 0.200 m farther, and what maximum force must you apply?

39. Food calories. The food calorie, equal to 4186 J, is a measure of how much energy is released when food is metabolized by the body. A certain brand of fruit-and-cereal bar contains 140 food calories per bar. (a) If a 65 kg hiker eats one of these bars, how high a mountain must he climb to “work off” the calories, assuming that all the food energy goes only into increasing gravitational potential energy? (b) If, as is typical, only 20% of the food calories go into mechanical energy, what would be the answer to part (a)?

40. A good workout. You overindulged on a delicious dessert, so you plan to work off the extra calories at the gym. To accomplish this, you decide to do a series of arm raises holding a 5.0 kg weight in one hand. The distance from your elbow to the weight is 35 cm, and in each arm raise you start with your arm horizontal and pivot it until it is vertical. Assume that the weight of your arm is small enough compared with the weight you are lifting that you can ignore it. As is typical, your

muscles are 20% efficient in converting the food energy they use up into mechanical energy, with the rest going into heat. If your dessert contained 350 food calories, how many arm raises must you do to work off these calories? Is it realistic to do them all in one session?

41. A 75 kg person is put on an exercise program by a physical therapist, the goal being to burn up 500 food calories in each daily session. Recall that human muscles are about 20% efficient in converting the energy they use up into mechanical energy. The exercise program consists of a set of consecutive high jumps, each one 50 cm into the air (which is pretty good for a human) and lasting 2.0 s, on the average. How many jumps should the person do per session, and how much time should be set aside for each session? Do you think that this is a physically reasonable exercise session?

42. Tall Pacific Coast redwood trees (Sequoia sempervirens) can reach heights of about 100 m. If air drag is negligibly small, how fast is a sequoia cone moving when it reaches the ground if it dropped from the top of a 100 m tree?

43. The total height of Yosemite Falls is 2425 ft. (a) How many more joules of gravitational potential energy are there for each kilogram of water at the top of this waterfall compared with each kilogram of water at the foot of the falls? (b) Find the kinetic energy and speed of each kilogram of water as it reaches the base of the waterfall, assuming that there are no losses due to friction with the air or rocks and that the mass of water had negligible vertical speed at the top. How fast (in m/s and mph) would a 70 kg person have to run to have that much kinetic energy? (c) How high would Yosemite Falls have to be so that each kilogram of water at the base had twice the kinetic energy you found in part (b); twice the speed you found in part (b)?

44. The speed of hailstones. Although the altitude may vary considerably, hailstones sometimes originate around 500 m (about 1500 ft) above the ground. (a) Neglecting air drag, how fast will these hailstones be moving when they reach the ground, assuming that they started from rest? Express your answer in m/s and in mph. (b) From your own experience, are hailstones actually falling that fast when they reach the ground? Why not? What has happened to most of the initial potential energy?

45. Pebbles of weight w are launched from the edge of a vertical cliff of height h at speed vo. How fast (in terms of the quantities just given) will these pebbles be moving when they reach the ground if they are launched (a) straight up, (b) straight down, (c) horizontally away from the cliff, and (d) at an angle θ above the horizontal? (e) How would the answers to the previous parts change if the pebbles weighed twice as much?

46. Volcanoes on Io. Io, a satellite of Jupiter, is the most volcanically active moon or planet in the solar system. It has volcanoes that send plumes of matter over 500 km high (see the accompanying figure). Due to the satellite’s small mass, the acceleration due to gravity on Io is only 1.81 m/s2 and Io has no appreciable atmosphere. Assume that there is no variation in gravity over the distance traveled. (a) What must be the speed of material just as it leaves the volcano to reach an altitude of 500 km? (b) If the gravitational potential energy is zero at the surface, what is the potential energy for a 25 kg fragment at its maximum height on Io? How much would this gravitational potential energy be if it were at the same height above earth?

47. Human energy vs. insect energy. For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.0-mm-long, 0.50 mg critter can reach a height of 20 cm in a single leap. (a) Neglecting air drag, what is the takeoff speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a 65 kg, 2.0-m-tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could he jump, and what takeoff speed would he need? (d) In fact, most humans can jump no more than 60 cm from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65 kg person? (e) Where does the flea store the energy that allows it to make such a sudden leap?

48. A 25 kg child plays on a swing having support ropes that are 2.20 m long. A friend pulls her back until the ropes are 42° from the vertical and releases her from rest. (a) What is the potential energy for the child just as she is released, compared with the potential energy at the bottom of the swing? (b) How fast will she be moving at the bottom of the swing? (c) How much work does the tension in the ropes do as the child swings from the initial position to the bottom?

49. Tarzan and Jane. Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 m that makes an angle of 45° with the vertical, steps off his tree limb, and swings down and then up to Jane’s open arms. When he arrives, his vine makes an angle of 30° with the vertical. Determine whether he gives her a tender embrace or knocks her off her limb by calculating Tarzan’s speed just before he reaches Jane. You can ignore air resistance and the mass of the vine.

50. A slingshot obeying Hooke’s law is used to launch pebbles vertically into the air. You observe that if you pull a pebble back 20.0 cm against the elastic band, the pebble goes 6.0 m high. (a) Assuming that air drag is negligible, how high will the pebble go if you pull it back 40.0 cm instead? (b) How far must you pull it back so it will reach 12.0 m? (c) If you pull a pebble that is twice as heavy back 20.0 cm, how high will it go?

51. When a piece of wood is pressed against a spring and compresses the spring by 5.0 cm, the wood gains a maximum kinetic energy K when it is released. How much kinetic energy (in terms of K) would the piece of wood gain if the spring were compressed 10.0 cm instead?

52. A 1.5 kg box moves back and forth on a horizontal frictionless surface between two different springs, as shown in the accompanying figure. The box is initially pressed against the stronger spring, compressing it 4.0 cm, and then is released from rest. (a) By how much will the box compress the weaker spring? (b) What is the maximum speed the box will reach?

53. A 12.0 N package of whole wheat flour is suddenly placed on the pan of a scale such as you find in grocery stores. The pan is supported from below by a vertical spring of force constant 325 N/m. If the pan has negligible weight, find the maximum distance the spring will be compressed if no energy is dissipated by friction.

54. A spring of negligible mass has force constant k = 1600 N/m. (a) How far must the spring be compressed for of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then drop a 1.20-kg book onto it from a height of 0.80 m above the top of the spring. Find the maximum distance the spring will be compressed.

55. A 1.50 kg brick is sliding along on a rough horizontal surface at 13.0 m/s. If the brick stops in 4.80 s, how much mechanical energy is lost, and what happens to this energy?

56. A fun-loving 11.4 kg otter slides up a hill and then back down to the same place. If she starts up at 5.75 m/s and returns at 3.75 m/s, how much mechanical energy did she lose on the hill, and what happened to that energy?

57. A 12.0 g plastic ball is dropped from a height of 2.50 m and is moving at 3.20 m/s just before it hits the floor. How much mechanical energy was lost during the ball’s fall?

58. You and three friends stand at the corners of a square whose sides are 8.0 m long in the middle of the gym floor, as shown in the accompanying figure. You take your physics book and push it from one person to the other. The book has a mass of 1.5 kg, and the coefficient of kinetic friction between the book and the floor is μk = 0.25. (a) The book slides from you to Beth and then from Beth to Carlos, along the lines connecting these people. What is the work done by friction during this displacement? (b) You slide the book from you to Carlos along the diagonal of the square. What is the work done by friction during this displacement? (c) You slide the book to Kim who then slides it back to you. What is the total work done by friction during this motion of the book? (d) Is the friction force on the book conservative or nonconservative? Explain.

