too t'',*.11."': i MECHANICS

EX E R.C IS ES

1. To bring a supertanker to a stop, its engines are rypicallycut offabout 25 km from port. \Vhy is it so difficult tostoP or turn a suPertanker?

2. In terms of impulse and momentum, why do airbags incars reduce the chances ofinjury in accidents?

3. Vhy do gymnasts use floor mats that are very thick?4. In terms of impulse and momentum, why are nylon

ropes, which stretch considerably under tension, favored

by mountain climbers?

5. Automobiles were previously manufactured to be as rigidas possible, whereas todays autos are designed to crumpleupon impact. V4ry?

6. In terms of impulse and momentum, why is it importantthat helicopter blades defect air downward?

7. A lunar vehicle is tested on Earth at a speed of 10 km/h.V{hen it travels as fast on the Moon, is its momentummore, Iess, or the same?

8. ifyou throw a raw egg against a wall, you'll break iu but,if you throw it with the same speed into a sagging sheet, itwont break. Explain, using boncepts from this chapter.

9. If a ball is projected upward from the ground with10 Lg ' m/s of momentum, what is the momentum ofrecoil of Earth? 1X4ry do we not feel this?

10. \X&y do 6-ounce boxing gloves hit harder than 16-ounce

gloves?

1 l. A boxer can punch a heary bag for more than an hourwithout tiring, but will tire quickly when boxing with anopponent for a few minutes. \Mhy? (Hint: -ff{hen theboxer's fist is aimed at the bag, what supplies the impulseto stop the punches?

-fiIhen the boxert fist is aimed at the

opponent, what or who supplies the impulse to stop thepunches that dont connect?)

12. Railroad cars are loosely coupled so that there is a notice-able time delay from the time the first car is moved untilthe last cars are moved from rest by the locomotive. Dis-cuss the advisability ofthis loose coupling and slackbetween cars from the point ofview of impulse andmomentum.

13. You are at the front ofa floating canoe near a dock. Youjump, expecting to land on the dock easily. Instead, youland in the water. Explain.

14. A fully dressed person is at rest in the middle of a pond onperfectly frictionless ice and must get to shore. How canthis be accomplished? Explain in terms of momentumconservation.

15. If you throw a ball horizontally while standing on rollerskates, you roll backward with a momentum that matchesthat of the ball. Vill you roll baclcrvard ifyou go throughthe motions of throwing the ball, but instead hold on toit? Explain in terms of momentum conservation.

16. The examples of the two previous exercises can be

explained both in terms of momentum conservation andin terms of Newtont third law. Explain your answers toExercises 14 ar.d 15 in terms of Newtont third law.

17. In the previous chapter, rocket propulsion was explainedin terms of Newton's third law. That is, the force that pro-pels a rocket is from the exhaust gases pushing against therocket, the reaction to the force the rocket exerts on theexhaust gases. Explain rocket propulsion in terms ofmomentum conserYation.

18. Your friend says that the law of momentum conservationis violated when a ball rolls down a hill and gains momen-tum. V/hat do you say?

19. The momentum of an apple falling to the ground is notconserved because the external force ofgravity acts on it.But momentum is conserved in a larger system. Explain.

20. Drop a stone from the top of a high cliff, IdentiS, the sys-

tem wherein the net momentum is zero as the stone falls.

21 . Bronco dives from a hovering helicopter and finds his

momentum increasing. Does this violate the conservationof momentum? Explain.

22. An ice sail craft is stalled on a

frozen lake on a windless day.

The skipper sets up a fan as

shown. If all the wind bouncesbackward from the sail, will thecraft be set in modon? If so, inwhat direction?

23. \fil your answer to the preced-ing exercise be different ifthe airis brought to a halt by the sail without bouncing?

24. Discuss the advisabiliry of simply removing the sail in the

preceding exercises.

25. \(hen vertically falling sand lands in a horizontally mov-ing cart, the cart slows. Ignore any friction beween thecart and the tracks.Cive rwo reasons

for this, one interms of a horizon-tal force acting onthe cart and one interms of momen-tum conservation.

26.|n a movie, the hero jumps straight down from a bridgeonto a small boat that continues to move with no changein velocity. \(hat physics is being violated here?

