c

Summary 281

SUMMARY The goals of Chapter 9 have been to introduce the ideas of impulse. momentum, and angular momentum and to learn a new prob lem-solving strategy based on conservation laws.

GENERAL PRINCIPLES

Law of Conservation of Momentum Solving Momentum Conservation Problems The total mo me ntum P = P I + P2 + ... of an isolated system is a constant. Thus

PREPARE Choose an iso lated syste m or a system that is isolated during at least part of the problem. Draw a visual overview of the system before and after the interact ion.

Conservation of Angular M omentum

SOLVE Write the law of conservati on of mo mentum in te rms of vector compo nents:

(p I.,)f + (P 2>:)r + ... = (p Ix), + (P2x)i + . The angular momentum L of a rotating object subject to zero ex ternal torque does not change. Thus (P ly)f + (P 2y )r + . .. = (P ly), + (P2y); + .

This can be wrilten in terms o f the mome nt o f inert ia and angular ve loc ity as

IMPORTANT CONCEPTS

Momentum p = 11/V /II P

Impulse i t = area under force curve ....

tk lmpu lse and momentum are related by the impulse­momentum theorem

I ;, I

This is an alternati ve state ment of Newton's second law.

Angular momentum L = Iw is the rotatio nal ana log of linear momentum p = mv.

APPLICATIONS

Collisions Two or more particles come together. In a perfectly inelastic colli sio n, they stic k together and move with a common fi nal ve locity.

Explosions Two or more particles move away from each other.

In terms of masses and ve loc ities, thi s is

m I(Vlx)r + m 2(v2~)r +

m I (V I)f + f1I 2(V2y)f +

ASSESS Is the result reasonable?

System A group of in teracting particles .

Isolated system A system o n which the net ex ternal force is zero.

Before-and-after visual overview

Defi ne the system.

Use two draw ings to show the system before and af ter the interactio n.

Li st known informat ion and ident ify what you are try ing to find.

= III I(V tx) i + J11 2(V2<) i +

= m I( VI.J i + 11I 2(V2)i +

Intcma l forces

/ '-1 <~

Two dimensions Both the x- and y-components of the total momentum P must be conserved, giving two s imultaneous equati o ns.

282 CHAPTER 9 Momentum

tMP)TM For homewo~k assig~ed on MasteringPhysics, go to ;.:;./ www.mastenngphyslcs.com

Problem difficulty is labeled as ! (straightforward) to nUl (challenging).

QUESTIONS

Conceptual Questions

I. Rank in order, from largest to smaUes t, the momenta PI. through P 5x of the objects presented in Figure Q9.1. Explain.

20g

87:; 109

20g

U I-- 2-m-I'-"

10 g

[2]1---- ~ DOO g

5 0.1 m/s

2 m/s

FIGURE 09 .1

2. Starting from rest, object 1 is subject to a 12 N force for 2.0 s. Object 2, with twice the mass, is subject to a 15 N force for 3.0 s. Which object has the greater final speed? Explain.

3. A 0.2 kg plast ic cart and a 20 kg lead cart can roll without fric­tion on a hor izontal surface. Equal forces are used to push both carts forward for a time of 1 s, starting from rest. After the force is removed at t = I s, is the momentum of the plast ic cart greater than, less than, or equal to the momentum of the lead cart? Explain.

4. Two pucks, of mass m and 4m, li e on a frictionless table. Equal forces are used to push both pucks forward a distance of I m. a. Which puck takes longer to travel the distance? Explain. b. Which puck has the greater momentum upon completing the

distance? Explain. 5. A stationary firecracker explodes into

three pieces. One piece travels off to the east; a second trave ls to the north. Which of the vectors of Figure Q9.5 could be the veloc ity of the third piece? Explain.

6. Two students stand at rest, facing each other on frict ion less skates. They then start tossing a heavy ball back and forth between them. Describe their subsequent motion.

N

7f.

A.

W E

D.

S C.

FIGURE 0 9 .5

7. Two particles collide, one of which was initiaUy moving and the other initially at rest. a. Is it possible for both particles to be at rest after the colli ­

sion ? Give an example in which this happens, or explain why it can't happen.

b. Is it possible for olle particle to be at rest after the coll ision? Give an example in which thi s happens, or explain why it can' t happen.

8. Automobiles are designed with "crumple zones" intended to collapse in a collision. Why would a manufacturer design part of a car so that it collapses in a colli sion ?

9. You probably know that it feels beller to catch a baseball if you are wearing a padded glove. Explain why thi s is so, using the ideas of momentum and impulse.

Problems labeled rli can be done on a Workbook Momentum

Worksheet; INT integrate significant material from earlier chapters;

BID are of biological or medical interest.

