16.

1.7.

48 CHAPTER 2 ONE-DIMENSIONAL KINEMATICS

A person on a trainpoline bounces straight upr,vard r,t'ith an ini-tial speed of 4.5 m/s. What is the person's speed u'hen she re-turns to her initial height?

After winning a baseball game, one player drops a g1ove, whileanother tosses a glorre into the air. How do the accelerations o{the two glor-es compare?

SECTION ?-I POSITIOhI, DISTANEE, ANED'5P!.ACEMTNT

1. . Referring to iiglre :*;$, you walk from your home to the li-brary, then to the park. (a) What is the distance traveled? (b)What.is your displacement?

f i&L{ef; t*3S Problems 1 and 4

o The two tennis players shown in fig.*r* ?*lr! walk to the net tocongratulate one another. (a) Find the distance traveled and thedisplacement of player A. (b) Repeat for player B.

P*i*{,$*.f 3"-t$ Problem 2

. The golfer in 9:ii*r* T..':l sinks the ball in two putts, as sholt'n.What are (a) the distance traveled by the ball, and (b) the dis-placement of the ball?

r-rr--itFr->;

Second pr-rtt ,t+.

18. A volcano shoots a lava bomb straight upward. Does the dis-placement of the lava bomb depend on (a) y6u11 choice of originfor your coordinate system, or (b) your choice of a positive di-rection? Explain in each case.

c In Figure 2-20, you r,r,alk from the palk to your friend's house,then back to yonr house. What is your (a) distance traveled, and(b) displacement?

c A jogger runs on the track shown in {:!gr;r* ?*?3. Negiecting thecLuvature of the corners, (a) what is the distance traveled andthe displacement in running from point A to point B? (b) Findthe distance and displacement for a complete circuit of the track.

r.- 100

ptaffi &Lgtu{$ &F'4m e#f-{{fr pTaiAL €x Fme ! 585

nre used to irtdicate thc leael of dfficttitty.

(The effects of.air resistance are to be ignored in this chapter.)

4.

rlSH** **3S Problem 5

6. '. *F A child rides a pony on a circular track r,t'hose radius is4.5 m. (a) Find the distance traveled and the displacement afterthe child has gone halfway around the track. (b) Does the dis-tance traveled increase, decrease, or stay the same when thechild completes one circuit of the track? Explain. (c) Does thedisplacement increase, decrease, or stay the same r,t-hen the childcompletes one circuit of the track? Explain. (d) Find the distanceancl displacement after a cornplete circuit of the track.

SEETIOI{ 2_2 AVERAGE SPEED AND V€LO{ITY7. . e6. Fredict/fixplain You drive your car iu a straight line at

15 m/s for 10 kilometers, then at25n;r/s for another 10 kilome-ters. (a) Is your average speed for the entire trip more than, lessthan, or equai to 20 m/s? (b) Choose ihe bcsf explanation fromamong the following:

I. More time is spent at 15 m/s than at 25 n/s.II. The average of 15 m/s and 25 m/s is 20 m/s.

III. Less time is spent at 15 m/s than at 25 m/s.

8. " CE PredictlExplain You drive your car in a straight line at15 m/s for 10 minutes, then at 25 n/s for another 10 minutes.(a) Is your at'erage speed fol the entire trip more than, less than,or equal to 20 m/s? (b) Choose the besf explanation from amongthe follou,'ing:

L Nlore time is required to drive at 15 m/s than at 25 n/s.II. Less distance is covered at 25 m/s than at 15 rn/s.

III. Equal time is spent at 15 m/s and 25 m/s.

9. " Joseph Deloacll of the United States set an Olympic record in1988 {or the 200-meter dash with a time of 19.75 seconds. Whaiwas his average speed? Gil.e your answer in meters per secondand miles per hour.

10. r Jn 1992 Zhuang Yoirg of China set a women's Olympic recordin the 100-meter freestvle sr,rrim with a time of 54.64 seconds.Wh;rt r.vas her averap;e speed ln m/s and milh?

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: ! ,3orn...rBr \_-*:._- !

2.

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S*{:}#R* t-t? Problem 3

:1. . BIO Kangaroos have been clocked at speeds of 65 km/h.(a) How far can a kangaroo hop in 3.2 minutes at this speed?(b) How long will it take a kangaroo to hop 0.25 km at this speed?

I l3cxbhey $weks; A severe storm on Janu ary 70,1992, caused a

cargo ship near the Aleutian Islands to spi1l 29,000 rubber.1ucks and other bath toys into the ocean. Ten months later hun-.lreds o{ rr-ibber ducks began to appear along the shoreline nearSrtka, Alaska, roughly 1600 miles away. What was the approxi-mate average speed o{ the ocean current that carried the ducksio shore in (a) m/s and (b) mi/h? (Rubber ducks from the samespi1l began to appear on the coast of Maine in July 2003.)

r Radio waves travel at the speed of light, approximately-86,000 rniles per second. How long does it take for a radio:nessage to travel from Ealth to the Moon and back? (See the in-.ide back cover for the necessary data.)

r lt was a dark and stormy night, when suddenly you saw a:1.rsh of iightning. Thlee-and-a-half seconds later you heard the:lrunder. Given that the speed of sound in air is abor-rt 340 m/s,.,ow far away was the lightning bolt?

. BIO Nq}rv* 3*r.pxRs*s The human nervous system can prop-.:gate nerve impulses at about 102 m/s. Estimate the time it:akes for a ner\re impulse generated when your finger touches a::ot object to travel to your brain.

. Estimate how fast your hair grotvs in miles per hour.

. . A finch rides on ihe back of a Galapagos tortoise, which'.'alks at the stately pace of 0.060 m/s. After 1.2 minutes the:nch tirc--s,of the tortoise's slow pace, and takes flight in the sameiirection for another 1.2 minutes at72n/s. Whatwas the aver-,'.ge speed of the finch for this 2.4-minute lnterr.al?

. . You jog at 9.5 km,u h for 8.0 km, then you jump into a car and,irir.e an additional 16 krn. With n hat ar.erage speed must you3rir.e your car if your average speed for the entire 24 km is to bell km/h?. . A dog runs back and forth between its tu'o orvvners, who are',ialking toward one another (frig*re 3*?4). The dog starts lun-:-,irlg when the owners are 10.0 m apart. If the dog runs with.'. speed of 3.0 m/s, at-rd the owners each walk with a speecl of-.3 m/s, how far has the dog traveled when the owners meet?