59. While a roofer is working on a roof that slants at 36° above the horizontal, he accidentally nudges his 85.0 N toolbox, causing it to start sliding downward, starting from rest. If it starts 4.25 m from the lower edge of the roof, how fast will the toolbox be moving just as it reaches the edge of the roof if the kinetic friction force on it is 22.0 N?

60. A block with mass 0.50 kg is forced against a horizontal spring of negligible mass, compressing the spring a distance of 0.20 m, as shown in the accompanying figure. When released, the block moves on a horizontal tabletop for 1.00 m before coming to rest. The spring constant k is 100 N/m. What is the coefficient of kinetic friction μk between the block and the tabletop?

61. A loaded 375 kg toboggan is traveling on smooth horizontal snow at 4.5 m/s when it suddenly comes to a rough region. The region is 7.0 m long and reduces the toboggan’s speed by 1.5 m/s (a) What average friction force did the rough region exert on the toboggan? (b) By what percent did the rough region reduce the toboggan’s (i) kinetic energy and (ii) speed?

62. A 62.0 kg skier is moving at 6.50 m/s on a frictionless, horizontal snow-covered plateau when she encounters a rough patch 3.50 m long. The coefficient of kinetic friction between this patch and her skis is 0.300. After crossing the rough patch and returning to friction-free snow, she skis down an icy, frictionless hill 2.50 m high. (a) How fast is the skier moving when she gets to the bottom of the hill? (b) How much internal energy was generated in crossing the rough patch?

63. (a) How many joules of energy does a 100 watt lightbulb use every hour? (b) How fast would a 70 kg person have to run to have that amount of kinetic energy? Is it possible for a person to run that fast? (c) How high a tree would a 70 kg person have to climb to increase his gravitational potential energy relative to the ground by that amount? Are there any trees that tall?

64. The engine of a motorboat delivers 30.0 kW to the propeller while the boat is moving at 15.0 m/s. What would be the tension in the towline if the boat were being towed at the same speed?

65. At 7.35 cents per kilowatt-hour, (a) what does it cost to operate a 10.0 hp motor for 8.00 hr? (b) What does it cost to leave a 75 W light burning 24 hours a day?

66. A tandem (two-person) bicycle team must overcome a force of 165 N to maintain a speed of 9.00 m/s. Find the power required per rider, assuming that each contributes equally.

67. An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 s, and it is driven by a motor that can provide up to 40 hp to the elevator. What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 kg.

68. U.S. power use. The total consumption of electrical energy in the United States is about 1.0 × 1019 joules per year. (a) Express this rate in watts and kilowatts. (b) If the U.S. population is about 310 million people, what is the average rate of electrical energy consumption per person?

69. Solar energy. The sun transfers energy to the earth by radiation at a rate of approximately 1.0 kW per square meter of surface. (a) If this energy could be collected and converted to electrical energy with 25% efficiency, how large an area (in square kilometers) would be required to collect the electrical energy used by the United States? (See the previous problem.) (b) If the solar collectors were arranged in a square array, what would be the length of its sides in kilometers and in miles? Does an array of these dimensions seem technologically feasible?

70. A 20.0 kg rock slides on a rough horizontal surface at 8.00 m/s and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is 0.200. What average thermal power is produced as the rock stops?

71. Horsepower. In the English system of units, power is expressed as horsepower (hp) instead of watts, where 1.00 hp = 746 W. Horsepower is often used for motors and automobiles. (a) What should be the power rating in horsepower of a 100 W lightbulb? (b) How many watts does a 75 hp motor produce? (c) Electrical resistors are rated in watts to indicate how much power they can tolerate without burning up. If a resistor is rated at 2.00 W, what should be its rating in horsepower? (d) Our sun radiates 3.92 × 1026 J of energy each second. What is its horsepower rating? (e) How many kilowatt-hours of energy is produced when you run a 25 hp motor for 90 minutes?

72. Maximum sustainable human power. The maximum sustainable mechanical power a human can produce is about 1/3 hp. How many food calories can a human burn up in an hour by exercising at this rate?

73. A typical flying insect applies an average force equal to twice its weight during each downward stroke while hovering. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 cm during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

74. When its 75-kW (100-hp) engine is generating full power, a small single-engine airplane with mass 700 kg gains altitude at a rate of 2.5 m/s. What fraction of the engine power is being used to make the airplane climb?

75. The power of the human heart. The human heart is a powerful and extremely reliable pump. Each day it takes in and discharges about 7500 L of blood. Assume that the work done by the heart is equal to the work required to lift that amount of blood a height equal to that of the average American female, approximately 1.63 m. The density of blood is 1050 kg/m3 (a) How much work does the heart do in a day? (b) What is the heart’s power output in watts? (c) In fact, the heart puts out more power than you found in part (b). Why? What other forms of energy does it give the blood?

76. At the site of a wind farm in North Dakota, the average wind speed is 9.3 m/s and the average density of air is (a) Calculate how much kinetic energy the wind 1.2 kg/m3 contains, per cubic meter, at this location. (b) No wind turbine can capture all of the energy contained in the wind, the main reason being that capturing all the energy would require stopping the wind completely, meaning that air would stop flowing through the turbine. Suppose a particular turbine has blades with a radius of 41 m and is able to capture 35% of the available wind energy. What would be the power output of this turbine, under average wind conditions?

77. Bumper guards. You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200 kg car moving at 0.65 m/s is to compress the spring no more than 7.0 cm before stopping. (a) What should be the force constant of the spring, and what is the maximum amount of energy that gets stored in it? (b) If the springs that are actually delivered have the proper force constant but can become compressed by only 5.0 cm, what is the maximum speed of the given car for which they will provide adequate protection?

78. Human terminal velocity. By landing properly and on soft ground (and by being lucky!), humans have survived falls from airplanes when, for example, a parachute failed to open, with astonishingly little injury. Without a parachute, a typical human eventually reaches a terminal velocity of about 62 m/s. Suppose the fall is from an airplane 1000 m high. (a) How fast would a person be falling when he reached the ground if there were no air drag? (b) If a 70 kg person reaches the ground traveling at the terminal velocity of 62 m/s, how much mechanical energy was lost during the fall? What happened to that energy?

79. A wooden rod of negligible mass and length 80.0 cm is pivoted about a horizontal axis through its center. A white rat with mass 0.500 kg clings to one end of the stick, and a mouse with mass 0.200 kg clings to the other end. The system is released from rest with the rod horizontal. If the animals can manage to hold on, what are their speeds as the rod swings through a vertical position?

80. Mountain climbing! A 75.0 kg mountain climber is holding his 60.0 kg partner over a cliff when he suddenly steps on frictionless ice at the horizontal top of the cliff, as shown in the accompanying figure. The rope has negligible mass and is held horizontally by the climber. There is no appreciable friction at the icy edge of the cliff. Use energy methods to calculate the speed of the climbers after the lower one has descended 1.50 m starting from rest.

81. More mountain climbing! What would be the speed of the climbers in the previous problem after they had moved 1.50 m if there were friction between the upper climber and the ice with a coefficient of kinetic friction of 0.250?

