Summary 315

SUMMARY The goals of Chapter 10 are to introduce the concept of energy and to learn a new problem­solving strategy based on conservation of energy.

GENERAL PRINCIPLES

Basic Energy Model Within a system, energy can be transformed between various forms.

Em:rgy is

within the ~ySlem .

I<: nvironment

System K _ U

. \ I Solving Energy Conservation Problems PREPARE Choose your system so thai it's isolated . Draw a before·and-after visual overview. Energy can be transferred into or Ollt of a sys tem

in two basic ways:

Work: The transfer of energy by mechan ical forces.

I i £-./ Woe' . \ hem £

"

SOLVE

.If the system is iso lated and there 's no friction , then mechanical energy is conserved:

Heal: The non mechanical transfer of energy from a honer to a colder object.

Energy i~ trrm,Iji' f(l'd \0

or from the 'y~lem rrom or \0 Ihe env ironment.

Conservation of Energy If the sys tem is iso lated but there 's friction present, then the [otal energy is conserved:

When work W is done on a system, the system's total energy changes by the amount of work done. In mathematical form, this is the work-energy equation: ASSESS Kinetic energy is always positive, as is the

change in thenna! energy.

A system is isolated when no energy is transferred inw or out of the system. This means the work is zero, giving the law of conservation of energy:

IMPORTANT CONCEPTS

Kinetic energy is an energy of motion:

I 2 I 2 K = 2111V + 21w ' . Tr~nslU1l onal .. ···,lI ...... Rotatlonu!

Potential energy is energy stored in a system of interacti ng objects.

Gravitational potential energy: Ug = mg)'

Elastic potential energy: I

U = - kx 2 , 2

Mechanical energy is the sum of a system's kinetic and potential energies:

Mechanical energy = K + U = K + Ug + Us

APPLICATIONS

Perfectly elastic collisions Both mechanical energy and momentum are conserved.

Before:

ObJecl2 mitla!ly at rest ( VI ) . J

~0'/K,

After: K,~ K,~~ (V(.r \ (\12.<\

m l - /1/ 2 (VLJ r = (V lx)i

tnl + 111 2

Thermal energy is the sum of the microscopic kinet ic and potential energies of all the molecules in an object. The hotter an object. the more thermal energy it has. When kinetic (s liding) friction is presen t, the increase in the thermal energy is ~Eth = A~x.

Work is the process by which energy is transferred to or from a system by the application of mechan ical forces.

If a particle moves through a displacement d while acted lIpon by a constan t force F, the force does work

I •

Fff --7 :' ~~

F,- FCOS(J"\ Only the component of the force pumllelto the displacement does work.

Power is the rate at which energy is transformed.

p = ~E +-- Amounl of energy lransfonned ~t 4----Time required 10 transfonll i(

... or at which work is done.

p = Jf"+---AmoulH of work done ~/ 4 " " " ''''' ' T i me required (0 do work

316 CHAPTER 10 Energy and Work

tMP)TM For homework assigned on MasteringPhysics, go to

!:!.!/ www.masteringphysics.com

Problem difficulty is labeled as I (straightforward) to 11 111 (challenging),

0,0

QUESTIONS

Conceptual Questions

I. The brake shoes of your car are made of a material that can tolerate very high temperatures without be ing damaged. Why is thi s so?

2. When YO LI pound a nail with a hammer, the na il gelS quite

warm. Describe the energy transformations that lead LO the add ition of thermal ene rgy in the na il.

For Questions 3 through to, give a specific example of a system with the energy transformation shown. In these questi ons, W is the work done on the syste m, and K, U, and £[11 are the kinetic, poten tial , and thermal energies of the system, respectively. Any energy not men­ti oned in the transfonn ation is assumed to remain constant; if work is not mentioned, it is assumed to be zero.

3. W --> K 4. W--> U 5. K-->U ~K-->W

7. U-K 8. W - 8£[h 9.U--> 4~ l QK--> 4~

II . A ba ll of puny is dropped from a he ight of 2 m onto a hard floor, where it sticks. What object or objects need to be included withi n the system if the system is to be iso lated during this process?

12. A 0.5 kg mass on a I-m-long string swings in a c ircle on a hori­zontal, frictionless tab le at a steady speed of 2 m/s. How much work does the tens ion in the string do on the mass during one revolution? Explain .