1.3 m/sj:, *-.*!-

ii.*.3.0 m/sd€Fw

,,,tt€

1.3 m/s cFl"€*-,1tr{

@

10.0 m

ri&#ffi.ffi X*X*!i Problem 19

. . *F You drive in a straight line at 20.0 m/s for 10.0 minutes, then.:i 30.0 m/s for another 10.0 minutes. (a) Is your average speed 25.0::r/s, more than 25.0 m/s, or less than 25.0 m,/s? Explain. (b) Ver-n- your answer to part (a) by calculating the a\rerage speed.

r o ln heavy rush-hour traffic you drive in a sh'aight line at 12 m/s-.rr 1.5 milrutes, then you have to stop for 3.5 minutes, and finallyi ou drirre at 15 m/s fc'rr another 2-5 rninutes. (a) Plot a position-'. ersus-time graph for this motion. Your plot should extend from. : 0 to t - 7 .5 minutes. (b) Use your plot from part (a) to calcu-laie the average velocity between / - 0 and t : 7.5 minutes.

. . $lit You drive in a straight line at 20.0 m/s for 10.0 miles, thenat 30.0 m/s for ar-rother 10.0 miles. (a) Is your average speedll.tl m/s, more than 25.0 m/s, or less than 25.0 m/s? Explain. (b)

\eritv your clr'rs\v€i' b part (a) bv calculatir-rg the average speed.

PROBLEMS AND CONCEPTUAL EXERCISES 49

. . *F Ar expectant father paces back and forth, producing the po-sition-versus-time graph shov,n in liigun* t*3$. Without perform-ing a calculatiory indicate whether the father's velocity is positive,negative, or zero on each of the following segments of ttre graptr:(a) A, (b) B, (c) C, and (d) D. Calculate the numerical value of thefather's velocity{or the segments (e) A, (0 B, (g) C, and (h) D, andshow that your lesults verify your arlswers to parts (a)-(d).

25.

orz345Time, f (s)

P!€WKffi 3*fti$ Problem23

.. The position of a palticle as a function of time is given byir : (-5 m/s)/ + (3 m/s2)t2. (a) Plot .r versus f for / : 0 tot : 2 s. (b) Find the average velocity of the particle from f : 0

to f : 1 s. (c) Find the averagespeedfrom f : 0 to I - 1 s.

. . The position of. a particle as a function of time is given byr : (6 m/s)l + (-2 mls2)t2. (a) Plot r versus f for / - 0 tot - 2s. (b) Find the average velocity of the particle from I : 0

to f - 1 s. (c) Find the averagespeedfrom / : 0 to I : 1 s.

.. &g!* A tennis player moves back and forth along the baselinewhile waiting for her opponent to serve, producing the position-versus-time graph shown in S!Xi;*n*:*3S. (a) Without per{orminga caiculation, indicate on which of the segments of the graph, A,B, or C, the piayer has the greatest speed. Calculate the player'sspeed for (b) segment A, (c) segment B, and (d) segment C, andshow that your results verify your answers to part (a).

o12345Time, f (s)

Sll&l$*.& 3*36 Problem 26

27. . . . On your wedding day you leave for the church 30.0 min-utes before the ceremony is to begin, which should be plenty oftime since the church is only 10.0 miles away. On the way, how-ever, you have to make an unanticipated stop for constructionu'ork on the road. As a result, yollr average speed for the first 15

minutes is only g.Q mi/h. What average speed do you need forthe rest of the trip to get you to the church on time?

23.

24.

26.

e

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kCj

o,

50 CHAPTER2 ONE.DIMENSIONALKINEMATICS

SECTION 2-3 INSTANTANEOUS VELOCITY

28. r €E The position-versus-time plot of a boat positioning itselfnext to a dock is shown in F$;;gur* ?*37. Rank the six points indi-cated in the plot in order of increasing value of the velocity o,

starting with the most negative. Indicate a tie with an equalsign.

F!l$#$*.fi ?*3? Problem 28

29. .. The position of a particle as a function of time is given byt - (2.0 m/s)f + ( 3.0 m/s3)t3. (a) Plot r versus I for time fromt - 0 to / : 1.0 s. (b) Find the average vekrcity of the particle'from f - 0.35 s to t - 0.45 s. (c) Find the average velocity from/ - 0.39 s to / - 0.41 s. (d) Do you expect the instantaneous ve-locity at t - 0.40 s to be closer to 0.54 m,/s, 0.56 rn/s, or 0.58

m/s? Explain.

30. .. The position of a particle as a function of time is given bvr : ( 2.00 m/s)/ + (3.00 m/s3)t3. (a) Plot x versus I for timefrom I : 0 to I : 1.00 s. $) Find the average velocity of theparticle from / - 0.150 s to t - 0.250 s. (c) Find the arrerage ve-locity from / - 0.190 s to / - 0.210 s. (d) Do you expect the in-stantaneous velocity at / : 0.200 s to be closer to -I.62 m/s, or

L.66 nls? Explain.

SECTION 2-4 ACCELERATION

31. . CE FredictlExplain Two bov('s shoot identical arrows lr,iththe same launch speed. To accomplish this, the string in bow 1

must be puiled back farther when shooting its arrow than thestring in bow 2. (a) Is the acceleration of the arrow shot by bow1 greater than, less thdn, or equal to the acceleration of thearrow shot by bow 2? (b) Choose the best explanation kornamong the following:

I. The arrow in bow 2 accelerates for a greater time.II. Both arrows start from rest.

III. The arrow in bow 1 accelerates for a greater time.

32. . A 747 airliner reaches its takeoff speed of 773 lr.i/h in 35.2 s.

What is the magnitude of its average acceleration?

33. r At the starting gun, a rr-rnner accelerates at 1.9 m/s2 for 5.2 s. Thenrnner's acceleration is zero for the rest of the race. \Alhat is thespeed of the runner (a) at / - 2.0 s, and (b) at the end of the race?