82. Ski jump ramp. You are designing a ski jump ramp for the next Winter Olympics. You need to calculate the vertical height h from the starting gate to the bottom of the ramp. The skiers push off hard with their ski poles at the start, just above the starting gate, so they typically have a speed of 2.0 m/s as they reach the gate. For safety, the skiers should have a speed of no more than 30.0 m/s when they reach the bottom of the ramp. You determine that for a 85.0-kg skier with good form, friction and air resistance will do total work of magnitude 4000 J on him during his run down the slope. What is the maximum height h for which the maximum safe speed will not be exceeded?

83. Rescue. Your friend (mass 65.0 kg) is standing on the ice in the middle of a frozen pond. There is very little friction between her feet and the ice, so she is unable to walk. Fortunately, a light rope is tied around her waist and you stand on the bank holding the other end. You pull on the rope for 3.00 s and accelerate your friend from rest to a speed of 6.00 m/s while you remain at rest. What is the average power supplied by the force you applied?

84. On an essentially frictionless horizontal ice-skating rink, a skater moving at 3.0 m/s encounters a rough patch that reduces her speed by 45% due to a friction force that is 25% of her weight. Use the work–energy principle to find the length f the rough patch.

85. Pendulum. A small 0.12 kg metal ball is tied to a very light (essentially massless) string 0.80 m long to form a pendulum that is then set swinging by releasing the ball from rest when the string makes a 45o angle with the vertical. Air drag and other forms of friction are negligible. What is the speed of the ball when the string passes through its vertical position, and what is the tension in the string at that instant?

86. A pump is required to lift 750 liters of water per minute from a well 14.0 m deep and eject it with a speed of 18.0 m/s. How much work per minute does the pump do?

87. A 350 kg roller coaster starts from rest at point A and slides down the frictionless loop-the-loop shown in the accompanying figure. (a) How fast is this roller coaster moving at point B? (b) How hard does it press against the track at point B?

88. Automobile air-bag safety. An automobile air bag cushions the force on the driver in a head-on collision by absorbing her energy before she hits the steering wheel. Such a bag can be modeled as an elastic force, similar to that produced by a spring. (a) Use energy conservation to show that the effective force constant k of the air bag is where m is the mass of the driver, v0 is the speed of the car (and driver) at the instant of the accident, and xmax is the maximum distance the bag gets compressed, which, in a severe accident, would be the distance from the driver’s body to the steering wheel. (b) Show that the maximum force the air bag would exert on the driver is (c) Now let’s put in some realistic numbers. Experimental tests have shown that injury occurs when a force density greater than 5.0 × 105 N/m2 acts on human tissue. As the accompanying figure shows, the force of the air bag acts mostly on the upper front half of the driver’s body, over an area of about 2500 cm2. Use this value to calculate the total force on the driver’s body at the threshold of injury. (d) Use your results to calculate the effective force constant k of the air bag and the maximum speed for which the bag will prevent injury to a 65 kg driver if she is 30 cm from the steering wheel at the instant of impact. Express the speed in m/s and mph. (e) How could you design a safer air bag for higher speed collisions? What things could you alter to do this? Would it be safe to make a stiffer air bag by inflating it more? Explain your reasoning.

89. In creating his definition of horsepower, James Watt, the inventor of the steam engine, calculated the power output of a horse operating a mill to grind grain or cut wood. The horse walked in a 24-ft diameter circle, making, according to Watt, 144 trips around the circle in an hour. (a) Using the currently accepted value of 746 watts for 1 horsepower, calculate the force (in pounds) with which Mr. Watt’s horse must have been pulling. (b) Calculate the power output, in hp, of a 70-kg human being who climbs a 3.0-m-high set of stairs in 5.0 seconds.

90. All birds, independent of their size, must maintain a power output of 10–25 watts per kilogram of body mass in order to fly by flapping their wings. (a) The Andean giant hummingbird (Patagona gigas) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat. (b) A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the steady power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings? Explain.

91. A 250 g object on a frictionless, horizontal lab table is pushed against a spring of force constant 35 N/cm and then released. Just before the object is released, the spring is compressed 12.0 cm. How fast is the object moving when it has gained half of the spring’s original stored energy?

92. Automobile accident analysis. In an auto accident, a car hit a pedestrian and the driver then slammed on the brakes to stop the car. During the subsequent trial, the driver’s lawyer claimed that the driver was obeying the posted 35 mph speed limit, but that the limit was too high to enable him to see and react to the pedestrian in time. You have been called as the state’s expert witness. In your investigation of the accident site, you make the following measurements: The skid marks made while the brakes were applied were 280 ft long, and the tread on the tires produced a coefficient of kinetic friction of 0.30 with the road. (a) In your testimony in court, will you say that the driver was obeying the posted speed limit? You must be able to back up your answer with clear numerical reasoning during cross-examination. (b) If the driver’s speeding ticket is $10 for each mile per hour he was driving above the posted speed limit, would he have to pay a ticket, and if so, how much would it be?

93. Bungee jump. A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 95.0 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 N. When you do this, what distance will the bungee cord that you should select have stretched?

94. Riding a loop-the-loop. A car in an amusement park ride travels without friction along the track shown in the accompanying figure, starting from rest at point A. If the loop the car is currently on has a radius of 20.0 m, find the minimum height h so that the car will not fall off the track at the top of the circular part of the loop.

95. A 2.0 kg piece of wood slides on the surface shown in the accompanying figure. All parts of the surface are frictionless, except for a 30-m-long rough segment at the bottom, where the coefficient of kinetic friction with the wood is 0.20. The wood starts from rest 4.0 m above the bottom. (a) Where will the wood eventually come to rest? (b) How much work is done by friction by the time the wood stops?

96. A 68 kg skier approaches the foot of a hill with a speed of 15 m/s. The surface of this hill slopes up at 40.0° above the horizontal and has coefficients of static and kinetic friction of 0.75 and 0.25, respectively, with the skis. (a) Use energy conservation to find the maximum height above the foot of the hill that the skier will reach. (b) Will the skier remain at rest once she stops, or will she begin to slide down the hill? Prove your answer.

97. Energy requirements of the body. A 70 kg human uses energy at the rate of 80 J/s on average, for just resting and sleeping. When the person is engaged in more strenuous activities, the rate can be much higher. (a) If the individual did nothing but rest, how many food calories per day would she or he have to eat to make up for those used up? (b) In what forms is energy used when a person is resting or sleeping? In other words, what happens to those 80 J/s. (c) If an average person rested and did other low-level activity for 16 hours (which consumes 80 J/s and did light activity on the job for 8 hours (which consumes 200 J/s) how many calories would she or he have to consume per day to make up for the energy used up?

98. The aircraft carrier USS George Washington has mass 1.0 × 108 kg. When its engines are developing their full power of 260,000 hp, the George Washington travels at its top speed of 35 knots. If 70% of the power output of the engines is applied to pushing the ship through the water, what is the magnitude of the force of water resistance that opposes the carrier’s motion at this speed?

99. Two paint buckets are connected by a lightweight rope passing over a pulley of negligible mass and friction. (a) As shown in the accompanying figure, the system is released from rest with the 12.0 kg bucket 2.00 m above the floor. Use energy conservation to find the speed with which this bucket strikes the floor. (b) Suppose the pulley had appreciable mass but no bearing friction. Would the bucket’s speed be greater than, less than, or the same as you found in part (a)? Explain your reasoning in terms of energy.