13. Particle A has less mass than particle B. Both are pushed fo r­ward across a fricti o nless surface by eq ual forces for I s. Bo th start from rest. a. Compare the amount of work done on each particle. That is,

is the work do ne on A greater than, less than, or equal to the work done on B? Ex plain .

b. Compare the impulses delivered to parti cles A and B. Ex plain .

c. Compare the final speeds of particles A and B. Explain. 14. The meani ng of the word "work" is quite different in phys ics

from it s everyday usage . Give an example of an actio n a person could do that " feel s like work" but that does not involve any work as we've defined it in thi s c hapter.

15. To cha nge a ti re, you need to use a jack to raise one corner of your car. While doing so, you happen to not ice that pushing the jack handle down 20 cm rai ses the car only 0 .2 c m. Use energy concepts 10 exp lai n why the handle must be moved so fa r to raise the car by such a small amount.

16. You drop two balls from a tower, one of mass 111 and the other o f mass 2m . Just before they hit the ground, which ball , if e ithe r, has the larger kinetic energy? Explain.

Problems labeled i'!.I can be done on a Workbook Energy Work­

sheet; INT integrate significant material from earlier chapters;

BIO are of biological or medical interest.

17. A roller coaster car ro ll s down a frictio nless track, reaching speed v at the bottom. a. If you want the car to go twice as fast at the bOllom, by what

factor must you increase the height of the track? b. Does your answer to part a depend on whether the track is

straight or not? Ex plain . 18. A spring gun shoots out a p lastic ba ll at speed v. The spring is

then compressed twice the di stance it was on the first shot. a. By what factor is the spring 's potential energy increased? b. By what factor is the ball 's speed increased? Explain.

19. Sandy and Chri s stand on the edge of a cliff and throw ident ical mass rocks at the same speed. Sandy throws her rock hor izon­tally while Chris throws hi s upward at an angle o f 45° to the hori zontal. Arc the rocks moving at the same speed when they hit the ground, or is one moving faster than the other? If one is

20.

2 1. BID

moving faster, which one? ExpJain. A solid cyl inder and a cyl indrical shell have the same mass, same radius, and turn on fricti o nless, horizon tal axles. (The cy li ndrical shell has lightweight spokes connect ing the she ll to the FIGURE Q10.20

axle.) A rope is wrapped aro und each cyl inder and tied to a block. The blocks have the same mass and are held the same he ight above the ground as shown in Figu re QIO.20. Both blocks are released s imultaneously. The ropes do not sli p. Which block hits the ground first? Or is it a tie? Explain. You are much more likely to be inj ured if you fall and your head strikes the gro und than if yo ur head strikes a gymnast ics pad. Use energy and work concepts to explai n why thi s is so.

Multiple-Choice Questions

22. II If you walk up a flight o f stairs at constant speed, gaining ver­ti cal he ight It, the work done on you (the system, of mass 11/) is A. +mgil, by the normal force of the stairs. B. -mgh, by the normal force of the stai rs. C. +mgh, by the grav itational force of the earth. D. - mgh, by the gravitat ional force of the earth.

23 . I You and a fri end each carry a 15 kg suitcase up two flights of stairs, walking al a constant speed. Take each sui tcase to be the system. Suppose you carry your suitcase up the stai rs in 30 s while your friend takes 60 s. Which of the following is true? A. You did more work, but both of you expe nded the same

power. B. You d id more work and expended more power. C. Both ofyoll did equal work, but you expended more power. D. Both of you did equal work, but you expended less power.

24. I A woman uses a pulley and a rope to raise a 20 kg weight to a height of2 m. If it takes 4 s to do thi s, about how much power is she suppl yi ng? A. lOOW B. 200W C. 300W D. 400W

25. I A hockey puck sl idi ng along frict ionless ice with speed I' to the ri ght coll ides with a horizontal spring and compresses it by 2.0 cm before comi ng to a momentary stop. What will be the spri ng's max imum compression if the same puck hits it at a speed or 21'? A. 2.0 em B. 2.8 em C. 4.0 em D. 5.6cm E. 8.0clll

PROBLEMS

Section 10.2 Work

I. II Duri ng an et iquette c lass, yo u wal k slowly and stead il y at 0.20 mls for 2.5 m with a 0.75 kg book balanced on top o r your head. How much work does your head do on the book?

2. II A 2.0 kg book is lying on a 0.75-m-high table. You pick it up and place it on a bookshe lf 2.3 III above the floor. a. How much work does gravity do on the book? b. How much work does your hand do on the book?

3. II The two ropes seen in Figure P I 0.3 are used to lower a 255 kg piano exactly 5 m from a second-s tory window to the ground. How much work is done by each of the three forces?