34. n A jet makes a landing traveling due east with a speed of115 m/s. If the jet comes to rest in 13.0 s, n'hat are the magni-tude and direction of its average acceleration?

35. . A car is traveling due north at 18.1 m/s. Find the velocity ofthe car after 7.50 s if its acceleration is (a) 1.30 m/s2 due north,or (b) 1.15 m/s2 due south.

36. .. A motorcycle moves according to the velocity-versus-timegraph shown in fl{gi*re ?-tf3. Find the average acceleration of themotorcycle during each of the following segments of the mo-tion: (a) A, (b) B, and (c) C.

o510 15

37.

Time, f (s)

d{{tsRg } ?$ Problem 3o

.. A person on horseback moves according to the velocity-versus-time graph shown in fi6ur* t*?$. Find the displacementof the person for each of the following segments of the motion:(a) A, (b) B, and (c) C.

o51015202sTime, t (s)

F9&8.t$qfi. t-*$ Problem 17

38. r r Running with an initial velocity of +11 m/s, a horse has anaverage acceleration of -1.81 m/s2. How long does it take forthe horse to decrease its velocity to +6.5 m/s?

39. r. $P Assume that the brakes in your car create a constant decel-eration o[ 4.2 m >) regardless of fiow fasr you r]re driving. lf youdouble your driving speed from 1,6 m/s to 32 rn/s, (a) does thetime required to come to a stop increase by a factor of two or a fac-tor of four? Explain. Verify your answer to part (a) by calculatingthe stopping times for initial speeds of (b) 16 m/s and (d 32rn/ s.

40. . . g P In the previous problem, (a) does the distance needed to

. stop increase by a factor of two or a factor of four? Explain. Ver-ify your answer to part (a) by calculating the stopping distancesfor initial speeds of (b) 16 m/s and (c) 32 m/s.

41. .. As a train accelerates away from a station, it reaches a speedof 4.7 m/ s in 5.0 s. If the train's acceleration remains constant,what is its speed after an additional 6.0 s has elapsed?

42. .. Aparticle has an acceleration of +6.21nf s2 for 0.300 s. Atthe end of this time the particle's velocity is +9.31 m/s. Whatwas the particie's initial velocity?

SECTION 2-5 MOTION WITH CONSTANTACCELERATION

43. . Landing with a speed of 81.9 m/s, and traveling due south, a

jet comes to rest in 949 m. Assuming the jet slows with constantacceleration, find the magnitude and direction of its acceleration.

44. r When you see a traffic light turn red, you apply the brakesuntil you come to a stop. If your initial speed was 12 m/s, and

15

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A9-O

2520

Is4ou

)

\.ou were heading due west, what was your average velocitvcluring braking? Assume constant deceleration.

-,i. Cf, € e Abal1 is releascd at the point r : 2 m on an inclined planervith a nonzero initial velocity. Aftel being released, the bailmor.es with constant acceleration. The acceleration and ilitial ve-iocity of the ball are described by one of the follo'iving four cases:

.ase 7, n > 0, zts > 0; case 2, n '> 0, -ttg < 0; case 3,

., { 0, ro } 0; case 4,n < 0, re { 0. (a) Irrwhich of these cases

rli11theballdefinitelypassr - 0 atsomelatertime? (b) Inwhich..i ihese cases is more information needed to determine lvhether:he ball will ctoss r : 0? (c) In r'r'hicfi of these cases'"vill tlre ball.rrme to l:est momentarily at some time during its motlon?

-: . r Suppose the car in Problem 44 comes to rest in 35 m. How:nuch time does this take?

- - r r Starting from rest, a boat increases its speed to 4.12 m/s \\,ith:.rnstant acceleration. (a) \\4rat is the boat's average speed? (b) If:: tz'rkes the boat 4.77 s to reach this speed, hol'v {ar has it traveled?

- r l r l$i gd$ A cheetah can accelerate from rest to 25.0 m/s in 6.22 s.

..ssr-uning constalt acceleration, (a) how {ar has the cheetah rr-rn il:-ris time? (b) After spritrting for just 3.11 s, is the cheetah's spe'ed

1.5 m /s, more than I2.5 m / s, or less than 72.5 m / s? Explain. (c)

\1 at is the cheetah's average speed for the first 3.11 s of its sprint?:.rr the secc'rnd 3.11 s of its splint? (d) Calculate the distance gor--:ed by the cheetah in the first 3.11 s and the second 3.11 s.

;::Tl**€ 2*6 APFLE{ATqGNS SF THEr iUA35*FqS *F tut*Tl*N

. \ child slides down a hill on a toboggarr with an acceleration, 1.8 m/s2. If she starts at rest, how far has she trar-eled ina) 1.0 s, (b) 2.0 s, and (c) 3.0 s?

: . : r'ie -}*{ercrs*g;r On a ride called the Detonator ai lVorlds of: -n in Kansas City, passengers acceierate straight dor,r'nward::.rrn rest to 45 mi/h in 2.2 seconds. What is the average accel-.:.rtion of the passengers orr this rlde?

The Detonator (ProbLem 50)

. :,.!r #;a&s Air bags are designed to deploy in 10 ms. Estimate::-c acceleration of the front surface of the bag as it expands. Ex-

--r.ess your answer in terms of the acceleration of gravity.g.

- ' .r:,r!** V*rc*+: In his novel From tlrc Eartlt to the Moon (7866),

,:les Verne describes a spaceship that is blasted out of a cannon,

-.:11ed the Coluntbiad,'i,r'ith a speed of 12,000 yards/s. The

-.,lutnbiad is 900 ft long, but part of it is packed w,ith powder', scr

::'e spaceship accelerates over a distance of only 700 {t. Estimate. -.e acceleration experienced by the occupants of the spaceshipr

-: -rring launch. Give your ans\ver in m/s2. (Verne realized that

PROBLEMS AND CONCEPTUAL EXERCISES 5I

the "travelers r,r.ou1d . . . encounter a violent recoil," but he prob-ably didn't know that people generally lose consciousness ifthey experlence accelerations greater tiran about 7 g - 70 nl s2 .)