10. In the early days of rocketry, some people claimed that rockets could­n' t fly in outer space as there was no air for the rockets to push against. Suppose you were an early investigator in the field of rocketry and met someone who made this argument. How would you convince the person that rockets could travel in space?

11. Two ice skaters, Megan and Jason, push off from each other on frictionless ice. Jason's mass is twice that of Megan. a. Which skater, if e ither, experiences the greater impulse dur­

ing the push? Explain. b. Which skater, if either, has the greater speed after the push­

off? Explain. 12. Suppose a rubber ball and a steel ball co llide. Which, if e ither,

receives the larger impulse? Explain. 13. While standing st ill on a basketball court, you throw the baU to

a teammate. Why do you not move backward as a result? Is the law of conservation of momentum violated?

14. To win a prize at the county fair, you ' re trying to knock down a heavy bowling pin by hitting it with a thrown object. Should you choose to throw a rubber ball or a beanbag of equal size and weight? Explain.

IS. Rank in order, from largest to smallest, the angular momenta L I through L5 of the balls shown in Figure Q9.15. Explain.

3m/, GI

m/' 02m/', ~kg~kgdkg 4m 2kg 4m _kg

UUU I 2 3

FIGURE 09.15

16. Figure Q9.16 shows two masses held together by a thread on a rod that is rotat­ing about its center with angular velocity w. If the thread breaks, the masses

4 5

will slide out to the ends of FIGURE 09.16

the rod. If that happens, will the rod 's angular velocity increase, decrease, or remain unchanged? Explain.

17. If the earth warms signi ficantly, the polar ice caps will melt. Water wiU move from the poles, near the earth 's rotation axis, and will spread out around the globe. In principle, this wiLl change the length of the day. Why? Will the length of the day increase or decrease?

18. The disks shown in Figure Q9.IS have equal mass . Is the angu­lar momentum o f disk 2, on the ri ght , larger than, smaller than , or equal to the angular momen tum of disk I? Explain.

FIGURE 09.18

Multiple-Choice Questions

19. 1 Curling is a sport played with 20 kg stones that s lide across an ice surface. Suppose a curl ing stone slidi ng at 1 mls strikes another stone and comes 10 rest in 2 ms. Approx imately how

much force is there on the stone during the impact? A. 200 N B. 1000 N C. 2000 N D. 10,000 N

20. 1 Two balls are hung from cords. The first ball , of mass 1.0 kg, is pulled to the side and released, reaching a speed of 2.0 m/s at

2 1.

the bottom o f its arc. Then, as shown in Figure Q9.20, it hits and sticks to anOlher ball. The speed of the pair just after the colli sion is 1.2 m/s. What is

the mass of the second ball? A. 0.67 kg 8. 2.0 kg C. 1.7 kg D. 1.0 kg

FIGURE 09 .20

I Fi gure Q9.2 1 shows two blocks s liding on a frictionless sur­face. Even tuall y the smaller block overtakes the larger one, co l­lides with it, and sticks. What is the speed of the two blocks after thecolli sion? A. vJ2 B. 4vJ5 c. Vj D. 5v,l4

~ ~

FIGURE 09.21

22. I Two frie nds are si tting in a stat ionary canoe. At I = 3.0 s the person at the fro nt lOsses a sack to the person in the rear, who catches the sack 0.2 s later. Which plot in Figure Q9.22 shows

A. B.

't=="" "C==", 3.0 3.2 3.0 3.2

C. D.

+-~"" +-==,., 3.0 3.2 3.0 3.2

FIGURE 09.22

Questions 283

the veloc ity of the boat as a funct ion of time? Pos itive ve loc ity is forward, negative veloc ity is backward. Neglect any drag force on the canoe from the water.

23. II Two blocks, with masses 1111 = 2.5 kg and 111 2 = 14 kg , approach each other along a horizonta l, fri c ti o nless track. The initial veloc ities o f the blocks are VI = 12.0 m/s to the right and V2 = 3.4 m/s to the lefL The two blocks then collide and sti ck toge ther. Which o f the graphs could represent the force of block 1 on block 2 during the coll ision?

30.00:;~(N) A.

20,000 10,000

o t(ms) o 10 20 30 40 50

:~(N) c.

100 o / (ms)

o 10 20 30 40 50

FIGURE 09.23

3~~(N) B.

20

10

o t(ms) o 10 10 30 40 50

300:;~(N) D.

2000

1000 o t (ms)

o 10 20 30 40 50

24. I A small puck is s liding to the ri ght with momen tum Pi on a hori zon tal, fri c ti o nless surface, as shown in Figure Q9.24. A force is applied to the puck for a short time and its momentum afterward is Pf. Whic h lettered arrow shows the directi on of the impulse that

-~--:--

25. was deli vered to the puck?