27. Suppose that there are three astronauts outside a space-

ship and that they decide to play catch. All the astronautsweigh the same on Earth and are equally strong. The first

t

astronaut throws the second one toward the third one and

the game begins. Describe the motion of the astronauts as

the game proceeds. How long is the duration of the game?

28. To throw a ball, do you exert an impulse on it? Do you

exert an impulse to catch it at the same speed? About howmuch impulse do you exert, in comparison, if you catch itand immediately throw it back again? (Imagine yourself

on a skateboard.)29 . lf yo,ul. friend pushes a lawnmower four times as far as

you do while exerting only half the force, which one ofyou does more work? How much more?

30. \X&ich requires more work: stretching a strong spring a

certain distance or stretching a weak spring the same dis-

tance? Defend your answer.

31 . Two people who weigh the same climb a flight of stairs.

The first person climbs the stairs in 30 s, while the second

person climbs them in 40 s. \X&ich Person does more

work? V4rich uses more power?

32. V&en a rifle with a longer barrel is fired, the force ofexpanding gases acts on the bullet for a longer distance.

\7hat effect does this have on the velociry of the emerging

bullet? (Do you see why long-range cannons have such

long barrels?)

33.Yonr friend says that the kinetic energy ofan object

depends on the reference frame of the observer. Explainwhy you agree or disagree.

34.Yotand a fight attendant toss a ball back and forth in an

airplane in flight. Does the KE of the ball depend on thespeed of the airplane? Carefully explain.

35. A baseball and a golf ball have the same momentum.\,Mhich has the greater kinetic energy?

36. At what point in its motion is the KE of a pendulum bobat a maximum? At what point is its PE at a maximum?tWhen its KE is at half its maximum value, how much PE

r"W&.?Vfr?", % MOMENTUM AND ENERGY tOI

42. Corsider the identical balls released from rest on tracks Aand B, as shown. -When they reach the right ends of the

tracks, which will have the greater speed? -ff4ry is this

question easier to answer than the similar one (Exercise

31) in Chapter 3?

does it possess?

37. A physics instructor demonsffates energy

conservation by releasing a heary pendu-lum bob, as shown in the sketch, allowingit to swing to and fro. -{4rat would hap-pen if, in his exuberance, he gave the boba slight shove as it left his nose? Explain.

38. \Mhy does the force of gravity do work on a car that rollsdown a hill, but no workwhen it rolls along a level part ofthe road?

39. On a playground slide, a child has potential energy thatdecreases by 1000 J while her kinetic energy increases by900 J. \Mhat other form ofenergy is involved, and howmuch?

40. Discuss the design ofthe roller coaster shownin the sketch in terms ofthe conservation ofenergy.

41. Suppose that you and two classmates are discussing thedesign of a roller coaster. One classmate says that each

summit must be lower than the previous one. Your otherclassmate says this is nonsense, for as long as the first one

is the highest, it doesnt matter what height the others are.

\flhat do you say?

43. If a golf ball and a Ping-Pong ball both move rvith the

same kinetic energy, can )rou sav which has the greater

speed? Explain in terms of the definition of KE. Similarly,in a gas"ous mixture of massive molecules and light mole-

cules with the same average KE, can vou sa1'rvhich have

the greater speed?

4,tr. Does a car burn more gasoline u'hen its ligl-rts are turned

on? Does the overall consumption of gasoline deperrd on

x.hether or not the ensine is running r'r'hile the lights are

or-r? Defend \-our ans\\'er..ii. This mav seem like an easy question for a physics tvPe to

ans\\'er: \\tir1'r u"hat force does a rock that weighs 10 Nstrike the ground if dropped from a rest position 1 0 m

high? In irct, the question canlrot be ansrvered unless 1'ouhave nore infbrmation. Vhat information?