.. FIG S;l*l*riai &f$tiqltt Approximately 0.1% of the bacteriain an adult human's intestines arc Escherichia coll. These bacte-ria have been observed to move rt ith speeds up to 15 pm/s andmaximum accelerations of 766 p.mf s2. Suppose an E. colibac-telir-rm in your intestines starts at rest and accelerates at156 pm/s2. How much (a) time and (b) distance are requiredfor the bacterium to reach a speed of 12 prnf s?

c s Two cars drive on a straight highrt'ay'. At time f : 0, car1 passes mile marker 0 traveling due east with a speed of20.0 m/s. At the same time, car 2 is 1.0 km east of mile malker0 traveling at 30.0 m/s due west. Car 1 is speeding up u,.ith anacceleration of magnitude 2.5 nf s2, and car 2 is slowing downr.vitlr an acceleration of magnitude 3.2 n f s2 . (a) Write r-versus-f equations of motion for both cars, taking east as the positivedirection. (b) At what time do the cars pass next to one another?

c 6 A n€e{siir{t* $tu"!le*r On October 9, 1992, a 27-pould mete-orite struck a car il Peekskill, Nl leaving a dent 22 crn deep in thetrr-rnk. If the meteorite struck the cal with a speed of 130 m/s, what'r'r'as the magnitude of its deceleratlon, assuming it to be constant?

" c A rocket blasts off and moves straight upwald from the1aunc1-r pad lr'ith constant accelcration. After 3.0 s the rocket is ata height of 77 m. (a) What are the magnitude and direction of therocket's acceleration? (b) What is iis speed at this tirne?

denly a ball ro11s out in front of you. You apply the brakes andbegin decelerating at 3.5 m1s2. (a) How far do you travei beforestopping? (b) lVhen -vou have trar.eled cmly half the distance inpart (a), is your speed 6.0 rn/ s, greatel than 6.0 m/s, or less thanLr.U m r sl Srrpport ) our answ('r with a calculation.

o o lf You are driving through to$.n at 76 m/ s when suddenlya car backs out of a driveway in front of you. Yor-r apply thebrakes and begin decelerating at 3.2 m1s2. (a) How much tirnedoes it ti:rke to stop? (b) After braking half the time found inpart (a), is yor-rr speed 8.0 m/s, greater than 8.0 m/s, or less than8.0 m/s? Support your ans\ver u'ith a calculation. (c) If the carbacking out was initially 55 m in front of you, what is the max-imum reaction time you can have before hitting the brakes andstiil avoid hiiting the car?

r o *P BEO r1?tr*gr;*{* &*e*1*::a*i$as \Mhen a chameleon capturesal insect, its tongue can extend 16 cm in 0.10 s. (a) Find the margni-tude of the tongue's acceleration, assuming it to be constant. (b) ful

the first 0.050 s, does the tongue extend E.0 cm, more than 8.0 cm,or less than 8.0 cm? Support your conclusion with a calculatic'rn.

53.

54.

55.

56.

58.

59.

1.

Iit,Iti-

It's not polite to reachl (Problenr 59)

52 CHAPTER2 ONE.DIMENSIONALKINEMATICS

60. .. *p Coasting due west on your bicycle at 8.4 m/s, you en-counter a sandy paich of road7.2 m across. When you leavethe sandy patch your speed has been reduced by 2.0 m/s to6.4 m/ s. (a) Assuming the sand causes a constant acceleration,what was the bicycle's acceleration in the sandy patch? Giveboth magnitude and direction. (b) How long did it take to crossthe sandy patch? (c) Suppose you enter the sandy patch with a

speed of only 5.4 m/s. Is.your final speed in this case 3.4 m/ s,

more than 3.4 m/s, or less than 3.4 m/ s? Explain.

61. . n BIO $xrvivix*g x L;arge Xleeelerat*** On |u1y 13, 1977, whlleon a test drive at Britain's Silverstone racetrack, the throttle onDavid Purley's car stuck wide open. The resulting crash subjected

Purley to the greatest't-force" ever survived by a human-he de-

celerated kom 173 km/h to zero in a distance of only about0.66 m. Calculate the magnitude of the acceleration experiencedby Purley (assuming it to be constant), and express yor.t answer inunitsof theaccelerationof graviryg - 9.8L mf s2.

62. o . $$$ A boat is cruising in a straight line at a constant speed of2.6 m/ s when it is shifted into neutral. After coasting 12 m theengine is engaged again, and the boat resumes cruising at the re-

duced constant speed of 1.6 n/s. Assuming constant accelera-

tion while coasting, (a) how long did it take for the bbat to coast

the 12 m? (b) \A4rat was the boat's acceleration while it wascoasting? (c) When the boat had coasted for 6.0 m, was its speed2.7 rn/s, more than 2.1rr./s, or less than 2.1 m/s? Explain.

63. r. Amodel rocket rises with constant acceleration to a height of3.2ll.:r, at which point its speed is 26.0 m/s. (a) How much timedoes it take for the rocket to reach this height? (b) What was themagnitude of the rocket's acceleration? (c) Find the height andspeed of the rocket 0.10 s after launch.

64. . " The infamous chicken is dashing toward home plate with a

speed of 5.8 m/s when he decides to hit the dirt. The chickenslides {or 1.1 s, just reaching the plate as he stops (safe, ofcourse). (a) What are the magnitude and direction of thechicken's acceleration? (b) How far did the chicken slide?

65. .. A bicyclist is finishing his repair of a flat tire when a friendrides by with a constant speed of 3.5 m/s. TWo seconds later thebicyclist hops on his bike and actelerates at 2.4m/s2 until he

catches his friend. (a) How much time does it take until he catcheshis friend? (b) How far has he traveled in this time? (c) \Atrat is hisspeed when he catches up?

56. ". A car in stop-and-go traffic starts at rest, moves forward13 m in 8.0 s, then comes to rest again. The velocity-versus-timeplot for this car is given in Flgare ?*3S" What distance does thecar cover in (a) the first 4.0 seconds of its motion and (b) the last2.0 seconds of its motion? (c) What is the constant speed V thatcharacterizes the middle portion of its motion?

o24Time (s)

sE{}!|$&.K X*31{} Problem 66

rco dgsl and a truck are heading directly toward one anotheron a straight and narrow street, but they avoid a head-on colli-sion by simultaneously applying their brakes at I - 0. The re-sulting velocity-versus-time graphs are shown in Flgurc 2*3i.What is the separation between the car and the truck when theyhave come to rest, given that at / : 0 the car is at x : 15 m andthe truck is at .x : -35 m? (Note that this information deter-mines which line in the graph corresponds to which vehicle.)