500 N 1.

2500 N ii' 223 N

FIGURE P10.3 FIGURE P10.4

4. 1 The two ropes shown in the bird' s-eye view of Figure PIOA are used to drag a crate exactly 3 m across the floor. How much work is done by each of the ropes on the crate?

5. II a. At the airport, you ride a "moving sidewa lk" that carri es you horizon tally fo r 25 m at 0.70 mls. Assumi ng that you were mov ing at 0.70 mls before stepping onto the movi ng sidewa lk and con ti nue at 0.70 mls afterward, how muc h work does the moving sidewal k do on you? Yo ur mass is 60 kg.

b. An escalator carries you from one leve l to the next in the airport termi nal. The upper level is 4 .5 III above the lower leve l, and the length of the escalator is 7.0 m. How much work does the up escalator do on you when you ride it from the lower level to the upper level?

c. How much work does the down escalator do on YOll when you ride it from the upper leve l to the lower level?

6. A boy flies a kite with the string at a 30° angle to the horizon-tal. The te nsion in the str ing is 4.5 N. How much work does the string do on the boy if the boy a. Stands still ? b. Walks a horizon tal distance of II m away from the kite? c. Walks a horizon tal distance of II m toward the kite?

Section 10.3 Kinetic Energy

7. I Which has the larger kinet ic energy, a 10 g bull et fired at 500 mls or a 10 kg bowling ba ll sliding at 10 m/s?

Problems 317

26. II A block s lides dow n a smooth ramp. starting fro m res t at a height h. When it reaches the bottom it 's moving at speed v . .It then continues to sl ide ti p a second smooth ramp. At what he ight is its speed equa l to vl2? A. h/4 B. hl2 C. 3h14 D. 2h

27. I A wrecking ba ll is suspe nded rrom a 5.0-m-long cable that makes a 300 angle with the verti cal. The ball is released and swings down. What is the ball's speed at the lowest po int? A. 7.7 m/s B. 4.4 mls C. 3.6 m/s D. 3. 1 m/s

8. II At what speed does a 1000 kg compact car have the same kinetic energy as a 20.000 kg truck going 25 km/hr?

9. I A car is travel ing atl O m/s. a. How fast would the car need to go to double its kinetic energy? b. By what factor does the car 's kinet ic energy increase if its

speed is doubled to 20 m/s? 10. III Sam 's job at the amusement park is to slow down and bring

to a SlOp the boats in the log r ide. I f a boat and its riders have a mass of 1200 kg and the boat drifts in at 1.2 mIs, how much work does Sam do to stop it?

II . III A 20 g plast ic ball is moving to the left at 30 m/s. How much work must be done on the ball to cause it to move (0 the right at 30 m/s?

12. III The turntable in a microwave oven has a moment of inertia of 0.040 kg · m 2 and is rotating once every 4.0 s. What is its kinetic energy?

13. 1111 An energy storage system based on a flywheel (a rotati ng d isk) can slOre a max imum of 4.0 MJ when the fl ywheel is rotating at 20,000 revo lutions per minute. What is the moment of inertia of the fl ywheel?

Section 10.4 Potential Energy

14. II The lowest po in t in Death Valley is 85.0 m be low sea level. The summit of nearby Mt. Whi tney has an elevation of 4420 m. What is the change in gravitati ona l poten ti al energy of an ener­get ic 65.0 kg hi ker who makes it from the floor of Death Valley to the top of Mt. Whi tney?

15. I a . What is the ki neti c energy of a 1500 kg car travel ing at a speed or 30 m/s ("'65 mph)?

b. From what he ight should the car be dropped (0 have this same amou nt of ki netic energy just before impact?

c. Does your answer to part b depend on the car's mass? 16. The world 's fastes t humans can reach speeds of about II m/s.

In order to increase hi s gravitational poten tial energy by an amount equal to hi s kineti c energy at full speed, how high wou ld such a sprinter need to cl imb?

17. I A 72 kg bike racer cl imbs a 1200-m-Iong section of road that has a slope of 4.3°. By how much does hi s grav itational poten­tial energy change during thi s climb?

18. II A 1000 kg wrecking ball hangs from a 15-m-long cable. T he ball is pu lled back un til the cable makes an angle of 2SO with the vertical. By how much has the grav itat ional potential energy of the ball changed?