100. A ball is thrown upward with an initial velocity of 15 m/s at an angle of 60.0o above the horizontal. Use energy conservation to find the ball’s greatest height above the ground.

101. Automotive power. A truck engine transmits 28.0 kW (37.5 hp) to the driving wheels when the truck is traveling at a constant velocity of magnitude 60.0 km/h (37.7 mi/h) on a level road. (a) What is the resisting force acting on the truck? (b) Assume that 65% of the resisting force is due to rolling friction and the remainder is due to air resistance. If the force of rolling friction is independent of speed, and the force of air resistance is proportional to the square of the speed, what power will drive the truck at 30.0 km/h? At 120.0 km/h? Give your answers in kilowatts and in horsepower.

102. Mass extinctions. One of the greatest mass extinctions occurred about 65 million years ago, when, along with many other life-forms, the dinosaurs went extinct. Most geologists and paleontologists agree that this event was caused when a large asteroid hit the earth. Scientists estimate that this asteroid was about 10 km in diameter and that it would have been traveling at least as fast as 11 km/s. The density of asteroid material is about 3.5 g/cm3, on the average. (a) What would be the approximate mass of the asteroid, assuming it to be spherical? (b) How much kinetic energy would the asteroid have delivered to the earth? (c) In order to put the amount of energy you found in part (b) in perspective, consider the following: the total amount of energy used in one year by the human race is roughly 500 exajoules. If this rate of energy use remained constant, how many years would it take the human species to use an amount of energy equal to the amount delivered by this asteroid?

103. Avoiding mass extinctions. It has been suggested that we can protect the earth from devastating asteroidal impacts such as the one discussed in the previous problem by using nuclear devices to alter the orbits of such asteroids around the sun so that they will miss our planet. If this is done very far from earth, it is necessary to move them only a few centimeters to spare the earth a mass extinction. How much energy would it take to move the asteroid that has been implicated in the dinosaur extinction by a few centimeters? To make the calculation reasonable, assume that we need to exert a force on the asteroid that will accelerate it uniformly from rest through a distance of 5.0 cm in 0.50 s. The energy we must give to the asteroid is the added kinetic energy from this motion. To see if it is feasible to do this, how many 1.0 megaton bombs would it take to accomplish the task?

104. The spring of a spring gun has force constant k = 400 N/m and negligible mass. The spring is compressed 6.00 cm, and a ball with mass 0.0300 kg is placed in the horizontal barrel against the compressed spring. The spring is then released, and the ball is propelled out the barrel of the gun. The barrel is 6.00 cm long, so the ball leaves the barrel at the same point that it loses contact with the spring. The gun is held so the barrel is horizontal. (a) Calculate the speed with which the ball leaves the barrel if you can ignore friction. (b) Calculate the speed of the ball as it leaves the barrel if a constant resisting force of 6.00 N acts on the ball as it moves along the barrel. (c) For the situation in part (b), at what position along the barrel does the ball have the greatest speed, and what is that speed?

105. A graph of the potential energy stored in a tendon as a function of the square of the distance x it has stretched from its equilibrium position is shown in the accompanying figure. (a) Does the tendon obey Hooke’s law? Explain your reasoning. (b) What is the force constant of the tendon? (c) How far must the tendon be stretched from its equilibrium position to store 10.0 J of energy? (d) What force is necessary to hold the tendon in place in part (c)? (e) Sketch a clear graph of the potential energy stored in the tendon as a function of x.

106. A sled with rider having a combined mass of 125 kg travels over the perfectly smooth icy hill shown in the accompanying figure. How far does the sled land from the foot of the cliff?

**Chapter 8**

Multiple-Choice Problems

1. A small car collides head-on with a large SUV. Which of the following statements concerning this collision are correct?

A. Both vehicles are acted upon by the same magnitude of average force during the collision.

B. The small car is acted upon by a greater magnitude of average force than the SUV.

C. The small car undergoes a greater change in momentum than the SUV.

D. Both vehicles undergo the same change in magnitude of momentum.

2. A ball of mass 0.18 kg moving with speed 11.3 m/s collides head-on with an identical stationary ball. (Notice that we do not know the type of collision.) Which of the following quantities can be calculated from this information alone?

A. The force each ball exerts on the other.

B. The velocity of each ball after the collision.

C. Total kinetic energy of both balls after the collision.

D. Total momentum of both balls after the collision.

3. A proton with a speed of 50000 km/s makes an elastic head on collision with a stationary carbon nucleus. We can look up the masses of both particles. Which of the following quantities can be calculated from only the known information? (There may be more than one correct choice.)

A. The velocity of the proton and carbon nucleus after the collision.

B. The kinetic energy of each of the particles after the collision.

C. The momentum of each of the particles after the collision.

4. In which of the following collisions would you expect the kinetic energy to be conserved? (There may be more than one correct choice.)

A. A bullet passes through a block of wood.

B. Two bull elk charge each other and lock horns.

C. Two asteroids collide by a glancing blow, but do not actually hit each other, and their only interaction is through gravity.

D. Two cars with springlike bumpers collide at fairly low speeds.

5. A rifle of mass M is initially at rest, but is free to recoil. It fires bullet of mass m with a velocity +v relative to the ground. After the rifle is fired, its velocity relative to the ground is

A. –√(m/M)v. B. –mv/(m + M) C. –mv/M D. –v

6. Two carts, one twice as heavy as the other, are at rest on a horizontal frictionless track. A person pushes each cart with the same force for 5 s. If the kinetic energy of the lighter cart after the push is K, the kinetic energy of the heavier cart is

A. (1/4)K B. (1/2)K C. K. D. 2K E. 4K.

7. A 70-kg wide receiver running west at 8 m/s collides head-on with, and is seized by, a 140-kg lineman lumbering eastward at 4 m/s. In this collision, the greater change in kinetic energy is
experienced by

A. the wide receiver.

B. the lineman

C. neither; they experience the same change in kinetic energy.

8. Two masses, M and 5M, are at rest on a horizontal frictionless table with a compressed spring of negligible mass between them. When the spring is released (there may be more than one correct choice)

A. the two masses receive equal magnitudes of momentum.

B. the two masses receive equal amounts of kinetic energy from the spring.

C. the heavier mass gains more kinetic energy than the lighter mass.

D. the lighter mass gains more kinetic energy than the heavier mass.

9. Cart A, of mass 1 kg, approaches and collides with cart B, which has a mass of 4 kg and is initially at rest. (See Figure 8.32.) When the springs have reached their maximum compression,

A. cart A has come to rest relative to the ground.

B. both carts have the same velocity.

C. both carts have the same momentum.

D. all the initial kinetic energy of cart A has been converted to elastic potential energy.

10. A glider airplane is coasting horizontally when a very heavy object suddenly falls out of it. As a result of dropping this object, the glider’s speed will

A. increase.

B. decrease.

C. remain the same as it was.

11. Which of the following statements is true for an inelastic collision? (There may be more than one correct choice.)

A. Both momentum and kinetic energy are conserved.

B. Momentum is conserved, but kinetic energy is not conserved.

C. Kinetic energy is conserved, but momentum is not conserved.

D. The amount of momentum lost by one object is the same as the amount gained by the other object.

E. The amount of kinetic energy lost by one object is the same as the amount gained by the other object.

12. Which of the following statements is true for an elastic collision? (There may be more than one correct choice.)

A. Both momentum and kinetic energy are conserved.

B. Momentum is conserved, but kinetic energy is not conserved.

C. Kinetic energy is conserved, but momentum is not conserved.

D. The amount of momentum lost by one object is the same as the amount gained by the other object.

E. The amount of kinetic energy lost by one object is the same as the amount gained by the other object.

13. Two lumps of clay having equal masses and speeds, but traveling in opposite directions on a frictionless horizontal surface, collide and stick together. Which of the following statements about this system of lumps must be true? (There may be more than one correct choice.)