FIGURE 09.24 1 A red ball , initially at rcst, is s imultane­ous ly hit by a blue ball traveling from weslto east at 3 m/s and a green ball travel ing east to west at 3 m/s. A ll three balls have equal mass. Afterward, the red ball is trave li ng south and the green ball is moving to the east. In whic h directi on is the b lue ball travel ing? A. West B. North C. Between north and west D. Between north and east E. Between south and west

26. 1 A 24 g, 3-cm-diameter thin , hollow sphere rotates at 30 rpm about a vert ical, fri ct ionless ax is thro ugh it s center. A 4 g bug stands at the top of the sphere. He then walks along the surface o f the sphere until he reaches its "equator." When he reaches the equator, the sphere is rotating at A. 15 rpm B. 24 rpm C. 30 rpm D. 37 rpm E. 45 rpm

27. I A 5.0 kg solid cy linde r o f radius 12 cm rotates with Wi = 3.7 radls about an ax is through its center. A torque of 0 .040 N · m is applied to the cy linder for 5.0 s. By how Illuc h does the cyl inder 's angular momen tum change? A. 0.12 kg · ml/s B. 0.20 kg · m2/s C. 0.38 kg · mlls D. 0.52 kg · m' !s E. 0.S8 kg · m2/s

284 CHAPTER 9 Momentum

PROBLEMS

Section 9. t Impulse

Section 9.2 Momentum and the Impulse-Momentum Theorem

I. ! At what speed do a bicycle and its rider, with a combined mass of 100 kg, have the same momentum as a 1500 kg car u'aveling at 5.0 m/s?

2. I A 57 g tennis ball is served at 45 m/s. Ir the ball started from rest, what impulse was applied to the ball by the racket?

3. II A student throws a 120 g snowball at 7.5 mls at the side of the schoolhouse, where it hits and sticks. What is the magn itude of the average force on the wall if the duration of the colli s ion is 0. 15 s?

4. 1111 In Figure P9.4, what value of Fn .... ~ gives a n impulse of 6.0 N · s?

F, /

¥-~-~~--I-~ I (ms) FIGURE P9 .4 246 10

5. 1 A sled and rider, g liding over horizon tal, fri ctionless ice at 4.0 mIs, have a combined mass of SO kg. The sled then slides over a rough spot in the ice, slowing down to 3.0 m/s. What impulse was deli vered to the sled by the friction force from the rough spot?

Section 9.3 Solving Impulse and Momentum Problems

6. II Use the impulse-momentum theorem to find how long a II stone fallin g straight down takes to increase it s speed from

5.5 mls to 10.4 m/s. 7. II a. A 2.0 kg object is moving to the right wi th a speed of II 1.0 mls when it experiences the force shown in Figure

P9.7a. What are the object's speed and direction after the force ends?

8.

iI

b. Answer th is question for the fo rce shown in Figure P9.7b.

(a)

F, (N)

2

I (s) 0 f-----1

1s

- 2

FIGURE P9 .7

1111 A 60 g tennis ball with an ini tial speed of 32 mls hit s a wa ll and rebounds with the same speed. Figure P9.S shows the force of the wall on the bail during the colli­

(b) F~ (N)

2

0

- 2

o

sion . What is the va lue of FIGURE P9 .8

Fmax , the maximum value of the contact force during the co lli sion?

I (s)

I (ms)

9. II A ch ild is sliding on a sled at 1.5 mls to the ri ght. YOll stop rI the sled by pushing on it for 0.50 s in a direction oppos ite to its

motion . If the mass of the child and sled is 35 kg, what average force do you need to apply to stop the sled? Use the concepts of impulse and momentum .

10. III An ice hockey puck slides along the ice at 12 mls. A hockey rI stic k del ivers an impul se of 4.0 kg' mIs, causing the puck to

move off in the opposite d irection with the same speed. What is the mass of the puc k?

II. I As part of a safe ty investi gation, two 1400 kg cars traveling II at 20 mls are crashed into different barri ers. Find the average

forces exerted on (a) the car that hits a line of water barrels and takes 1.5 s to stop, and (b) the car that hi ts a concrete barrier and takes 0. 1 s to stop.

12. II In a Little League baseball game, the 145 g ball ente rs the rI strike zone with a speed of 15.0 m/s. The batter hits the ball ,

and it leaves hi s bat with a speed of 20.0 mls in exac tl y the opposite direct ion. a. What is the mag nitude of the impulse delivered by the bat to

the baB? b. If the bat is in contact with the ball for 1.5 ms, what is the

magnitude of the average force exerted by the bat on the ball ?

Section 9.4 Conservation of Momentum

13. 1111 A small, 100 g cart is moving at 1.20 mls on an air track II when it co llides with a larger, 1.00 kg cart at rest. After the col-

li s ion, the small cart reco il s at 0.850 m/s. What is the speed of the large cart after the collis ion?