'i6. In the absence ofair resistance, a ball throrvn velticall)'ups'ard s'ith a certain initial KE r'vill return to its originalIer-el rvith the same KE. rWhen air resistance is a factor

atlicting the ball, rvill it return to its original level rvithrhe sami, less, or more KE? Does y'our ans\Yer contradict

the larv ofenergy conservation?4-. \bu're on a rooftop ar-rd you thror'"'one ball dou'nrvard to

the ground below and another uprvard. The second ball,

afier rising, falls and also strikes the ground belorv. Ifairresistance can be neglected, and if vour downr'r'ard and

uprvard initial speeds are the samer how rvill the speeds ofthe balls compare upon striking the ground? (Use the idea

ol'energl conservariotr to arrive al votlr ansiret')48. Vhen a driver applies brakes to keep a car going dor'vnhill

at constant speed arrd co.nstant kir-retic energ)i the poten-

tial energy of the car decreases. Where does this energr-

go? \ffhere does most of it appear in a h1'brid vehicle?

49. Does the KE of a car change more u'hen it goes from 10

to 20 km/h or r.vhen it goes from 20 to 30 km/h?

50. Can something have energ,v without having momentum?Explain. Can something have momentum r,r'ithout having

energv? Explain.51. Vhen the mass of a moving obiect is doubled with rro

change in speed, by what factor is its momentumchanged? By what factor is its kinetic energl' cl-ranged?

52. \[hen the r.elocity of an object is dor-rbled, by rvhat factor

is its momentum changed? 81'what factor is its kineticenergv changed?

53. V'hich, if either, has greater momentr.lm: a 1-kg ball mor-ing at 2 m/s or a 2-kg ball moving at 1 mi s? \(hich has

greater kinetic energy?

54. Iftrvo objects have equal kinetic energies, do thev neces-

sarily have the same momentum? Defend )rour answer.

4. Calculate the force of graviry berw^een Earth and the

Sun (theSun's mass : 2.0 X lOru kg;average Earrh-Sun

disrance : 1.5 X l01l m).5. Calculate the force of graviry beflveen a newbo-rn baby

(m,a,ss : 3Ig) and rhi planet Mars (mass : 6'4 x102r ks), when Mars is at its closest to Earth (distance :5.6 x"1010 m).

,.:'. l-ttt::,;,., t-,.; ,., GRAVITY, PROJECTILES, AND SATELTITES 129

6. Calculate the force of gravity between a newborn baby

of mass 3 kg and the obstetrician of mass 100 kg, who is

0.5 m from the baby. Vhich exerts more gravitationalforce on the baby, Mars or the obstetrician? By howmuch?

EXER(ISES

1. Comment on whether or not the following label on a

consumer product should be cause for concern.

CAUTION: The mass ofthis product pwlls on euery other

mass in tlte uniuerse, utith an attrdctingforce that is propor-tional to the product ofthe masses and inaersely proportionaltu the sqaare of the distance between them.

2. Gravitational force acts on all bodies in proportion to

their masses. \Vhy, then, doesnt a heavy body fall faster

than a light body?

3. Is the force ofgravity stronger on a piece ofiron than it is

on a piece of wood if both have the same mass? Defendyour answer.

4. Is the force of graviry stronger on a crumPled piece ofpaper than on an identical piece ofpaper that has notbeen crumpled? Defend your answer.

5. A'friend ,rp thrt "r,.o.r*rr,.

in orbit are weightless

because they're beyond the pull of Earth's graviry. Correctyour friendt ignorance.

6. Somewhere between Earth and the Moon, gravity fromthese two bodies on a space pod would cancel. Is thislocation nearer to Earth or to the Moon?

7. Is the acceleration due to graviry more or less atop Mt.Everest than it is at sea level? Defend your answer.

8. An astronaut lands on a planet that has the same mass as

Earth but twice the diameter. How does the astronaut's

weight differ from that on Earth?

9. An astronaut lands on a planet that has rwice Eartht and

rwice Eartht diameter. How does the astronaut's weightdiffer from that on Earth?

10. If Earth somehow expanded to a larger radius, with nochange in mass, how would your weight be affected? Howwould it be affected if instead Earth shrunk? (Hint: Letthe equation for gravitational force guide your thinki^ng.)

t t. A small light soulce located 1 m in f,ont of a 1 - m2 -

opening illuminates a wall behind. If the wall is 1 mbehind the opening (2 m from the light source), the illu-minated area covers 4 m2. How many square meters willbe illuminated if the wall is 3 m from the light source?