A. The momentum of the system is conserved during the collision.

B. The kinetic energy of the system is conserved during the collision.

C. The two masses lose all their kinetic energy during the collision.

D. The velocity of the center of mass of the system is the same after the collision as it was before the collision.

14. A heavy rifle initially at rest fires a light bullet. Which of the following statements about these objects is true? (There may be more than one correct choice.)

A. The bullet and rifle both gain the same magnitude of momentum.

B. The bullet and rifle are both acted upon by the same average force during the firing.

C. The bullet and rifle both have the same acceleration during the firing.

D. The bullet and the rifle gain the same amount of kinetic energy.

15. You drop an egg from rest with no air resistance. As it falls,

A. only its momentum is conserved.

B. only its kinetic energy is conserved.

C. both its momentum and its mechanical energy are conserved.

D. its mechanical energy is conserved, but its momentum is

Problems

1. For each case in Figure 8.33, the system consists of the masses shown with the indicated velocities. Find the net momentum of each system.

2. For each case in Figure 8.34, the system consists of the masses shown with the indicated velocities. Find the x and y components of the net momentum of each system.

3. Three objects A, B, and C are moving as shown in Figure 8.35. Find the x and y components of the net momentum of the particles if we define the system to consist of (a) A and C, (b) B and C, (c) all three objects.

4. A 1200 kg car is moving on the freeway at 65 mph. (a) Find the magnitude of its momentum and its kinetic energy in SI units. (b) If a 2400 kg SUV has the same speed as the 1200 kg

5. The speed of the fastest-pitched baseball was 45 m/s and the ball’s mass was 145 g. (a) What was the magnitude of the momentum of this ball, and how many joules of kinetic energy did it have? (b) How fast would a 57 gram ball have to travel to have the same amount of (i) kinetic energy, and (ii) momentum?

6. Some useful relationships. The following relationships between the momentum and kinetic energy of an object can be very useful for calculations: If an object of mass m has momentum of magnitude p and kinetic energy K, show that (a) K = (p2/2m) and (b) p = √(2mK) (c) Find the momentum of a 1.15 kg ball that has 15.0 J of kinetic energy. (d) Find the kinetic energy of a 3.50 kg cat that has 0.220 kg m/s of momentum.

7. The magnitude of the momentum of a cat is p. What would be the magnitude of the momentum (in terms of p) of a dog having three times the mass of the cat if it had (a) the same speed as the cat, and (b) the same kinetic energy as the cat?

8. Two figure skaters, one weighing 625 N and the other 725 N, push off against each other on frictionless ice. (a) If the heavier skater travels at 1.50 m/s how fast will the lighter one travel? (b) How much kinetic energy is “created” during the skaters’ maneuver, and where does this energy come from?

9. Recoil speed of the earth. In principle, any time someone jumps up, the earth moves in the opposite direction. To see why we are unaware of this motion, calculate the recoil speed of the earth when a 75 kg person jumps upward at a speed of 2.0 m/s.

10. On a frictionless air track, a 0.150 kg glider moving at 1.20 m/s to the right collides with and sticks to a stationary 0.250 kg glider. (a) What is the net momentum of this two-glider system before the collision? (b) What must be the net momentum of this system after the collision? Why? (c) Use

your answers in parts (a) and (b) to find the speed of the gliders after the collision. (d) Is kinetic energy conserved during the collision?

11. Baseball. A regulation 145 g baseball can be hit at speeds of 100 mph. If a line drive is hit essentially horizontally at this speed and is caught by a 65 kg player who has leapt directly upward into the air, what horizontal speed (in cm/s) does he acquire by catching the ball?

12. You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a 0.400 kg ball that is traveling horizontally at 10.0 m/s. Your mass is 70.0 kg. (a) If you catch the ball, with what speed do you and

the ball move afterwards? (b) If the ball hits you and bounces off your chest, so that afterwards it is moving horizontally at 8.00 m/s in the opposite direction, what is your speed after the collision?

13. On a frictionless, horizontal air table, puck A (with mass 0.250 kg) is moving to the right toward puck B (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck B has a velocity of 0.650 m/s to the right. (a) What was the speed of puck A before the collision? (b) Calculate the change in the total kinetic energy of the system that occurs during the collision.

14. Block A in Figure 8.36 has mass 1.00 kg, and block B has mass 3.00 kg. The blocks are forced together, compressing a spring S between them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block B acquires a speed of 1.20 m/s (a) What is the final speed of block A? (b) How much potential energy was stored in the compressed spring?

15. Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1° from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniel’s velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

16. You (mass 55 kg) are riding your frictionless skateboard (mass 5.0 kg) in a straight line at a speed of 4.5 m/s when a friend standing on a balcony above you drops a 2.5 kg sack of flour straight down into your arms. (a) What is your new speed, while holding the flour sack? (b) Since the sack was dropped vertically, how can it affect your horizontal motion? Explain. (c) Suppose you now try to rid yourself of the extra weight by throwing the flour sack straight up. What will be your speed while the sack is in the air? Explain.

17. A 4.25 g bullet traveling horizontally with a velocity of magnitude 375 m/s is fired into a wooden block with mass 1.12 kg, initially at rest on a level frictionless surface. The bullet passes through the block and emerges with its speed reduced to 122 m/s. How fast is the block moving just after the bullet emerges from it?

18. A ball with a mass of 0.600 kg is initially at rest. It is struck by a second ball having a mass of 0.400 kg, initially moving with a velocity of 0.250 m/s toward the right along the x axis. After the collision, the 0.400 kg ball has a velocity of 0.200 m/s at an angle of 36.9° above the x axis in the first quadrant. Both balls move on a frictionless, horizontal surface. (a) What are the magnitude and direction of the velocity of the 0.600 kg ball after the collision? (b) What is the change in the total kinetic energy of the two balls as a result of the collision?

19. Combining conservation laws. A 5.00 kg chunk of ice is sliding at 12.0 m/s on the floor of an ice-covered valley when it collides with and sticks to another 5.00 kg chunk of ice that is initially at rest. (See Figure 8.37.) Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

20. Combining conservation laws. A 15.0 kg block is attached to a very light horizontal spring of force constant 500.0 N/m and is resting on a frictionless horizontal table. (See Figure 8.38.) Suddenly it is struck by a 3.00 kg stone traveling horizontally at 8.00 m/s to the right, whereupon the stone rebounds at 2.00 m/s horizontally to the left. Find the maximum distance that the block will compress the spring after the collision.