14. II A man standing on very sl ic k ice fires a rifle hori zontall y. II The mass of the man together with the rifle is 70 kg , and the

mass of the bullet is 10 g. If the bullet leaves the muzz le at a speed of 500 mis, what is the final speed of the man?

15. III A 2.7 kg bloc k of wood sits on a table. A 3.0 g bullet, fired rI horizontall y at a speed of 500 mIs, goes completely through the

block, emerging at a speed of 220 m/s. What is the speed of the block immediately after the bu llet ex its?

16. I A strong man is compress ing a lightwe ight spring between II two weights. One weight has a mass of2.3 kg, the other a mass

of 5.3 kg. He is holding the weights stationary, but then he loses hi s grip and the weigh ts fly off in oppos ite directions. The lighter of the two is sho t out at a speed of 6.0 m/s. What is the speed of the heavier weight?

17. II A 10,000 kg railroad car is roLling at 2.00 m/s when a 4000 II kg load of g rave l is suddenly dropped in . What is the car 's

speed j ust after the gravel is loaded? 18. I A 5000 kg open train car is roll ing on fricti on less rail s at 11 22.0 mls when it starts pouring rain. A few mi nutes later, the

car's speed is 20.0 mls. What mass of water has coll ected in the car?

19. II A 50.0 kg archer, standing on fri cti onless ice, shoo ts a 40 g rI arrow at a speed of 60 m/s. What is the recoil speed of the

arc her? 20. II A 9.5 kg dog lakes a nap in a canoe and wakes up to find the rJ canoe has drifted out onto the lake but now is stationary. He

walks along the length of the canoe at 0.50 mIs, relat ive to the water, and the canoe simultaneously moves in the opposite direc~ tion at 0. 15 mls. What is the mass of the canoe?

Section 9.5 Inelastic Collisions

2 1. II A 300 g bird flying along at 6.0 mJs sees a 10 9 insec t head­g ing straight toward it with a speed of 30 m/s. The bird opens its

mouth wide and enjoys a nice lunch. What is the bi rd 's speed immediately after swallowing?

22. 11 A 7 1 kg baseball player j umps straight up to catch a line ra drive. If the 140 g ball is movi ng hori zontall y at 28 m/s. and the catch is made when the ballplayer is at the highest point of hi s leap, what is hi s speed immediately after stopping the ball?

23. !II A kid at the j unior high cafeteria wants to propel an empty ra miLk carton along a lunch tab le by hitti ng it wilh a 3.0 g spit ball. If he wants the speed of the 20 g carton just after the spit ball hits it to be 0.30 mis, at what speed should hi s spit ball hit the carton?

24. 1 The parking brake on a 2000 kg Cadillac has failed, and it is ra roll ing slowly, at I mph. toward a group of small children. See­ing the situation, you realize you have just enough time 10 drive your 1000 kg Vol kswagen head-on into the Cadillac and save the children. With what speed should you impact the Cadillac to bring it to a halt?

25. J A 2.0 kg block slides along a frictionless surface at 1.0 mls. A rJ second block, sliding at a faster 4.0 mIs, collides with the first

from behind and sticks to it. The final veloc ity of the combined blocks is 2.0 m/s. What was the mass of the second block?

Section 9.6 Momentum and Collisions in Two Dimensions

26. 11 11 A 20 g ball of c lay traveling eas t at 3.0 m/s coll ides with a PI 30 g ball of clay travel ing north at 2.0 m/s. What are the speed

and the directi on of the resulting 50 g ball of clay? 27. II Two part icles coll ide and

bounce apart. Figure P9.27 shows the ini tial momenta of both and the final momentum of particle 2. What is the final nlO men­tum of particle I? Show your answer by copying the fi gure and drawing the final momentum vector on the fi gure.

Py (kg 'm/s)

4

3

- 2 -3 - 4

FIGURE P9.27

28. 11 A 20 g ball of c lay travel in g eas t at 2.0 mls collides with a D 30 g ball of clay traveling 30° south of west at 1.0 mls. What

are the speed and direction of the resulting 50 g blob of clay? 29. 11 A firecracker in a coconut blows the coconut into three pieces. PI Two pieces of equal mass fl y off south and west , perpendicular

to each other, at 20 m/s. The third piece has twice the mass as the other two. What are the speed and direct ion of the third piece?

Section 9.7 Angular Momentum

30. III What is the angular momentum of the moon around the INT eanh? The moon's mass is 7.4 X 1022 kg and it orb its

3.8 X 108 m from the earth . 3 1. III A little girl is go ing on the merry-go-round for the first time,

and wan ts her 47 kg mother to stand nex t to her on the ride, 2.6 m from the merry-go-round 's center. If her mother 's speed is 4.2 mls when the ride is in motion , what is her angular momen­tum around the cen ter of the merry-go-round?