Fi4URg ?-3: lrobiem o7

e r o 11 2 physics lab, students measure the time it takes a smallcart to siide a distance of 1.00 m on a smooth track inclined at anangle 0 above the horizontal. Their results are given in the {ol-lowing table.

0 10.00 20.00 30.0'

time, s 1.08 0.770 0.640

(a) Find the magnitude of the cart's acceleration for each ang1e.(b) Show that your iesults for part (a) are in close agreementwith the formula, a - g sin 6. (We will derive this formula inChapter 5.)

sEcTtoN 2-7 FREELY FALLTNG OBJECTS

'\f Qa# ?f)42t4 Z€A{ 6 st{r/ r'\r ^e{r{JI{.{aLE 5€(t^j}5-"

67.

-10

-15

68.

>\

oO

u (m/s)

(Probtems 71 and72)

. CE At the edge of a roof you throw ball 1 upward with an ini-:.al speed osi d mom€nt later you throw bail 2 downward with:re same initial speed. The balls land at the same time. Which of-re following statements is true for the instant just before the:alls hit the ground? A. The speed of ball 1 is greater than the.'-.eed of ball 2; B. The speed of ball 1 is equal to the speed of balll: C. The speed of ball 1 is less than the speed of ball 2.

. Legend has it that Isaac Nelt'ton was hit on the head by a falling,'-3p1e, thus triggering his thoughts on gravity. Assuming the story:.r be true, estimate the speed of the apple when it struck Newton.

. The cartoon shows a car in free fall. Is the statement made in

.re cartoon accurate? Justify your answer.

. Referring to the cartoon in Problem 71, how long would it

.-:ke for the car to go ftom 0 to 30 mi/h?

. ,iorrc9**os $wleap Michael Jordan's vertical leap is reported to:'e -lB inches. What is his takeoff speed? Give your answer inleters per second.

. BIO Gulls are often observed dropping clams and other.r.elLfish from a height to the rocks below, as a means of open--r.g the shells. If a seagull drops a she1l from rest at a height of-- m, how fast is the shel1 moving when it hits the rocks?

. \ r.olcano launches a lava bomb straight upward with an ini-

. :l speed of 28 m/s. Taking upward to be the positive direc-

. ..n, find the speed and direction of motion of the lava bomb (a)

1.,-r seconds and (b) 3.0 seconds after it is launched.

. Ar: fix€xete:;res€r*a€ &"93*e;sii::* The first active volcano ob-.:n-ed outside the Earth was discovered in 7979 on Io, one of-re moons of Jupiter. The volcano was observed to be ejecting::aierial to a height of about 2.00 ,, 105 m. Given that the accel-

=:ation of gravity on Io is 1.80 m/s2, find the initial velocity of::-re ejected material.

. BIO &4eas$y* Y*:tir R.eiaeti&s* l*lime Here's something you,,r try at home-an experiment to measure your reaction time.

lave a friend hold a ruler by one end, letting the other endr.ang down vertically. At the lower end, hold your thumb and:idex finger on either side of the ru1er, ready to grip it. Have. our friend release the ruler without warning. Catch it as:-'-iickly as you can. If you catch the ruler 5.2 cm from the lower

=rd, what is your reaction time?

How fast are your reactions? (Problem 77)

o o (E Predict/Explain A carpenter on the roof of a building.rccidentally drops her hammer. As the hammer falls it passes

PROBLEMS AND CONCEPTUAL EXERCISES 53

two windows of equal height, as shown in fligxr* ?^3?, (a) Is theincrense in speed of the hammer as it drops past window 1

greater than, less than, or equal to the increase in speed as itdrops past window 2? (b) Choose the besl explnnation Irornamong the following:

I. The greater speed at window 2 results in a greater increasein speed.

II. Constant acceleration means the hammer speeds up thesame amount for each window.

III. The hammer spends more time dropping past window 1.

PX{:}Wffi;n }*S:A Problem 78

r. Cf Predict/f;xplain !:igrxr* ?*33 shows a z-versus-f plot forthe hammer dropped by the carpenter in Problem 78. Noticethat the times when the hammer passes the two windows areindicated by shaded areas. (a) Is the area of the shaded regioncorresponding to window 1 greater than, less than, or equal tothe area of the shaded region corresponding to window 2? (b)Choose the basl explanation from among the following:

I. The shaded area for window 2 is higher than the shadedarea for window 1.

II. The windows are equally tal1.

III. The shaded area for window 1 is wider than the shadedarea for window 2.

Window 1 Window 2

P$SWie€ 3*33 Problem 79

80. .. Ct Aball is thrown straight upward with an initial speed u6.

When it reaches ihe top of its flight at height /r, a second ball isthrown straight upward with the same initial speed. Do theballs cross paths at height 1ft, above lh:, or below ]ft?

81. .. Bill steps off a 3.0-m-high divingboard and drops to the waterbelow. At the same time, Ted jumps upward with a speed of4.2 m/ s from a 1.0-m-high diving board. Choosing the origin tobe at the water's surface, and upward to be the positive r direc-tion, write l-versus-f equations of motion for both Bill. and Ted.

79.

54 CHAPTER 2 ONE-DIMENSIONAL KINEMATICS

82.

83.

.. RePeat the previous problem, this time with the orlgin 3'0 m

above the water, and wlih downward as the positive;r direction'

.. On a hot summer day in the state of Washington while

kayaking, I saw several swimmers jump from a railroad bridge

inio the bnohomish River below' The swimmers stepped off the

bridge, and I estimated that they hit the water 1 5 s later' (a)

Ho# ngn was the bridge? (b) How fast were the swimmers

morring"when they hit the r'r'ater? (c) What would the swim-

-".r' Jrop time be if the bridge were twice as high?

84. .. X{!!Sest 3€a€*:: F&lr*s!*t*lr The world's highest fountain of

water ii located, appropriately enough' in Forintain Hills' Arizona'

The fountain rises tt a height of 560 ft (5 feet higher than the Wash-

ington Monument). (a) Vi4rat is the initial speed of the water? (b)

Hiw long does it take for water to reach the top of the for'rntain?