21. Three identical boxcars are coupled together and are moving at a constant speed of 20.0 m/s on a level track. They collide with another identical boxcar that is initially at rest and couple to it, so that the four cars roll on as a unit. Friction is small enough to be neglected. (a) What is the speed of the four cars? (b) What percentage of the kinetic energy of the boxcars is dissipated in the collision? What happened to this energy?

22. On a highly polished, essentially frictionless lunch counter, a 0.500 kg submarine sandwich moving 3.00 m/s to the left collides with a 0.250 kg grilled cheese sandwich moving 1.20 m/s to the right. (a) If the two sandwiches stick together, what is their final velocity? (b) How much mechanical energy dissipates in the collision? Where did this energy go?

23. An astronaut in space cannot use a scale or balance to weigh objects because there is no gravity. But she does have devices to measure distance and time accurately. She knows her own mass is 78.4 kg, but she is unsure of the mass of a large gas canister in the airless rocket. When this canister is approaching her at 3.50 m/s she pushes against it, which slows it down to 1.20 m/s (but does not reverse it) and gives her a speed of 2.40 m/s (a) What is the mass of this canister? (b) How much kinetic energy is “lost” in this collision, and what happens to that energy?

24. On a very muddy football field, a 110-kg linebacker tackles an 85-kg halfback. Immediately before the collision, the linebacker is slipping with a velocity of 8.8 m/s north and the halfback is sliding with a velocity of 7.2 m/s east. What is the velocity (magnitude and direction) at which the two players move together immediately after the collision?

25. A 5.00 g bullet is fired horizontally into a 1.20 kg wooden block resting on a horizontal surface. The coefficient of kinetic friction between block and surface is 0.20. The bullet remains embedded in the block, which is observed to slide 0.230 m along the surface before stopping. What was the initial speed of the bullet?

26. You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass 80.0 kg, is given a push and slides eastward. Abigail, with mass 50.0 kg, is sent sliding northward. They collide, and after the collision Sam is moving at 37.0° north of east with a speed of 6.00 m/s and Abigail is moving at 23.0° south of east with a speed of 9.00 m/s (a) What was the speed of each person before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

27. A hungry 11.5 kg predator fish is coasting from west to east at 75.0 m/s when it suddenly swallows a 1.25 kg fish swimming from north to south at 3.60 m/s. Find the magnitude and direction of the velocity of the large fish just after it snapped up this meal. Neglect any effects due to the drag of the water.

28. Bird defense. To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600 gram falcon flying at 20.0 m/s ran into a 1.5 kg raven flying at 9.0 m/s. The falcon hit the raven at right angles to its original path and bounced back with a speed of 5.0 m/s. By what angle did the falcon change the raven’s direction of motion?

29. Accident analysis. Two cars collide at an intersection. Car A, with a mass of 2000 kg, is going from west to east, while car B, of mass 1500 kg, is going from north to south at 15 m/s. As a result of this collision, the two cars become enmeshed and move as one afterwards. In your role as an expert witness, you inspect the scene and determine that, after the collision, the enmeshed cars moved at an angle of 65° south of east from the point of impact. (a) How fast were the enmeshed cars moving just after the collision? (b) How fast was car A going just before the collision?

30. A hockey puck B rests on frictionless, level ice and is struck by a second puck A, which was originally traveling at 40.0 m/s and which is deflected 30.0° from its original direction. Puck B acquires a velocity at a 45.0° angle to the original direction of A. The pucks have the same mass. (a) Compute the speed of each puck after the collision. (b) What fraction of the original kinetic energy of puck A dissipates during the collision?

31. A 0.300 kg glider is moving to the right on a frictionless, horizontal air track with a speed of 0.80 m/s when it makes a head-on collision with a stationary 0.150 kg glider. (a) Find the magnitude and direction of the final velocity of each glider if the collision is elastic. (b) Find the final kinetic energy of each glider.

32. On a cold winter day, a penny (mass 2.50 g) and a nickel (mass 5.00 g) are lying on the smooth (frictionless) surface of a frozen lake. With your finger, you flick the penny toward the nickel with a speed of 2.20 m/s. The coins collide elastically; calculate both their final velocities (speed and direction).

33. Nuclear collisions. Collisions between atomic and subatomic particles are often perfectly elastic. In one such collision, a proton traveling to the right at 258 km/s collides head-on and elastically with a stationary alpha particle (a helium nucleus, having mass 6.65 × 10–27 kg). Consult Appendix E as needed. (a) Find the magnitude and direction of the velocity of each particle after the collision. (b) How much kinetic energy does the proton lose during the collision? (c) How can the collision be elastic if the proton loses kinetic energy?

34. On an air track, a 400.0 g glider moving to the right at 2.00 m/s collides elastically with a 500.0 g glider moving in the opposite direction at 3.00 m/s. Find the velocity of each glider after the collision.

35. Blocks A (mass 2.00 kg) and B (mass 10.00 kg) move on a frictionless, horizontal surface. Initially, block B is at rest and block A is moving toward it at 2.00 m/s. The blocks are equipped with ideal spring bumpers, as in Example 8.9. The collision is head-on, so all motion before and after the collision is along a straight line. (a) Find the maximum energy stored in the spring bumpers and the velocity of each block at that time. (b) Find the velocity of each block after they have moved apart.

36. Two identical objects traveling in opposite directions with the same speed V make a head-on collision. Find the speed of each object after the collision if (a) they stick together and (b) if the collision is perfectly elastic.

37. A catcher catches a 145 g baseball traveling horizontally at 36.0 m/s (a) How large an impulse does the ball give to the catcher? (b) If the ball takes 20 ms to stop once it is in contact with the catcher’s glove, what average force did the ball exert on the catcher?

38. A block of ice with a mass of 2.50 kg is moving on a frictionless, horizontal surface. At t = 0, the block is moving to the right with a velocity of magnitude 8.00 m/s. Calculate the magnitude and direction of the velocity of the block after each of the following forces has been applied for 5.00 s: (a) a force of 5.00 N directed to the right; (b) a force of 7.00 N directed to the left.

39. Biomechanics. The mass of a regulation tennis ball is 57 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0 ms contact time throughout this problem. The fastest-known served tennis ball was served by “Big Bill” Tilden in 1931, and its speed was measured to be 73.14 m/s (a) What impulse and what force did Big Bill exert on the tennis ball in his record serve? (b) If Big Bill’s opponent returned his serve with a speed of 55 m/s what force and what impulse did he exert on the ball, assuming only horizontal motion?

40. To warm up for a match, a tennis player hits the 57.0 g ball vertically with her racket. If the ball is stationary just before it is hit and goes 5.50 m high, what impulse did she impart to it?

41. A 150 gram baseball is hit toward the left by a bat. The magnitude of the force the bat exerts on the ball as a function of time is shown in Figure 8.40. (a) Find the magnitude and direction of the impulse that the bat imparts to the ball. (b) Find the magnitude and direction of the ball’s velocity just after it is hit by the bat if the ball is initially (i) at rest, and (ii) moving to the right at 30.0 m/s.

42. Your little sister (mass 25.0 kg) is sitting in her little red wagon (mass 8.50 kg) at rest. You begin pulling her forward and continue accelerating her with a constant force for 2.35 s, at the end of which time she’s moving at a speed of 1.80 m/s (a) Calculate the impulse you imparted to the wagon and its

passenger. (b) With what force did you pull on the wagon?