Problems 285

32. II What is the ang ular momentum about the ax le of the 500 g rotating bar in Figure P9.32?

120 rpm

2.0m

A,I, ~ ~ \

Axle

FIGURE P9.32 FIGURE P9.33

33. 1111 What is the angular momentum about the axle of the 2.0 kg, 4.0-cm-diameter rotating di sk in Figure P9.33?

34. 1 Di vers change their body position in mida ir while rotat ing BID abou t their center of mass . In one di ve, the di ver leaves the

board with her body nearly straight, then tucks into a somer­sault position. If the moment of inertia of the diver in a straight position is 14 kg· m2 and in a tucked position is 4.0 kg· m2

, by what factor is her angular veloci ty when tucked greater than when straight?

35. II Ice skaters often end the ir performances with spin turns, BID where they spin very fast aboutlheir center of mass with their

arms fo lded in and legs together. Upon ending, their arms ex tend outward, procla iming their finish. Not quite as not ice­ably, one leg goes out as we ll. Suppose that the moment of iner­ti a of a skater with arms out and one leg extended is 3.2 kg· m2

and for arms and legs in is 0.80 kg' m2. If she starts out spin­

ning at 5.0 revIs, what is her angular speed (in rev/s) when her arms and one leg open outward?

General Problems

36. III What is the impulse on a 3.0 kg particle that experiences the force described by the graph in Figure P9.36?

F, (N)

1500 -

1000

500

o +-+-~+-~~, I' (m" - 500 2 4 8 10

FIGURE P9.36 - 1000

37. 111 A 600 g a ir-track glider coll ides with a spri ng at one end of li the track. Figure P9.37 shows the glider's velocity and the force

exerted on the gl ider by the spring. How long is the glider in con tact with the spring?

'\ (m/s) F, (N)

3 36 j /\ 0 I (s)

0 f--------< I (s)

-3 ., FIGURE P9.37

286 CHAPTER 9 Momentum

38. II Far in space, where grav ity is negligible, a 425 kg rocket II traveling at 75.0 m1s in the positive x-direction fires its engines.

Figme P9.38 shows the thrust force as a function of time. The mass losl by the rocket during these 30.0 s is neg ligible. a. What impulse does the engine impart to the rocket? b. At what time does the rocket reach its maximum speed?

What is the maximum speed?

l~r~ 1 O~'(')

o to 20 30 O L----+-~--- t (ms)

FIGURE P9 .38 FIGURE P9 .39

39. 11111 A 200 g baH is dropped from a height of 2.0 m, bounces on a II hard floor, and rebounds to a height of 1.5 m. Figure P9.39 INT shows the impulse recei ved from the floOl". What maximum

force does the floor exert on the ball? 40. 11111 A 200 g baU is dropped from a height of 2.0 m and bounces rJ on a hard floor. The force on the baB from the floor is shown in INT Figllre P9.40. How high does the baU rebound?

l~r /\ o~'(m')

o 2 4 6 8

~;~kL(N) 1200

~ I(S) o 0.2 0.4 0.6

FIGURE P9 .40 FIGURE P9 .41

41. III Figure P9.4l is a graph of the force exerted by the floor on a g woman making a vertical jump. At what speed does she leave INT the ground?

Hint: The force of the floor is not the on ly force acting on the woman.

42. II A sled slides along a horizontal surface for which the coeffi­rJ cient of kinetic friction is 0.25. Its ve locity at point A is 8.0 m/s INT and at point B is 5.0 m/s. Use the impul se-momentum theorem

to find how long the sled takes to travel from A to 8. 43. III A 140 g baseball is moving horizontally to the right at35 m/s

when it is hi t by the bat. The ball flies off to the left at 55 mIs, at an angle of 25° above the horizontal. What are the magnitude and direction of the impulse that the bat delivers to the baH?

44. II Squids rely on jet propulsion, a versatile technique to move rJ around in water. A 1.5 kg squid at rest suddenly expels 0.10 kg of BID water backward to quickly get itself movi ng forward at 3.0 m/s.

If other forces (such as the drag force on the squid) are ignored, what is the speed with which the squid expels the water?

45. III The flowers of the bunchberry plant open with asto nishing BID force and speed, causing the pollen grai ns to be ejected out of

the flower in a mere 0.30 ms at an accelerat ion of 2.5 X 104 m/s2• If the acceleration is constant, what impulse is deli vered to a pollen grain with a mass of 1.0 X 10- 7 g?