19. II How far must you stre tch a spring with k = 1000 Nlm to store 200 J of energy?

318 CHAPTER 10 Energy and Work

20. II How much energy can be stored in a s pr ing wi th a spring

constant of 500 N/m if its max imum possible stretch is 20 em? 2 1. 11111 The elast ic energy stored in your tendons can contri bute lip to BID 35% of your energy needs whe n running. Sports sc ie nti sts have

studied the change in length of the knee ex tensor tendon in sprinters and no n a thle te s . They find (on average) th at th e

sprinters' tendons stretch 4 1 mm, while nonathletes ' stretch only 33 nun. The spring constant for the tendon is the same for both groups, 33 N/mm. What is the d ifference in maximum

stored energy between the sprinters and the non athletes?

Section 10.5 Thermal Energy

22. I Mari ssa drags a 23 kg duFrel bag 14 In across the gym nOOf.

If the coefficient of ki netic friction between the floor and bag is

O.IS , how muc h thennal energy does Marissa create?

23. II Mark pushes hi s broken car ISO m down the block to hi s friend's house. He has to exert a 110 N hori zontal force to push

the car at a constant speed. How muc h thermal e nergy is c reated in the tires and road during thi s short trip?

24. III A 900 N cra te s lides 12 m down a ramp that makes an ang le of 3So with the horizontal. If the crate slides at a constant speed,

how much thermal energy is created? 2S. III A 2S kg c hild slides down a playground slide at a COllSIanI

speed. The slide has a he ight o f 3.0 m and is 7 .0 m long. Us ing energy cons iderat ions, rind the magn itude of the kinetic fric tion

force act ing o n the child .

Section 10.6 Using the Law of Conservation of Energy

26. II A boy reaches o ut of a w indow and tosses a ball strai ght up II wi th a speed of 10 m/s. The ball is 20 m above the gro und as he

releases it. Use conservation of energy to rind a. The ball 's max imum he ight above the ground. b. The ball 's speed as it passes the window on its way down.

c. The speed of impact on the ground.

27. II a . With w hat minimum speed must you toss a 100 g ball

II stra ig ht up to just barely hit the I O-m-high ce ili ng of the gy mnasium if you release the ball 1.5 III above the floor? Solve thi s problem lIsing energy.

b. With what speed does the ball hit the floor? 2S. III What minimum speed does a 100 g puck need to make it to the

rJ top of a frictionless ramp that is 3.0 m long and inclined at 200 ? 29. II A car is parked at the to p o f a 50-m-high hill. It s lips ou t o f

II gear and rolls down the hill. How fast will it be going at the bot-tom? (Ignore fri ct ion.)

30. III A 1500 kg car is approaching the hill shown in Figure PI 0.30

& at 10 mls when it suddenly runs out of gas .

a. Can the car make it to the top of the hill by coast ing? b . If yo ur answer to part a is yes. what is the car ' s speed after

coast ing down the other s ide?

FIGURE Pl0 .30

3 1. II A 10 kg runaway grocery cart runs into a spring w ith spr ing

rI consta nt 250 N/m an d compresses it by 60 em. What was the

speed o f the cart j ust before it hit the spring? 32. II As a 15,000 kg jet lands on an a ircraft carrier, its tail hook

g snags a cable to s low it down. The cable is attached to a spring with spring constant 60,000 N/m. If the spri ng stretches 30 m to

stop the plane, what was the plane 's land ing speed?

33 . II Your friend 's Fr isbee has become s tuc k 16 m above the

II ground in a tree. You want to di slodge the Frisbee by throw ing a rock at it. The Frisbee is stuck pre tty tight, so you fi g ure the rock needs to be traveling at least 5.0 mls w he n it hits the Fris­

bee. If you release the rock 2.0 m above the ground, with what

minimum speed must you throw it? 34. II A fireman of mass SO kg slides down a pole. When he reaches

ti the bottom, 4.2 m below hi s starting point, hi s speed is 2.2 m/s.

By how much has thermal energy increased during hi s s lide?

3S. II A 20 kg chi ld s lides down a 3.0-m-high playground s lide.

II She starts from rest, and her speed at the bottom is 2.0 m/s. a. What e nergy trans fe rs and trans formations occur during the

s lide? b. What is the total change in the the rm al energy of the slide

and the seat of her pants? 36 . II A hockey puck is g iven an initi a l speed of 5.0 m/s. If the

II coeffic ient of kinetic friction be twee n the puck an d the ice is 0.05 , how far does the puck slide before com ing to rest? Solve

thi s problem using conservation o f e nergy.