43. Bone fracture. Experimental tests have shown that bone will rupture if it is subjected to a force density of 1.0 × 108 N/m2. Suppose a 70.0 kg person carelessly roller-skates into an overhead metal beam that hits his forehead and completely stops his forward motion. If the area of contact with the person’s forehead is 1.5 cm2 what is the greatest speed with which he can hit the wall without breaking any bone if his head is in contact with the beam for 10.0 ms?

44. A bat strikes a 0.145 kg baseball. Just before impact, the ball is traveling horizontally to the right at 50.0 m/s and it leaves the bat traveling to the left at an angle of 30° above horizontal with a speed of 65.0 m/s (a) What are the horizontal and vertical components of the impulse the bat imparts to the ball? (b) If the ball and bat are in contact for 1.75 ms, find the horizontal and vertical components of the average force on the ball.

45. Calculate the location of the center of mass of the earth-moon system (that is, find the distance of the center of mass from the earth’s center). Use data from Appendix E and assume the orbital radius of the moon is equal to the distance between the centers of the earth and the moon. What can you say about the position of the center of mass with respect to the earth’s surface?

46. Two small-sized objects are placed on a uniform 9.00 kg plastic beam 3.00 m long, as shown in Figure 8.41. Find the location of the center of mass of this system by setting x = 0 at (a) the left end of the beam and (b) the right end of the beam. (c) Do you get the same result both ways?

47. Detecting planets around other stars. Roughly 500 planets have so far been detected beyond our solar system. This is accomplished by looking for the effect the planet has on the star. The star is not truly stationary; instead, it and its planets orbit around the center of mass of the system. Astronomers can measure this wobble in the position of a star. (a) For a star with the mass and size of our sun and having a planet with five times the mass of Jupiter, where would the center of mass of this system be located, relative to the center of the star, if the distance from the star to the planet was the same as the distance from Jupiter to our sun? (b) If the planet had earth’s mass, where would the center of mass of the system be located if the planet was just as far from the star as the earth is from the sun? (c) In view of your results in parts (a) and (b), why is it much easier to detect stars having large planets rather than small ones?

48. Three odd-shaped blocks of chocolate have the following masses and center-of-mass coordinates: (1) 0.300 kg, (0.200 m, 0.300 m) (2) 0.400 kg, (0.100 m, –0.400 m) (3) 0.200 kg, (–0.300 m, 0.600 m). Find the coordinates of the center of mass of the system of three chocolate blocks.

49. A machine part consists of a thin, uniform 4.00-kg bar that is 1.50 m long, hinged perpendicular to a similar vertical bar of mass 3.00 kg and length 1.80 m. The longer bar has a small but dense 2.00-kg ball at one end (Fig. 8.42). By what distance will the center of mass of this part move horizontally and vertically if the vertical bar is pivoted counterclockwise through 90° to make the entire part horizontal?

50. Changing your center of mass. To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0 cm long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0 kg person, the mass of the upper leg would be 8.60 kg, while that of the lower leg (including the foot) would be 5.25 kg. Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

51. A 1200 kg station wagon is moving along a straight highway at 12.0 m/s. Another car, with mass 1800 kg and speed 20.0 m/s has its center of mass 40.0 m ahead of the center of mass of the station wagon. (a) Find the position of the center of mass of the system consisting of the two automobiles. (b) Find the magnitude of the total momentum of the system from the given data. (c) Find the speed of the center of mass of the system. (d) Find the total momentum of the system, using the speed of the center of mass. Compare your result with that of part (b).

52. Walking in a boat. A 45.0 kg woman stands up in a 60.0 kg canoe of length 5.00 m. She walks from a point 1.00 m from one end to a point 1.00 m from the other end. If the resistance of the water is negligible, how far does the canoe move during this process?

53. A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity of magnitude 1600 m/s relative to the rocket. (a) What is the thrust of the rocket? (b) Would the rocket operate in outer space, where there is no atmosphere? If so, how would you steer it? Could you brake it?

54. A rocket is fired in deep space, where gravity is negligible. If the rocket has an initial mass of 6000 kg and ejects gas at a relative velocity of magnitude 2000 m/s, how much gas must it eject in the first second to have an initial acceleration of 25.0 m/s2.

55. A rocket is fired in deep space, where gravity is negligible. In the first second, it ejects 1/160 of its mass as exhaust gas and has an acceleration of 15.0 m/s2. What is the speed of the exhaust gas relative to the rocket?

56. In outer space, where gravity is negligible, a 75,000 kg rocket (including 50,000 kg of fuel) expels this fuel at a steady rate of 135 kg/s with a speed of 1200 m/s relative to the rocket. (a) Find the thrust of the rocket. (b) What are the initial acceleration and the maximum acceleration of the rocket? (c) After the fuel runs out, what happens to this rocket’s acceleration? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning. (d) After the fuel runs out, what happens to the rocket’s speed? Does it (i) remain the same as it was just as the fuel ran out, (ii) suddenly become zero, or (iii) gradually drop to zero? Explain your reasoning.

57. A 70-kg astronaut floating in space in a 110-kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s2 when he fires one of the MMU’s thrusters. (a) If the speed of the escaping N2 gas relative to the astronaut is 490 m/s how much gas is used by the thruster in 5.0 s? (b) What is the thrust of the thruster?

58. In 1.00 second an automatic paintball gun can fire 15 balls, each with a mass of 0.113 g, at a muzzle velocity of 88.5 m/s. Calculate the average recoil force experienced by the player who’s holding the gun.

59. In a volcanic eruption, a 2400-kg boulder is thrown vertically upward into the air. At its highest point, it suddenly explodes (due to trapped gases) into two fragments, one being three times the mass of the other. The lighter fragment starts out with only horizontal velocity and lands 274 m directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

60. A 5.00 kg ornament is hanging by a 1.50 m wire when it is suddenly hit by a 3.00 kg missile traveling horizontally at 12.0 m/s. The missile embeds itself in the ornament during the collision. What is the tension in the wire immediately after the collision?

61. A stone with a mass of 0.100 kg rests on a frictionless, horizontal surface. A bullet of mass 2.50 g traveling horizontally at 500 m/s strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 300 m/s (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

62. A steel ball with a mass of 40.0 g is dropped from a height of 2.00 m onto a horizontal steel slab. The ball rebounds to a height of 1.60 m. (a) Calculate the impulse delivered to the ball during the impact. (b) If the ball is in contact with the slab for 2.00 ms, find the average force on the ball during the impact.

63. A movie stuntman (mass 80.0 kg) stands on a window ledge 5.0 m above the floor (Fig. 8.45). Grabbing a rope attached to a chandelier, he swings down to grapple with the movie’s villain (mass 70.0 kg), who is standing directly under the chandelier. (Assume that the stuntman’s center of mass moves downward 5.0 m. He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is μk = 0.250 how far do they slide?

64. Tennis, anyone? Tennis players sometimes leap into the air to return a volley. (a) If a 57 g tennis ball is traveling horizontally at 72 m/s (which does occur), and a 61 kg tennis player leaps vertically upward and hits the ball, causing it to travel at 45 m/s in the reverse direction, how fast will her center of mass be moving horizontally just after hitting the ball? (b) If, as is reasonable, her racket is in contact with the ball for 30.0 ms, what force does her racket exert on the ball? What force does the ball exert on the racket?