5m?10m?12. The intensity of light from a central source varies

inversely as the square ofthe distance. Ifyou lived on a

planet only halfas far from the Sun as our Earth, howwould the light intensity compare with that on Earth?

How about a planet ten times farther away than Earth?

13. The planet Jupiter is more than 300 times as masslve as

Earth, so it mieht seem that a body on the surface of

-lupiter rvould lr'eigh 300 times as much as it would weigh

on Earth. But it so happens that a bodl' would scarcelv

u.eigh three times as much on the surface of Jupiter as itr.'ould on the surface of Earth. Can you think of an

expianation fbr rvhv this is so? (Hint: Let the terms inthe equation fbr gravitational force guide,vourthinking.)

14. \\'hv do the passengers in hieh-altitude jet planes feel the

sensation of u'eight n-hile passengers in an orbiting space

vel-ricle. such as a space shuttle, do not?

i i. Iivou s'ere in a car that drove offthe edge ofa cliff, rvhy

u'ould vou be momentarilv rveightless? \fould gravitl' sti1I

be acting on vou?

16. \\'hat nr.o fbrces act ot1 vou rvhile you are in a movingelevator? \\then are these forces ofequal magnitude and

u'hen are thev not?17. Ifvou trere in a freely falling elevator and you dropped a

pencil, it u'ould hover in front of 1'sL' Is there a force ofgravin'acting on the pencil? Defend lour answer.

18. Your fi'iend says that the primary reason astronauts inorbit feel weightless is that they are beyond the main pullof Earth's gravity. Vhy do you agree or disagree?

19. Explain u,hy the foilowing reasoning is wrong' "The Srrn

attracts all bodies on Eartir. At rridnight, u'hen the Sun is

directly belorv, it pulls on you in the same direction as

Earth pul1s on vou; at noon, when the Sun is directlvoverhead, it pr-r11s or-r vou in a direction opposite to Earth's

pull on you. Therefore, 1'ou should be somer.vhrt heavier

at n.ridnight and somervhat lighter at noon."20. V'hich requiles more fuel-a rocket going from Earth to

the Moon ol a rocket coming fi'om the N4oon to Earth?

v'hy?2 i . Some people dismiss the validiry of scientific theories b,v

saying they are "on11l' theories. The lau' of universal

gravitatior.r is a theory. Does this mean that scientists stil1

doubt its validiry? Explain.22. Suppose you roil a ball offa tabletop. Will the time to hit

the floor depend on the speed of the ball? (\Mill a faster

ball take a longer tin're to hit the floor?) Defend youranswer.

23. A heaw crate accidentally f-a11s from a high-flying airplanejust as it flies directly above a shiny red Porsche smartlv

r30 t:,

".'1...':' . MECHANICS

parked in a car iot. Relative to the Porsche, where will thecrate crash?

24.In the absence of air resistance, why does the horizontalcomponent of a projectilet motion not change, while thevertical component does?

25. Atwhat point in its trajectory does a batted baseball haveits minimum speed? If air resistance can be neglected,how does this compare with the horizontal component ofits velocity at other points?

26. Afriend claims that bullets fired by some high-poweredrifles travel for many meters in a straightJine path beforethey start to fall. Another friend disputes this claim andstates that all bullets from any rifle drop beneath a

straight-line path a vertical disrance giienbylsl andthat the curved path is apparent for low velocities and less

apparent for high velocities. Now itt your rurn: \fi[ allbullets drop the same vertica.l distance in equal times?Explain.

27 . Two golfers each hit a ball at the same speed, but one at60'with the horizontal and the other at 30'. !7hich ballgoes farther? \flhich hits the ground first? (Ignore airresistance.)

28. A park ranger shoots a monkey hanging from a branch ofa tree with a tranquilizing dart. The ranger aims directlyat the monkey, not realizing that the dart will follow aparabolic path and thus will fall below the monkey. Themonkey, however, sees the dart leave the gun and lets goof the branch to avoid being hit. \7ill the monkey be hitanyway? Does the velociry of the dart affect your answer,assuming that it is great enough to travel the horizontaldistance to the ffee before hitting the ground? Defendyour answef.