Section 10.7 Energy in Collisions

37. II A 50 g marb le moving at 2.0 mls strikes a 20 g marble at rest.

What is the speed of each marble immediate ly after the co lli ­

sion ? Assume the colli s io n is perfec tl y e lastic and the marbles collide head-on .

3S. II Balli , w ith a mass of 100 g and traveling at 10 mIs, collides

head-on with ball 2, which has a mass of 300 g and is initi ally at

rest. What are the fin al ve loc iti es of each ball if the colli sion is (a) perfectly elastic? (b) perfectly inelasti c?

39. I An a ir-trac k g lider undergoes a perfectl y ine last ic co lli s io n with an identi cal gl ider that is initially at rest. What fraction of the first glider's initi a l kinetic e nergy is trans formed into ther­

mal energy in thi s coll is ion? 40. I Two ba ll s undergo a perfec tl y e last ic head-on colli sion, w ith

one ball ini tia ll y at rest. If the incom ing ball has a speed of 200 mIs, what are the fin al speed and directio n of each ball if a. The incoming ball is milch more mass ive (han (he stati onary

ball? b. The stati o nary ball is milch more mass ive than the incom ing

ball?

Section 10.8 Power

4 1. II a. How much work must you do to push a 10 kg block of stee l

across a stee l tabl e at a steady speed of 1.0 mls for 3.0 s?

The coeffic ien t o f kinet ic fr ic ti on for steel on steel is 0.60. b. What is your power output while doing so?

42. I a. How much work does an e levator motor do to lift a 1000 kg e levator a height of 100 m?

b. How much power must the motor suppl y to do thi s in 50 s

at constant speed? 43 . II I A 1000 kg sports car accele rates fro m 0 to 30 m/s in 10 s.

What is the average power of the eng ine?

44. II I In just 0.30 S, you compress a spring (spring constant 5000 N/m), which is initially at its equilibrium length , by 4 .0 cm. What is

your average power output? 45 . II I In the w inte r sport o f c urling, players g ive a 20 kg stone a

push across a sheet of ice. A curle r accelerates a stone to a speed o f 3.0 m/s over a time of 2 .0 s .

a. How muc h force does the curler exe rt on the stone?

b . What average powe r does the curle r use to bring the sto ne

up to speed?

46. II A 7 1 0 kg car dri ves at a constan t speed of 23 m/s. It is subject to a drag force of 500 N. What power is required from the car's engine to drive the car a. On leve l ground? b. Up a hill with a slope of 2.oo?

47. III An e levator we ighing 2500 N ascends at a constant speed of 8.0 mls. How much power must the motor supply to do thi s?

General Problems

48. II A 2.3 kg box, start ing from rest, is pushed up a ramp by a to N fo rce paralle l to the ramp. The ramp is 2.0 m long and tilted at 17°. The speed of the box at the top of the ramp is 0.80 m/s. Consider the system to be the box + ramp + earth . a. How much work W does the force do on the system? b. What is the change !3. K in the kineti c energy of the system? c. What is the change !3. Ug in the gravitat ional potential energy

of the system? d. What is lhe change LlE(h in the thermal energy of lhe system?

49. 1 A 55 kg skateboarder wants rJ to j ust make it to the upper

edge of a "half-pipe" with a radius of 3.0 m, as shown in Figure P IO.49. What speed Vi

does he need at the bottom if FIGURE P10 .49

he is to coast all the way up? a. Firsl do the calcul at ion treat ing the skateboarde r and board

as a point part icle. with the entire mass nearly in contac t with the half-p ipe.

b. More reali sticall y, the mass of the skateboarder in a deep crouch might be thought of as concentrated 0.75 m from t.he hal f-pipe. Assumi ng he remains in that pos ition all the way up, what Vi is needed to reach the upper edge?

50. !II Fleas have remarkable jumping ab ility. A 0.50 mg flea, j ump­a ing straight up, would reach a height of 40 em if there were no BID air res istance. In reality, air resistance limits the height to 20 cm.

a. What is the nea's ki neti c energy as it leaves the ground? b. At its highest point, what fraction of the initia l ki netic

energy has been converted to potential energy? 5 1. III A marble sl ides without friction in a vertical plane around the PI inside of a smooth , 20-em-di ameter hori zonta l pipe. The mar­INT ble's speed at the bottom is 3.0 mls; thi s is fas t enough so that

the marble makes a complete loop, neve r los ing contact with the pipe. What is its speed at the top?