35...WlonSlycalledforafoul,anangrybasketballplayerthrowsthe bal iraight down to the floor' If the bali bounces straight

up and returis to the floor 2'8 s after first striking it' what was

the ball's greatest height above the floor?

86. r r To celebrate a victory, a pitcher throws her glove straight up-

ward with an initial sp"ea tt 6'0 m/s (a) How long does it take

for the glove to return to the pitcher? (b) How long does it take

for the glove to reach its maximum height?

87. .. $$s Standing at the edge of a cliff 32'5 m high' you drop a

ball. Later, yo.ithrow a seiond ball downward with an initial

speed of ii.o */.. (a) Which ball has the greater inctense in

,i"ua *n"rl it reaches the base of the cliff' or do both ba11s

sp""d ,rp by the same amount? &) Verify your answer to part

(a) with a calculation.

88. rr You shoot an arrow into the air' TWo seconds later (2'00 s)

the arrow has gone straight upward to a height oI 30'0 m above

its launch pJint. (a) \Arhat was the arrow's initial speed?

G) Hou, long did it take for the arrow to first reach a height of

15.0 m above its launch Point?

89. I o While riding on an elevator descending with a constant

speed of a.0 m7s, you accidentally drop a book from under

y'on, ur*. (a) How long does it take for ll: !:tlk to reach the el-'evator

iloot,L.2 m below your arm? &) What is the book's

speed relative to you when it hits the elevator floor?

90. . I A hot-air balloon is descending at a rate of 2 0 m/s when a

passenger drops a ,camera lf the camera is 45 m above the

gro.,rli*n"tl ii is dropped, (a) how long does jt take for the cam-

Jra to reach the ground, and (b) what is its velocity just before it

lands? Let upwuid b" th. positive direction for this problem'

91.rr&Fstandingsidebyside,youandafriendstepoffabridgeat different times and fall for 1'6 s to the water below Your

friend goes first, and you follow after she has dropped a dis-

tance oT 2.0 m' (a) When your friend hits the water' is the sepa-

ration between the two of you 2'0 m, less than 2'0 m/ or more

than 2.0 m? G) Verify your answer to part (a) with a calculation'

92. r - A model rocket blasts off and moves uPward with an accel-- --

eration of 12 ml s2 until it reaches a height of 26 m' at which

point its engine shuts off and it continues its flight in free {all

ia) What iJ the maximum height attained by the rocket?

(b) What is the speed of the rocket just before it hits the ground?

(c) \A4rat is the total duration of the rocket's flight?

jJ.looX{i&xi*agthe,,Kigk$tc.ik*l.Ayoung-womanatacarni-val steps uf to the "irigh striker," a popular test of strength

where the contestant hiti one end of a lever with a mallet' pro-

pelling a small metai plug upward toward a beli' She girres the

inallet'a mighty t*it-tg u"Asends the plug to the top of the

striker, wheie it rings the bell' Figt.era ?*3;{ shows the corre-

sponding position-vJrsus-time plot for the plug' Using the in-

ri&gl&H **3* Problem93

formation given in the plot, answer the following questions:

(a) \A4rat is the average speed of the piug during its uplt'ard

journey? (b) By how *n.h do"t the speed of the plug decrea.se

durini its upward journey? (c) What is the initial speed of the

plug?"(Assume the plug to be in free fall during.its upward mo-

ilonl wlth no effects of air resistance or friction')

94. ... While sitting on a tree branch 10'0 m above the ground'

you drop a chestnut. When the chestnut has fallen 2 5 m' you

inro* u second chestnut straight down' What initial speed

must you give the second chestnut if they are both to reach the

ground at the same time?

GENERAL PROBLEMS

95. . In a well-known Jules Verne novel, Phileas Fogg travels

around the woridin B0 days' What was Mr' Fogg's approximate

average speed during his adventure?

96. . An astronaut on the Moon drops a rock straight downward

from a heieht of 1.25 m. If the acceleration of gravity on the Moon

rs I.62 t.t:.is2, what is the speed of the rock just before it lands?

97. . You jump from the top of a boulder to the ground 1'5 m

belort'. Estimate your deceieration on landing'

98. . ,& $au6:ers**eies4lat*x$a}t Geologists have learned of periods

in the past when the Strait of Gibraltar ciosed ciff' and the

Medite;ranean Sea dried out and become a desert Later' when

the strait reopened, a massive saltwater waterfall was created'

According to geologists, the water in this waterfall was super-

sonic; thai is, it tett wlttr speeds in excess of the speed of sound'

Ignoring air resistance, what is the minimum height necessarl"

to create a supersonic waterfall? (The speed of sound may be

taken to be 340 m/s.)

gg.r.CEAttheedgeo{aroofyoudropballAfromrest'andthenthrow ball B downward with an initial velocity of o6' Is the in-

crease in speed iust before the balls land more for ball A' more

for bali B, or the same for each ball?

3.0

tr

E 2.0booE

0.10

0

Time

F!*qt&6 X*$& I'roblem 100

0.20 0.30 0.40 0.50 0.60

Time (s)

Plug is launched

E

25

20

10

. " €& Suppose the hvo barlls described in Problem 99 are releasecl.rt the same time, r'r'itl-r ball A dropped from rest and ball B thror,r,n.1or,r,nward r,tith an initial speed lLt. ldentift, which of the fiver-lots shown in !i;tiri'r !l*lrl:li correslronds to (a) ball A and (b) ball B.

. . Astronauts on a distant planet thror,rr a rock straight uplr.ard:nd record its motion r,r,ith a r.ideo camera. After digitizing their', ideo, they are arble to produce the graph of height, .ry, versus.irne, /, shor,r.n in li:il,rr;: I l,i. (a) What is the acceleratlon of grarr-iv on this planet? (b) What uras the initial speed of the rock?

v (m)

30

15

r (s)r23.+5678

iril ;:i;,iltll lii....lll: T)roblem 101

- r o i-i;l's1F'Jhra,er NASAoperates a2.2-second drop tou.eratthe

-,ler-ur Resealch Center in Cler,-eland, Ohio. At this facilit]', ex-::r'imental packages are dropped from the top of the to\^,er, on::'e Eth floor of the building. During their 2.2 seconds of free fa11,

.',periments experience a microgravitv environment similar to::-:t of a spacecraft in orbit. (a) \{hat is the drop distance of a

- l-s tolr'er'? (b) Hon, fast are the experiments trar.eling nhen. -..v hit the air bags at the bottom of the tc-rwer? (c) If the exper-::rental package comes to rest o\.er a distance of 0.75 m upon--itilg the air bags, rvhat is the average stoppir-rg acceler:ation?