46. II a. With what speed arc poUen grains ejected from a bunch­II berry fl ower? See Problem 45 for information. BID b. Suppose that 1000 ejected pollen gra ins slam into the

abdomen of a 5.0 g bee that is hovering just above the flower. If the collision is perfectly inelastic, what is the bee's speed immediately afterward? Is the bee likely to nOlice?

47. III A ten nis player swings her 1000 g racket with a speed of 11 10 m/s. She hits a 60 g tennis ball that was approach ing her at a

speed of 20 tn/so The ball rebounds at 40 m/s. a. How fast is her racket moving immediately a fter the impact?

You can ignore the interaction of the racket with her hand for the brief duration of the collis ion.

b. Jfthe tennis ball and racket are in con taci for 10 ms, what is the average force that the racke t exerts on the ball ?

48 . A 20 g ball of clay is thrown horizontall y at 30 m/s toward a rJ 1.0 kg block sitti ng at rest on a fri ctionless surface. The clay

hits and Slicks to the block. a. What is the speed of the block and clay right after the

coll ision? b. Use the block's initial and final speeds to calculate the

impulse the clay exerts on the block. c. Use the clay's ini tia l and final speeds to calculate the

impulse the block exerts on the clay.

d. Docs 1 blockonctay = - l clayonbtock? 49. II Dan is glid ing on hi s skateboard at4.0 m/s. He suddenly jumps II backward off the skateboard , kicking the skateboard forward at

8.0 m/s. How fast is Dan going as hi s feet hit the ground? Dan 's mass is 50 kg and the skateboard's mass is 5.0 kg.

50. I James and Sarah stand on a stationary cart with frictionless II wheels. The total mass of the cart and riders is 130 kg. At the

same instant, James throws a 1.0 kg balJ to Sarah at 4.5 mIs, while Sarah throws a 0.50 kg ball to James at 1.0 m/s. James's throw is to the right and Sarah 's is to the left. a. While the two balls are in the air, what are the speed and

direction of the cart and its riders? b. After the balls are caught, what are the speed and direction

of the cart and riders? 5 1. III Ethan, whose mass is 80 kg, stands ar one end of a very long, rJ stationary Wheeled cart that has a mass of 500 kg. He then

starts sprinting toward the ot her end of the cart. He soon reaches his top speed of 8.0 mIs, measured relative to the carl. What is the speed of the cart when Ethan has reached his top speed?

52. II The cars of a long coal train are filled by pulling them under rJ a hopper, from which coal fall s into the cars at a rate of

10,000 kg/so Ignoring friction due to the rail s, what is the aver­age force that the engine must exert on the coal train to keep it moving under the hopper at a speed of 0.50 m/s?

53. II Three identical train cars, coupled together, are rolling east at

rJ 2.0 m/s. A fourth car trave ling east at 4.0 m/s catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train . What is the speed of the ri ve­car train?

54. I A liD kg li nebacker ru nning at 2.0 m1s and an 82 kg quarler­rJ back running at 3.0 mls have a head-on colli sion in midair. The

linebacker grabs and holds onto the quarterback. Who ends up moving forward after they hi t?

55 . II Most geologists believe that the dinosaurs became ex tinct rJ 65 million years ago when a large comet or asteroid struck the INT earth, throwing up so much dust that the sun was blocked out

for a period of many months. Suppose an astero id with a diame­ter of 2.0 km and a mass of 1.0 X I Ot3 kg hits the earth with an impact speed of 4 .0 X 104]n/s. a. What is the earth' s recoil speed after such a co lli sion? (Use a

reference frame in which the earth was initially at rest.) b. What percentage is thi s of the earth 's speed around the sun?

(Use the astronom ical data inside the back cover.)

56. III At the center of a 50-m-diameter c ircular ice rink , a 75 kg ~ skater trave ling north at 2.5 mls collides with and holds onto a INT 60 kg skater who had been heading west at 3.5 m/s.

a. How long will it take them to glide to the edge of the rink? b. Where will they reach it? Give your answer as an angle

north of west. 57. II Two ice skaters, with masses of 50 kg and 75 kg, are at the II center of a 60-m-diameter circular rink. The skaters push off IN! against each other and glide to opposite edges of the rink. If the

heavier skater reaches the edge in 20 s, how long does the lighter skater take to reach the edge?

58. III One billiard baJJ is shot east at 2.00 m/s. A second, ident ical PI biUiard ball is shot west at 1.00 m/s. The balls have a glancing

collision, not a head-on collis ion, deflecting the second ball by 900 and sending it north at 1.41 m/s . What are the speed and direct ion of the first ball after the collision?