52. 11 A 20 kg child is on a swing thal hangs from 3.0-m-long chains, ra as shown in Figure PI 0.52. What is her speed Vi at the bottom of the arc if she swings out to a 45° angle before reversing di rection?

53.

FIGURE Pl0 .52 FIGURE Pl0 .53

Suppose you li ft a 20 kg box by a he ight of 1.0 m. a. How much work do you do in lifting the box?

Inslead of lifti ng the box straighl up, suppose you push it up a I .O-m-high ramp that makes a 30° degree angle with the hori­zontal, as shown in Figure P I0.53. Being clever, you choose a ramp with no friction.

Problems 319

b. How much force F is required to push the box straight up the slope at a constant speed?

c. How long is the ramp? d. Use your force and di slance results to calculate the work

you do in pushing the box up the ramp. How does this com­pare to your answer to part a?

54. II A cannon tilted up at a 30° angle fi res a cannon ball at 80 mls II from atop a 10-m-high fortress wal l. What is the ball's impact

speed on the ground be low? Ignore aiT res istance. 55. I The sledde r shown in Figure P10.55 starts from the top of a a frictionless hill and slides dow n in to the valley. What inilial

speed Vi does the sledder need to j ust make it over the next hill ?

t" 42m 30m

-- ------------------------------FIGURE Pl0 .55

56. 11111 In a phys ics lab experiment, a spri ng clamped to the table PI shoots a 20 g ball hori zontall y. When the spring is compressed INT 20 cm, the ball travels horizontall y 5.0 m and lands on the noor

1.5 m below the point at which it left the spri ng. What is the spring constant?

57. 11 11 A 50 g ice cube can sl ide without friction up and down a 30° II slope. The ice cube is pressed aga inst a spring at the bottom of

the slope, compress ing the spri ng 10 cm. The spring constant is 25 N/m. When the ice cube is released, what distance will it travel up the slope before reversing di rection?

58. 11111 The max imum energy a bone can absorb without breaki ng is BID surprisingly small. For a healthy human of mass 60 kg, experi­

mental data show that the leg bones can absorb about 200 J . a. From what max imum height could a person jump and land

ri gidly upright on both fee t without break ing hi s legs? Assume that all the energy is absorbed in the leg bones in a ri gid landi ng.

b. People j ump from much greater heights than thi s; explain how thi s is poss ible.

Hint: Think aboul how people land when they jump from greater heights .

59. II In an amusement park water sl ide, people slide down an II essen tially fr ict ionless tube. They drop 3.0 m and ex it the slide. INT moving horizontaUy, 1.2 m above a swimm ing poo l. What hori-

zontal distance do they trave l from the ex it point before hitti ng the water? Does the mass of the person make any di fference?

60. II The 5.0-m-long rope in PI Figure P IO.60 hangs verti ­

cally from a tree ri ght at the edge of a rav ine. A woman wants to use the rope to swing to the other side of the rav ine. She ru ns as fast as she can, grabs the rope, and swings a lit over the rav ine. a. As she swings, what energy

conversion is taki ng place?

FIGURE Pl0 .60

b. When she's directly over the far edge of the rav ine, how much higher is she Ihan when she started?

c. Given your answers to parts a and b, how fast must she be run ning when she grabs the rope in order to swing all the way across the rav ine?

320 CHAPTER 10 Energy and Work

61. III YOLI have been asked to design a "ballistic spring sys tem" to II measure the speed of bullets. A buJiet of mass m is fired into a INT block of mass M . The block, with the embedded bullet , then

slides across a frict ionless table and collides with a hori zontal spring whose spring constant is k. The oppos ite end of the spring is anchored to a waH. The spring's maximum compress ion d is measured. a. Find an expression for the bullet's initial speed VB in terms

of 111, M, k, and d. Hint: This is a two-part problem. The buLlet 's colli s ion with the block is an inelasti c colli sion . What quantity is conse rved in an inelasti c co lli s ion ? Subsequently the block hilS a spring on a fri ct ionless surface. What quantity is conserved in thi s collis ion? b. What was the speed of a 5 .0 g bullet if the block 's mass is

2.0 kg and if the spring, with k = 50 N/m, was compressed by 10 em?

c. What fraction of the bullet's initi al kinetic energy is "lost"? Where d id it go?