: r r lr' A youngstel bounces straight up and donn on a tram---..lirre. Suppose she doubles her initial speed from 2.0 m/s io= I n"r/s. (a) 81, n'hat factol does her time in the air i.creasc? (b)

: . n'hat factor does her maximum height increase? (c) Verif)'.,ur .rns\ rers to parts (a) and (b) vr.ith an explicit calculation.

- , . ,\t the l8th green of the U.S. Open vor-r need to make a 20.5-ft: -iit to win the toumament. When vou hit the ba1l, giving it an ini-

-''1 speed o11..57 m/s, it stops 6.00 fi short of the ho1e. (a) Assum-- g the deceleratitin car-rsed bv the grass is constant, r,r'hat shor,rld

: -.e rnitial speed have been to just make the putt? (b) What initial.:eed do you need to make the remaining 6.00-ti pr-rtt?

: ' I ri: A popular entertainment at some carnir.als is the blan-..-:t toss (see photo, p. 39). (a) If a persor-r is thror'r,'n to a maxi-:r-r,rm height of 28.0 ft above the blanket, hor,v long does she-:end in the air? (b) Is the amount of time the person is above

rreight of 14.0 ft more than, less than, or etluarl to the amLrunt: trrne the person is belorv a height of 14.0 ft? Explain. (c) Ver-r', \.our ans\\'er to part (b) r.r'ith a calcr,rlation.

- , . Refelling to Conceptual Checkpoint 2 5, find the separa-:-.rn betn:een the rocks at (a) t - 1.0 s, (b) t - 2.0 s, and (c)

: - 3.0 s, where time is measr-rred from the instant the second

-.ck is dlopped. (d) Verify that the separation increases lin-..',rlv rvith time.

- , . ::r' A glar-rcous-rvinged gu11, ascending straight upr,r'ard atr.ll-) m/s, drops a she1l r'vhen it is 12.5 m above the ground. (a)

,\ hat are the rnagnitr-rde and direction of the shell's accelera-:.rn just after it is release.ll (b) Find the maximum height

.,:ror,er the ground reached bv- the sheil. (c) Hou' long does it,..ke for the shell to reach the ground? (d) l{hat is the speecl of

.-',e shell at ihls time?

PROBLEMS AND CONCEPTUAL EXERCISES 55

108. " e A doctor; preparing to gir.e a patient an injection, squirts a

small .rmount of liquid straight upward from a syringe. lf iheliquld emerges r,r'ith a speed of 1.5 m,/s, (a) hor,r'iong does ittake for it to return to the level of the sylinge? (b) What is themaximum height of the liquid above the svringe?

109. e' A hot-air"ba11oon has just lifted off and is rising at the con-stant rate of 2.0 m/s. Suddenly one of the passengels realizesshe has left her camera on the ground. A fliend picks it up andtosses it straight upn,ard with an initial speed of 13 m /s. lf thepassenger is 2.5 m above her friend when the camera is tossed,holv high is she r,r'hen the camela reaches her?

110. e . 6 In the prcvious problem, wl-rat is the minimum initlal speedof tlre camera if it is to just reach the passen ger? (Hirrt: When thecamera is thror,r,'n lrrith its minimum speed, its speed on reach-lng the passenger is the same as the speed of the passenger.)

111. . o, &ld Faitir.i:n1 Watching Old Faithful er.upt, you noticethat it takes a time I for water to emer€ie from the base of thege1'ser and reach its maximum height. (a) What is the height oftire geyser, and (b) n'hat is the initial speed of the water? Eval-uate your expressions for (c) the heiglrt and (d) the initialspced for a measurec'l time of 1.65 s.

112. o6 o ll:ir A ball is thlor.l'n upr,r,arrd rt'ith an initial speed 16.

When it reaches the top of its flight, at a height ir, a se'cond baliis throlr.n uplr,ard with the same initial velocity. (a) Sketch anr-r,ersus-f plot for cach ba11. (b) Frum your graph, decider.r.hether the ba11s cross paths at lL / 2, above h /2, ctr below lt /2.(c) Fincl the height where the paths cross.

113. 6 6 o Weights are tied to each end of a 20.0-cm string. You holdone n'eight in your hand and let the other hang veltically a

height ft above the floor. When you release the weight in your'hand, the trvolveights strike the ground one after the otherr,vith ar-rdible thuds. Find the rralue of lr for which the time be-tr'r,een release ancl the first tl-rt d is equel to the time betweenthe first thud and the second thud.

114. s 4 s A ball, clropped from rcst, covers tfuee-quarters of the dis-tance to the ground in the last second of its fall. (a) From whatheig}rt rvas the ball clropped? (b) What'l,"as the total time of fa11?

115. 6 , o A stalactite on the roof of a cave drips water at a steadyrate to a pool 4.0 m belovr'. As one drop of r,r'ater hits the poo1,a second drop is in the air, and a third is just detaching fromthe stalactite. (a) What are the position and velocity of the sec-ond drop lr,hen the first drop hits the pool? (b) Hor,r,' manvdrops psr minute fa1l into the pool?

116. * * o You drop a ski gior-e from a height /t ontrr fresh sns14r, afldit sinks b a depth r1 befole cLrming t() rrst. {a) In terms of g and/r, r.r'hat is the speed of tl-re g1or.e I,r'hen it reaches the snow? (b)lVhat are the magnitr-rde and direction of the glove's accelera-tion as it mor.es through the snolt', assuming it to be constant?Gir.e yor-rr ans\ver in terms ol g,lL, and d.

117 . * ", To find the height of an overhead por,r,'er line, vou throw a

ball straight upu.ard. The ball passes the lirre on the wav upafter 0.75 s, and passes it again on the u'ay dou,'n 1.5 s after itrvas tossed. What are tire height of the power lirre anil the ini-tial speed of the ball?