59. 1111 A \0 g bullet is fired into a 10 kg wood block that is at rest PI on a wood table. The block, with the bullet embedded, slides INT 5.0 cm across the table. What was the speed of the bullet? 60. III You are part of a search- P;\Clwge

~~ PI and-rescue mi ssion that has ","" INT been called out to look for a

lost explorer. You ' ve found the mi ss ing explorer, but you·re separated from him by a 200-m-high cliff and a 30-m-wide raging ri ver, as shown in Figure P9.60. To save his life, you need to get a 5.0 kg package of emergency sup-plies across the ri ver. Unfor[U-

200m

nately, you can ' t throw the FIGURE P9 .60

package hard enough to make it ac ross. Fortunately, you happen to have a

Explorer

30m

1.0 kg rocket intended for launching flares. Improvising quickly, you auach a sharpened stick to the front of the rocket. so that it will impale itself into the package of supplies, then fire the rocket at ground level toward the supplies. What minimum speed must the rocket have just before impact in order to save the explorer's life?

61. 1111 A 1500 kg weather rocket accelerates upward at 10.0 mls2.1t PI explodes 2.00 s after liftoff and breaks into two fragments, one tNT twice as mass ive as the other. Photos reveal Ihat Ihe li ghter

fragment traveled straight up and reached a maximum height of 530 m. What were the speed and direction of the heavier frag­ment just after the explosion?

62. 1I Two 500 g blocks of wood are 2.0 m apart on a frictionless rJ table. A to g bullet is fired at 400 m/s toward the blocks. It

passes all the way through the first block, then embeds itsel f in the second block. The speed of the fi rst block immediately afterward is 6.0 m/s. What is the speed of the second block after the builet stops?

63. I A 500 kg cannon fires a 10 kg can nonball with a speed of & 200 m/s re lative to the muzzle. The cannon is on wheels that

roll without friction. When the can non fires, what is the speed of the cannonball relati ve to the earth?

64. III Laura, whose mass is 35 kg, j umps hori zonta ll y otf a 55 kg Ii canoe at 1.5 m/s relat ive to the canoe. What is the canoe 's speed

just after she jumps? 65. II A spaceship of mass 2.0 X 106 kg is crui sing at a speed of g 5.0 X 106 m/s when the antimatter reactor fails, blowing the

ship into three pieces. One section, having a mass of 5.0 X 105 kg, is blown straight backward with a speed of

Problems 287

2.0 X lOt) m/s. A second piece, with mass 8.0 X 105 kg, con tin ­ues forward at 1.0 X 106 m/s. What are the direction and speed of the third piece?

66. III A proton is shot at 5.0 X \07 m/s toward a go ld target. The

lSI nucleus of a gold atom, with a mass 197 times that of the pro­ton , repels the proton and denects it straight back with 90% of its initial speed. What is the recoil speed of the gold nucleus?

67. III Figure P9.67 shows a coLli sion between three balls of clay. g The three hit simultaneously and stick together. What are the

speed and direction of Ihe resulting blob of clay?

6B.

lSI

~o 4.0m/~

FIGURE P9 .67

3.0m/s --o30g

+2.0 m/s 0 20g

III The carbon isotope 14C is used for carbon dating of archeolog­ical artifacts. 14C (mass 2.34 X 10- 26 kg) decays by the process known as beta decay in which the nucleus emits an electron (the beta particle) and a subatomic particle called a neutrino. In one such decay, the electron and the neutrino are emitted at right angles to each other. The electron (mass 9. 11 X 10- 31 kg) has a speed of 5.00 X 107 mls and the neutrino has a momentum of 8.00 X 10-24 kg ' m/s . What is the recoil speed of the nucleus?

69. III A I.O-m-Jong mass less rod is pivoted at one end and swings II around in a circle on a frictionless table. A block with a hole

through the center can sl ide in and out along the rod. InitiaUy, a small piece of wax holds the block 30 cm from the pivot. The block is spun at SO rpm, then the temperature of the rod is slowly increased. When the wax melts, the block slides out to the end of the roo. What is the final angular speed? Gi ve your answer in rpm.

70. II A 200 g puck revolves in a circle on a frictionless table at the II end of a 50.0-cm-long string. The puck's angular momentum INT about the center of the circle is 3.00 kg· m2/s. What is the ten-

sion in the string? 7 1. III Figure P9.7 1 shows a 100 g II puck revo lving in a 20-cm­INT radius c ircle on a frictionless

table. The string passes through a hole in the center of the table and is tied to two 200 g weights. a. What speed does the puck

need to support the two weights?

100,

20cm

200 g

200 g Pin-i

FIGURE P9 .71

b. The lower weight is a lighl bag filled with sand. Suppose a pin pokes a hole in the bag and the sand slowly leaks out while Ihe puck is revolving. What will be the puck' s speed and the radius of its trajectory after all of the sand is gone?