62. III A new event, shown in g Figure PI 0.62, has been INT proposed for the Winter

O lym pics. An athlete will sprint 100 m, starting from rest, then leap onto a 20 kg FIGURE Pl0.62

bobs led. The person and bobsled will then slide down a 50-m-Iong ice-covered ramp, sloped at 200

, and in to a spring with a care full y ca librated spring constan t of 2000 N/m. The athlete who compresses the spring the farthest wins Lhe gold medal. Lisa, whose mass is 40 kg, has been training for thi s even t. She can reach a maximum speed of 12 mls in the 100m dash. a. How far will Lisa compress the spring? b. The Olympic committee has very exac t specifications about

the shape and angle of the ramp. Is thi s necessary? If the committee asks your opinion, what factors about the ramp will you tell them arc important?

63. II Boxes A and B in Figure PIO.63 II have masses of 12.0 kg and 4.0 kg,

respectively. The two boxes are released from rest. Use conserva­tion of energy to find the boxes' speed when box B has fallen a di s­tance of 0.50 m. Assume a frict ion- FIGURE Pl0 .63

less upper surface. 64. 1111 What would be the speed of the boxes in Problem 63 if the rJ coefficient of kinetic fri ct ion between box A and the surface it

slides on we re 0.20? Use conservat ion of energy. 65. 11111 A 20 g ball is fired hori zontally with initial speed Vi toward a 11 100 g ball that is hanging motionless from a 1.0-m-long string.

The balls undergo a head-on, perfect ly elastic colli sion, after which the 100 g ball swings out to a max imum angle 8max = 500

•

What was Vi? 66. II Two coupled boxcars are rolling along at 2.5 mls when they INT collide with and couple to a third, stationary boxcar.

a. What is the fi nal speed of the three coupled boxcars? b. What fraction of the cars ' initial kinetic e nergy is trans­

formed into thermal energy? 67. A fi sh scale, consist ing of a spring with spring constant k =

rJ 200 N/m, is hung vert ically from the ceiling. A 5.0 kg fi sh is

attached to the end of the unstretched spring and then released. The fi sh moves downward until the spring is fully stretched, the n starts to move back up as the spring begins to contract. What is the maximum di stance through which the fi sh falls?

68. I A 70 kg human sprinter can accelerate from rest to 10 mls in BID 3.0 s. During the same interval, a 30 kg greyhound can acce ler­

ate from rest to 20 m/s. Compute (a) the change in kinet ic energy and (b) the average power output for each.

69. III A 50 g ball of c lay travel ing aI speed Vi hi ts and s ti cks to a INT 1.0 kg block sitting at rest on a frictionless surface.

a. What is the speed of the block after the coll ision? b. Show that the mechanica l energy is 1101 conserved in thi s

colli s ion. What percentage of the ball 's initial kinetic energy is " lost"? Where did thi s kinetic energy go?

70. II A package of mass m is released from rest at a warehouse load­WJl all ing dock and slides down a 3.0-m-high frictionless chute to a INT waiting truck. Unfol1Lmately. the truck dri ver wen t on a break

without having removed the previous package, of mass 2m, from the bottom of the chute as shown in Figure PIO.70. a. Suppose the packages st ick together. What is their common

speed after the coll ision? b. Suppose th e colli s ion be tween the pac kages is perfectly

e lastic. To what he ight does the package of mass II!

rebound?

m

FIGURE Pl0 .70

30mI ~II

7 1. 1111 A 50 kg sprinter, starti ng from rest, runs 50 m in 7.0 s at con­INT stan t acce leration.

a. What is the magnitude of the horizontal force acting on the sprinter?

b. What is the sprinter 's average power output during the first 2.0 s of hi s run?

c. What is the sprinter 's average power output during the fin al 2.0 s?

72. II Bob can throw a 500 g rock with a speed of 30 mls. He INT moves hi s hand forward 1.0 m while doing so.

a. How muc h force, ass umed to be constant, does Bob apply to the rock?

b. How muc h work does Bob do on the rock? 73 . III A 2.0 hp electric motor on a water well pumps water from 10 m

below the surface. The density of water is 1.0 kg per L. How many liters of water can the motor pump in I h?

74. II The human heart has to pump the average adult 's 6.0 L of BID blood through the body every minute. The heart must do work

to overcome frict ional forces that resist the blood flow. The average blood pressure is 1.3 X 10 ..\ N/m 2 •

a. Compute the work done moving the 6.0 L of blood com­ple te ly through the body, assuming the blood pressure always takes its average value.

b. What power output must the heart have to do thi s task once a minute?