29. Aprojectlle is fired straight upward at 141 m/s. Howfast is it moving ar rhe insranr it reaches the top of itstrajectory? Suppose instead that it were fired upward at45'. -ffhat would be its speed at the top of itstrajectory?

30. \[4ren you jump upward, your hang time is the time yourfeet are off the ground. Does hang time depend on yourvertical componenr of velociry when you jump, your hor-izontal component of velocity, or both? Defend youranswer.

31 . The hang time of a basketball player who .jumps a verticaldistance of 2 feet (0.6 m) is about 213 second. V4rat willbe the hang time if the player reaches the same heightwhile jumping4feer (1.2 m) horizontally?

32. Since the Moon is gravitationally attracted to Earth, whydoesnt it simply crash into Earth?

33. Does the speed of a falling object depend on its mass?

Does the speed of a satellite in orbit depend on its mass?

Defend your answers.

34.lf yon have ever watched the launching of an Earth satel-lite, you may have noticed that the rocket starts verricallyupward, then departs from avertical course and continuesits climb at an angle. \(hy does it start vertically? \X4-ry

' does it not continue vertically?35.If a cannonball is fired from a tall mountain, gravity

changes its speed all along its trajectory. But if it is firedfast enough to go into circular orbit, gravity does notchange its speed at all. Explain.

36. A satellite can orbit at 5 km above the Moon, but not at5 km above Earth. \(/hy?

37. \Mould the speed of a satellite in close circular orbitaboutJupiter be greater than, equal to, or less than8 km/s?

38. V4ry are satellites normally sent into orbit by firing themin an easterly direction, the direction in which Earthspins?

39. Of all the United States, why is Hawaii the most efficientlaunching site for nonpolar satellites? (Hint: Look at thespinning Earth from above either pole and compare ir to aspinning turntable.)

40. Earth is closer to the Sun in December than it is in June.In which of these rwo months is Earth moving fasteraround the Sun?

41. \7[rat is the shape of the orbit when the velociry of thesatellite is everywhere perpendicular to the force ofgraviry?

42. A comm,tnications satellite with a 24-hosr period hoversover a fixed point on Earth. \X4ry is it placed in orbit onlyin the plane of Eartht equator? (Hint: Think of the satel-lite's orbit as a ring around Earth.)

43.If a flight mechanic drops a wrench from a high-flyingjumbo jet, it crashes to Earth. If an astronaut on theorbiting space shuttle drops a wrench, does it crash toEarth also? Defend your answer.

44. How could an astronaut in a space shuttle "drop" anobject vertically to Earth?

45. If you stopped an Earth satellite dead in its tracks, itwould simply crash into Earth. Why, then, dont the com-munications satellites thar "hover motionless" above thesame spot on Earth crash into Earth?

46. The orbital velocity of Earth about the Sun is 30 km/s. IfEarth were suddenly stopped in its tracks, it would simplyfall radially into the Sun. Devise a plan whereby a rocketloaded with radioactive wastes could be fired into the Sun

for permanent disposal. How fast and in what directionwith respect to Earth's orbit should the rocket be fired?

47 . In an accidental explosion, a satellite breaks in half whilein circular orbit about Earth. One half is broughtmomentarily to rest. \7hat is the fate of the half broughtto rest? \7hat happens to the other hal0

48. If Pluto were somehow stopped short in its orbit, it wouldfall into, rather than around, the Sun. How fast would itbe moving when it hit the Sun?

49. At which of the indicated positions does the satellite inelliptical orbit experience the greatest gravitational force?

Have the greatest speed? The greatest velociry? The great-

est momentum? The greatest kinetic energy? The greatest

gravitational potential energy?

PRSmLmftfi$

1. S Consider a pair of planets for which the distance

between them is decreased by a factor of 5. Show that the

force beween them becomes 25 times greater.

2. g Many people mistakenly believe that the astronauts

who orbit Earth are "above gravity." Earth's mass is

6 x lO24 kg, and its radiusls 6.38 X 106 m (6380 km).Use the inverse-square law to show that in space-shuttle

territory, 200 kilometers above Eartlis surface, the force

of gravity on a shuttle is about 94Vo of thar. at Earthtsurhce.