72. II A 2.0 kg, 20-cm-diameter turntable rotates at 100 rpm on frictionless bearings. Two 500 g blocks fall from above, hit the turntable simultaneously at opposite ends o f a diagonal , and st ick. What is the turntable's angular speed, in rpm, j ust after thi s event?

73. II Joey, from Example 9.10, stands at rest at the outer edge of the frictionless merry-go-round of Figure 9.28. The merry-go­round is also at rest. Joey then begins to run around the perime­ter of the merry~go-round , finally reaching a constant speed, measured relalive to the ground, of 5 .0 m/s. What is the final angular speed of the merry·go-round?

288 CHAPTER 9 Momentum

74. III A 3.0-m-diameter merry-go-ro und with a mass of 250 kg is spi nning at 20 rpm . John runs aro und the merry-go-round at 5.0 mIs, in the same direct ion that it is turning, and j umps onto the outer edge. John 's mass is 30 kg. What is the merry-go­round's angular speed, in rpm, after John jumps on?

75. II I Disk A, with a mass of 2.0 kg and a radius of 40 em, rotates ciock-wise about a fri ctionless vertical axle at 30 rev/s. Disk 8 , also 2.0 kg but wilh a rad ius of 20 em, rotates counterclockwise about that same ax le, bUi at a greater height than disk A, at 30 rev/s. Disk 8 slides down the axJe unt il it lands on top of disk A, after which they rotate together. ~fter (he collision, what is their common angular speed (in rev/s) and in which directi on do they rotate?

Passage Problems

Hitting a Golf Ball

Consider a go lf club hiui ng a go lf ball. To a good approx imat ion, we can model thi s as a co lli sion between (he rapid ly moving head of the golf club and the stat ionary golf baU, ignoring the shaft of the club and the go lfer.

Stop to T hink 9. t: F. The cart is initia ll y movi ng in the negat ive x-d irect ion, so (Px); = -20 kg ' m/s. After it bounces, (Px)f = 10 kg ' mls. Thus ilp ~ ( 10 kg ' mls) - (-20 kg ' mls) ~ 30 kg ' mls.

Stop to Think 9.2: B. The clay ball goes from (vx); = v to (V.I.)f = 0, so Jday = ~Px = -//lV. The ru bber ball rebounds, go ing from (Vt)i = v to (Vx)f = -v (same speed, opposite direction). Thus J rubher = ~Px =

-2mv. The rubber ba ll has a greater momentum change, and thi s requires a greater impulse.

Stop to Think 9.3 : Right end. The ball s started at rest, so the total momentum of the system is zero. It 's an isolated system, so the total momentum after the explos ion is still zero. The 6 g ball has momen­tum 6v. The 4 g ball, with velocity - 2v, has momentum - Sv. The combined momentum of these two balls is -2v. in order for P to be zero, the third ball must have apos;r;ve momentum (+2v) and thus a positive veloc it y.

A golf ball has a mass of 46 g. Suppose a 200 g club head is mov­ing at a speed of 40 mls j ust before striking the gol f bal l. After the coll ision, the golf ball 's speed is 60 m/s.

76. I What is the momentum of the club + ball system ri ght before the collision? A. 1.8 kg ' mls B. 8.0 kg ' mls C. 3220 kg' mls D. 8000 kg ' mls

77. I immediately after the colli sion, the momentum of the club + ball system will be

A. Less than before the colli sion. B. The same as before the colli sion. C. More than before the co lli sion.

78. I A man ufacturer makes a golf ball that compresses more than a traditional go lf balJ when struck by a club. How wiU this affect (he average fo rce during the colli sion?

A. The force wi ll decrease. B. The force wi ll not be affected. C. The force will inc rease.

79. I By approx imate ly how much does the club head slow down as a result of hilling the ball ? A. 4 mls B. 6 mls C. 14 mls D. 26 mls

Stop to Think 9.4: B. !fhe momentum of parti cle 1 is (DAD kg)(2.5 mls) ~ 1.0 kg ' ml<. while that of panic le 2 is (0.80 kg)( - 1.5 m/s) = - 1.2 kg ' m/s . The total momentum is then 1.0 kg ' m/s - 1.2 kg ' m/s = - 0.2 kg ' m/s. Because it 's negati ve, the total momentum, and hence the fi nal velocity of the parti cles, is directed to the left.

Stop to Think 9.5: B. Angular momentum L = Iw is conserved. Both boys have mass m and initi all y stand di stance R12 from the ax is. Thus the initia l moment of inerti a is Ii =

Idisk + 2 X I11(RI2f = I di$k + "ir1lR2• The Fi nal moment of inerti a is

I f = Idi.~'.:. + 0 + mR2. because the boy standing at the ax is contributes not hi ng to the moment of inerti a. Because I f> I ; we must have wr<wi '