118. r " * Sr-rppose the first rock in Conceptual Checkpoint 2-5

dlops through a height lu before the second rock is releasedfrom rest. Shorv that the separation between the rocks, S, isgir ert br the i,'llr'11 i11g e\Irre\sion:

S-lt t rV4lriIn this result, the time i is measrired flom the time the secondrock is dropped.

l

.r-l.J11

I r-u

... 11 s

tht-:'1C11

:.Pel.

ief-,:rr11.

: atl:\'

)be

:heu: ir-i-

:1Ol'e

'6 CHAPTER 2 ONE-DIMENSIONAL KINEMATICS

119. ... An arrow is fired with a speed of 20.0 m/s at a block ofStyrofoam lesting on a smooth surface. The arrort peneftatesa certain distance into the block be{ore coming to rest relativeto it. During this process the arrow's decelera'tion has a mag-nitude of 1550 m/s2 and the block's acceleration has a magni-tude of 450 m/s2. (a) How long does it take for the arror'r, tostop moving with respect to the bkrck? (b) What is the com-mon speed of the arrow and block when this happens? (c)

How far into the block does the arrow penetrate?

120. ... Sitting in a second-story apartment, a physicist notices a

ball moving straight upward just outside her windon'. The ballis visible for 0.25 s as it moves a distance o{ 1.05 m from thebottom to the top of the window. (a) Hovr'1ong does it take be-

{ore the ball reappears? (b) What is the greatest height of theball above the top of the windo'rv?

121. ... €iie* ffaaaetra'afic Fc*trlltr;e Fr*ll,l Kiraenr.aties In this prob-lem we show how the kinematic equations of motion can be

used to derive the quadratic formula. First, consider an objectwith an initial position :16, an initial velocity z,9, and an accel-

eration a. To find the time when this object reaches the positionr - 0 we can use the quadratic formula, or apply the follow-ing two-step procedure: (a) Use Equatic'rn 2-I2 Io sho-w thatthe velocity of the object when it reaches r : 0 is given

-

by a : +f u62 2axs. (b) Use Equation 2-7 to show that the

time corresponding to the velocity found in part (a) is

,, t-2 "t - -u0+V '0' t"o

. rrrTo complete our derivation, shovr

that the ,"rrrifof purt (b) is the same as applyiug the quadraticformula to x - xs .tnl - )att : 0.

=ASSAS€ pffi.0Sn-C&d5

Barnl-Apollo 75 Lands on the MoonThe first word spoken on the surface of the Moon atter Apollo15 landed on July 30, 797I, was "Baml" This was James Irwin'sinvoluntary reaction to their rather bone-jarring touchdown."We did hit harder than any o{ the other flights!" says Irwin."And I was startled, obviously,,when I said, 'Raml"'. The reason for the "firm touchdown" of Apollo 15, as pilotDavid Scott later characterized tt, was that the rocket enginewas shut off a bit earlier than planned, when the lander wasstill 4.30 ft above the lunar surface and moving downwardwith a speed of 0.500 ftls. From that point on the iander de-scended in lunar free fal1, with an acceleration of 1.62 m/s2. Asa result, the landing speed of Apollo 15 was by far the largest ofany of the Apollo missions. In comparison, Neil Armstrong'slanding speed on Apollo 11 was the lowest at 7.7 ft/s-hedidn't shut off the engine until the footpads were actually onthe surface. Apollos 12, 11, and 17 all landed with speeds be-tween 3.0 and 3.5 ftls.

To better understand the descent of Apollo 15, we show itstrajectory during the final stages of landing in Figa:r* ?*37 {*}. InFiga;r* ?*3? {b} we show a variety of speed-versus-time p1ots.

1.22. . How long did it take for the lander to drop the final 4.30 ft tothe Moon's surface?

A. 1.18 s

C. 1.78 s

B. 1.37 s

D. 2.36 s

123. ,. What was the impact speed of the lander when it toucheddown? Give your answer in feet per second (ft/s), the same

units used by the astronauts.

9.0 9.5 10

Time (s)

10.5

Time (s)

(b)

silstiR!c X*3? Prciblems 722, 123,724, and 725

124. o Which of the sp.eed-versus-time plots in Figure 217 (b) cor-rectiy represents the speed of the Apollo 15 lander?

ABCD125. . Suppose, instead of shutting off the engine, the astronauts

had increased its thrust, giving the lander a small, but con-stant, upward acceleration. Which speed-versus-time plot inFigure 2-37 (b) would describe this situation?

ABCDf $"*T E ffi,4(T I V E F Rffi h I. ffi ftd 5

126. .. &e$*ra'[*g tar Sx*xaapB* ?*S Suppose the speeder (red car)is traveling with a constant speed of 25 m/s, and that the max-imum acceleration of the police car (blue car) is 3.8 m/s2. If thepolice car is to start from rest and catch the speeder in 15 s orIess, what is the maximum head-start distance the speeder can

. have? Measure time from the nioment the police car starts.

127. .. *R*{erringte: #xamp}* **S The speeder passes the posi-tion of the police car with a constant speed of 15 mr's. Thepolice car immediately starts from rest and pursues thespeeder with constant acceleration. What acceleration mustthe poiice car have if it is to catch the speeder in 7.0 s? Measureiime from the moment the police car starts.

128. .. llF K*:fcrring tar iflxaltapi* 3*t? (a) In Exan-rple 2-12, thebag of sand is released at 20.0 m ancl reaches a maximum heightof 22 nl.. If the bag had been released at 30.0 m instead, witheverything else remaining the same, would its maximum heightbe 32 m, greater t1.ran 32 m, or less than 32 m? (b) Find the speedof the bag just before it lands when it is released from 30.0 m.

129. .. R*derring *,* $ix;*xepile **3? Suppose the balloon is de-scending with a constant speed of 4.2 m/s when the bag ofsand comes loose at a height of 35 m. (a) How long is the bagin the air? (b) What is the speed of the bag when it is 15 mabove the ground?

5 3.0

bo

4.0

1.0

€oop.It

A. 2.41 {t/s

C. 9.95 tt/ s

B. 6.78 lt/ s

D. 10.6 ftls

Trajectory withengine still on