3. 3 The mass of a certain neutron star is 3.0 x t030 kg(1.5 solar masses) and its radius is 8000 m (8 km). Show

that the force ofgravity at the surface ofthis condensed,

burned-out star is about 300 billion times that of Earth.4. a Aball is thrown horizontally from a cliffat a speed of

10 m/s. Show that its speed one second later is 14.1 m/s.

5. n An airplane is flying horizontally with speed 1000

km/h (280 m/s) when an engine falls off. Neglecting airresistance, assume it takes 30 s for the engine to hit theground.a. Show that the altitude of the airplane is 4500 m.b. Show that the horizontal distance that the aircraftengine falls is 8400 m.c. If the airplane somehow continues to fly as if nothinghad happened, where is the engine relative to the airplaneat the moment the engine hits the ground?

6. # A cannonba.ll shot with an initial velocity of 141 m/s at

an angle of 45' follows a parabolic path and hits a balloonat the top ofits trajectory. Neglecting air resistance,

show that the cannonball hits the balloon at a speed of100 m/s.

7. * Acertain satellite has a kinetic energy of 8 billionjoules at perigee (the point at which it is closest to Earth)and 5 billion joules at apogee (the point at which it is far-thest from Earth). As the satellite travels from apogee toperigee, how much work does the gravitational force do

on it? Does its potential energy increase or decrease dur-ing this time, and by how much?

,;, GRAVITY. PROIECT LES. AND SATELLITES

The greatest total energl.? The greatest acceleration?B

50. A rocket coasts in an elliptical orbit around Earth. Toattain the greatest arnount ofKE for escape using a given

amount of fuel, should it fire its engines at the aPogee orat the perigee? (Hint: Let the formula Fd : LKE be yourguide to thinking. Suppose the thrust.Fis brief and of thesame duration in either case. Then consider the distance

d the rocket would travel during this brief burst at the

apogee and at the perigee.)

BEGINNER ii:i INTERMEDIATE I EXPERT

8. . The fbrce that pulls moving objects into circular paths

is caIled centripetalJbrcr and is given bv the equation17tU-

F, - -:;, where zei is the mass of an object moving in .rd

circular path at speed z and distirnce / from the center ofthe circular path. For the Moon circling Earth, glar-irr-

supplies the cer-rtripetal force. Equate centripetal force togravitational fbrce and shorv t1.rat rhe speed of the Moor.r

Telrin its orbit about Earth is zr :

, , , where M is the

mass oiEarth and /is t['re distance between the centers ofthe -\{oor.r and Earth.

c). :: Calculate the speed in m/s at which Earth revolves

about the Sun. You firav assume the orbit is nearly circular'19. ,,', The lv{oon is about 3.8 X 10) km from Earth. Shou'

that its average orbital speecl about Earth is 1026 m/s.1 1. I The force-of graviry o,, yo., b1' Earth is GtnMl d2,

rvhere G is the universal gravitatior-ral constant, rz is t'ottrmass,.Mis the mass of Earth, and /is ]'our distance frot-t-t

Earth's cer-rter.

a. Use Nervton's second laro, to show- that your gravira-

tional acceleration tor.vard llarth at clistance / from itscentcri\ a - LMld:.b. Horv does this equation slrpport tbe finding that the

acceleration due to gravitv doesn't depend on the mass ofan objecr in frec lalll

12. r) The gravitational field about a massive object is

defir-red to be the gravitational force per mass on an objectin the vicir-rity of the massive ob ject. -l-he s,vmbol for the

gravitational field is boldfaced g (rvith magnitude the

same as the rnagnitude of gravitational accelelation at

that point, g).a. Show that the gravitational field a distanc e d fromEarth's center it Clltld2, rvhere G is the universal gravita-

tionai constant and -M is the mass of F,arth.

b. The value ofg at Earth's surface is about 9.8 N/kg.Show that the value ofg at a distance from Earth's center

that is four times Earth;s radius would be 0.6 m/s2.

6-?41r.?:) ?;'t/. lll