Hint: When the heart contracts , it applies fo rce to the blood. Pressure is just forcelarea, so we can write work = (pressure) (area)(di stance). But (area)(d istance) is j ust the blood vol ume passing through the heart.

Passage Problems

Tennis Ball Testing

A tennis ball bouncing on a hard surface compresses and then rebounds. The detail s of the rebound <rrc specified in te nnis regula­tions. Tennis balls, to be acceptable for tournament play, must have a mass of 57.5 g. When dropped from a height of 2.5 m on to a concrete surface, a ball must rebound to a height of 1.4 m. During impacl, the ball compresses by approximately 6 mm.

75. I How fast is the ball moving when it hilS the concrete surface? (Ignore air resistance.) A. 5 m/s B. 7 m/s C. 25 mls D. 50 In/s

76. I tr the ball accelerates unifollnly when it hits the floor, what is its approximate acceleration as it comes to rest before rebounding? A. 1000 m/s 2 B. 2000 mls 2 C. 3000 m/s 2 D. 4000 mls 2

77. I The balrs kinetic energy just after the bounce is less than j ust before the bounce. In what form does this lost energy end up? A. Elastic potenti al energy B. Gravitational poten tial energy C. Thermal energy D. Rotational kineti c energy

78. I By approximately what percent does the kineti c energy decrease? A. 35% B. 45% C. 55% D. 65%

79. I When a tennis ball bounces from a racket, the ball loses approximately 30% o f it s kine ti c energy to thermal energy. A ball that hit s a racket at a speed of 10 m/s will rebound with approximately what speed? A. S.5 mls B . 7.0 Tn/s C. 4.5 Tn/s D. 3.0 mls

Stop to Think 10.1: D. Since the child slides at a constant speed, hi s kinetic energy doesn' t change. But hi s gravitational potential energy decreases as he descends. It is transformed into thermal energy in the slide and hi s bottom.

Stop to Think 10.2: C. W = Fdcos8. The 10 N force at 90° does no work at al l. cos 60° = -1, so the 8 N force does less work than the 6 N force.

Stop to Think 10.3: B > D > A = C. K =( 1/2)/IIv2. Using the given masses and velocities, we find K A = 2.0 J, KB = 4.5 J, Kc = 2.0 J. Ko = 4.0 J.

Stop to Think 10.4: (Ugh > (Ugh = (Ug)~ > (Ug)l' Gravitational potenti al energy depends only on he ight, not speed.

Problems 321

Work and Power in Cycling

When yo u ride a bicycle at constant speed, a lmost all of the energy YO Ll expend goes into the work you do against the drag force of the air. In thi s problem, ass ume that all of the energy expended goes into working against drag. As we saw in Section 5.7, the drag force on an object is approximately proporti onal to the square of it s speed with respec t to the air. For thi s problem. assume that F ex v 2 exactly and that the air is motionless with respect to the ground un less noted otherwise. Suppose a cycl ist and her bicycle have a combined mass of 60 kg and she is cycl ing aJ ong at a speed of 5 mls.

80. I If the drag force on the cycli st is ION, how much energy does she use in cycl ing I km? A.6kJ B. 10kJ C. 50kJ D. 100kJ

8 1. I Under these conditions, how much power does she expend as she cycles? A. lOW B. SOW C. IOOW D. 200W

82. I If she doubles her speed to 10 mis, how much energy does she lise in cycling I km ? A. 20 kJ B. 40 kJ C. 200 kJ D. 400 kJ

83. I How much speed? A. IOOW

power does she expend when cycl ing at that

B. 200W C. 400 W D. IOOOW 84. I Upon reduc ing her speed back down to 5 mIs, she hits a

headwind of 5 m/s . How much power is she expending now? A. 100 W B. 200 W C. 500 W D. 1000 W

Stop to Think 10.5: D. The potential energy of a spring depends on the square of the displacement x, so the energy is pos iti ve whether the spring is compressed or extended. Furthermore, if the spring is compressed by twice the amount it had been stre tched, the energy will increase by a factor of 22 = 4. So the energy will be 4X IJ = 4J.

Stop to Think 10.6: Pu > PA = Pc > Pr>' The power here is the rate at which each runner's internal chemical energy is converted into gravitational poten tial energy. The change in gravitational potential energy is I1IgLly, so the power is mg LlylLl t. For runner A, the ratio 1II !!. )'/!!.1 equals (SO kg)( IO m)/( IO s) = SO kg ' m/s. For C. it 's the same. For B, it 's 100 kg · mis, while for D the ratio is 64 kg ' m/s.