65. Two identical masses are released from rest in a smooth hemispherical bowl of radius R, from the positions shown in Fig. 8.46. You can ignore friction between the masses and the surface of the bowl. If they stick together when they collide, how high above the bottom of the bowl will the masses go after colliding?

66. Two identical 1.50 kg masses are pressed against opposite ends of a light spring of force constant 1.75 N/cm, compressing the spring by 20.0 cm from its normal length. Find the speed of each mass when it has moved free of the spring on a frictionless horizontal lab table.

67. A rifle bullet with mass 8.00 g strikes and embeds itself in a block with a mass of 0.992 kg that rests on a frictionless, horizontal surface and is attached to a coil spring. (See Figure 8.47.) The impact compresses the spring 15.0 cm. Calibration of the spring shows that a force of 0.750 N is required to compress the spring 0.250 cm. (a) Find the magnitude of the block’s velocity just after impact. (b) What was the initial speed of the bullet?

68. A 5.00 g bullet traveling horizontally at 450 m/s is shot through a 1.00 kg wood block suspended on a string 2.00 m long. If the center of mass of the block rises a distance of 0.450 cm, find the speed of the bullet as it emerges from the block.

69. Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan’s weight is 800 N, Jane’s weight is 600 N, and that of the sleigh is 1000 N. They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity (relative to the ice) of 5.00 m/s at 30.0° above the horizontal, and Jane jumps to the right at 7.00 m/s at 36.9° above the horizontal (relative to the ice). Calculate the sleigh’s horizontal velocity (magnitude and direction) after they jump out.

70. Animal propulsion. Squids and octopuses propel themselves by expelling water. They do this by taking the water into a cavity and then suddenly contracting the cavity, forcing the water to shoot out of an opening. A 6.50 kg squid (including the water in its cavity) that is at rest suddenly sees a dangerous predator. (a) If this squid has 1.75 kg of water in its cavity, at what speed must it expel the water to suddenly achieve a speed of 2.50 m/s to escape the predator? Neglect any drag effects of the surrounding water. (b) How much kinetic energy does the squid create for this escape maneuver?

71. The objects in Figure 8.48 are constructed of uniform wire bent into the shapes shown. Find the position of the center of mass of each one.

72. Origin of the moon. Astronomers believe that our moon originated when a Mars-sized body collided with the earth over 4.5 billion years ago, knocking off matter that condensed to form the moon. To get some idea of how such a collision would affect the earth, assume that the collision occurred head-on (with the two bodies traveling in opposite directions), that the Mars-sized body merged completely with the earth and was originally the same mass as Mars, and that the earth’s

orbital speed was the same as it is now. Assume further that the speed of the colliding body was equal to the earth’s escape velocity of 11 km/s (although it could have been greater). (a) By how much (in m/s) did such a collision change the earth’s speed? (b) How much thermal energy would such a collision create? (c) To how many megaton bombs is the energy in part (b) equivalent? (One ton of TNT releases 4.184 × 109 J of energy.)

73. Changing mass. A railroad hopper car filled with sand is rolling with an initial speed of 15.0 m/s on straight, horizontal tracks. You can ignore frictional forces on the railroad car. The total mass of the car plus sand is 85,000 kg. The hopper door is not fully closed so sand leaks out the bottom. After 20 min, 13,000 kg of sand has leaked out. Then what is the speed of the railroad car?

74. Forensic science. Forensic scientists can measure the muzzle velocity of a gun by ﬁring a bullet horizontally into a large hanging block that absorbs the bullet and swings upward. (See Figure 8.49.) The measured maximum angle of swing can be used to calculate the speed of the bullet. In one such test, a rifle fired a 4.20 g bullet into a 2.50 kg block hanging by a thin wire 75.0 cm long, causing the block to swing upward to a maximum angle of 34.7° from the vertical. What was the original speed of this bullet?

75. A 20.0-kg lead sphere is hanging from a hook by a thin wire 3.50 m long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00-kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

76. A blue puck with mass 0.0400 kg, sliding with a velocity of magnitude 0.200 m/s on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass m, initially at rest. After the collision, the velocity of the blue puck is 0.050 m/s in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision; and (b) the mass m of the red puck.

77. The structure of the atom. During 1910–1911, Sir Ernest Rutherford performed a series of experiments to determine the structure of the atom. He aimed a beam of alpha particles (helium nuclei, of mass 6.65 × 10–27 kg) at an extremely thin sheet of gold foil. Most of the alphas went right through with little deflection, but a small percentage bounced directly back. These results told him that the atom must be mostly empty space with an extremely small nucleus. The alpha particles that bounced back must have made a head-on collision with this nucleus. A typical speed for the alpha particles before the collision was 1.25 × 107 m/s, and the gold atom has a mass of 3.27 × 10–25 kg. Assuming (quite reasonably) elastic collisions, what would be the speed after the collision of a gold atom if an alpha particle makes a direct hit on the nucleus?

78. Rocket failure! Just as it has reached an upward speed of 5.0 m/s during a vertical launch, a rocket explodes into two pieces. Photographs of the explosion reveal that the lower piece, with a mass one-fourth that of the upper piece, was moving downward at 3.0 m/s the instant after the explosion. (a) Find the speed of the upper piece just after the explosion. (b) How high does the upper piece go above the point where the explosion occurred?

79. In a common physics demonstration, two identical carts having rigid metal surfaces and equal speeds collide with each other. Each cart has a piece of Velcro at one end and a spring at the other end. For each collision shown in the figure, find the magnitude and direction of the velocity of each cart after the collision. Are both collisions elastic? What causes one of them to be inelastic, and what happens to the “lost” kinetic energy?

80. A 7.0 kg shell at rest explodes into two fragments, one with a mass of 2.0 kg and the other with a mass of 5.0 kg. If the heavier fragment gains 100 J of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?

81. Radioactive beta decay. Beta decay is a radioactive decay in which a neutron in the nucleus of an atom breaks apart (decays) to form a proton, an electron, and an antineutrino. The electron is also known as a beta particle. The proton remains in the nucleus, while the electron and antineutrino shoot out. Before the existence of the antineutrino was suspected, it was assumed that only the electron was emitted. Assuming (incorrectly) that there is no antineutrino, and that the neutron is initially at rest inside the nucleus, find (a) the ratio of the speed of the electron to the speed of the proton just after the decay and (b) the ratio of the kinetic energy of the electron to that of the proton just after the decay. Look up the necessary masses in Appendix E. (c) Use the results from parts (a) and (b) to explain why it is the electron, and not the proton, that shoots out of the nucleus.

82. A 15.0 g acorn falls from rest from the top of a 35.0-m-high oak tree. When it is halfway to the ground, a 135 g bird gliding horizontally at 75.0 cm/s scoops it up with its beak. Find (a) the horizontal and vertical components of the bird’s velocity, and (b) the speed of the bird and the angle its velocity makes with the vertical, just after the bird scoops up the acorn.

83. Accident analysis. A 1500 kg sedan goes through a wide intersection traveling from north to south when it is hit by a 2200 kg SUV traveling from east to west. The two cars become enmeshed due to the impact and slide as one there after. On-the-scene measurements show that the coefficient of kinetic friction between the tires of these cars and the pavement is 0.75, and the cars slide to a halt at a point 5.39 m west and 6.43 m south of the impact point. How fast was each car traveling just before the collision?

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