58 CHAPTER 2 Motion in One Dimension

SUMMARY The goal of Chapter 2 has been to describe and analyze linear motion.

GENERAL STRATEGIES

Proble m-Solving Strategy Visual Overview Our general problem-solving strategy has three parts:

A visual overview consists of several pieces that completely specify a problem. This may include any or all of the elements below:

PREPARE Set lip the problem: l\Iotion diagram Pictorial representation Graphicall't!presentation List or values

Draw a picture.

Collect necessary information.

Do preliminary calculat ions.

SOLVE Do the necessary mathematics or reasoning.

ASSESS Check your answer to see if it is complete in all detail s and makes physical sense.

IMPORTANT CONCEPTS

Velocity is the rate of change of position:

~x II = ­, ~f

Acceleration is the rate of change of veloc ity:

~ " (/ = --' , ~f

The units of acceleration are m/s2.

y ~ YI' (1',)1' If

ij

, ta ta

0 Yi' (v)",;

A position-versus-time graph plots position on the vertical ax is against time on the horizontal axis.

x(m)

6

4

2

An object is speeding up if V.I and ax have the same sign, slowing down if they have opposite signs.

o +"-~~-~-~-~ I (s) 0246810

APPLICATIONS

Uniform motion An object in uniform motion has a constant velocity. Its velocity graph is a horizontal line; its position graph is linear.

Kinematic equation for uniform motion:

Xf= Xi + VA' 61

Uniform motion is a special case of constant­acceleration motion, with a.l , = O.

Motion with constant acceleration An object with constant acceleration has a constantly chang ing velocity. Its velocity graph is linear; its position graph is a parabola.

Kinematic equations for motion with constant accelerat ion:

(V,)r = (V)i + (/x 61

Xf = Xi + (v,,), D.I + ta,( D.t )2

(vJl = (v.T)? + 2ax D. X

Known ,. , Yi= Om

(I',)f (r,)i = 0 m/s

I; = Os

(I , = !8 m/sl

I f = 150 s

Find

" (V,)I andYr

A velocity-versus-time graph plots velocity on the vertical axis against time on the hori zontal axis.

Vt

(m/s)

30

10

Acce)emliQn j, the slope or The velOCiTY graph.

0+--_-_-_-="- , (s) o 2 3 4

Free fall Free fall is a special case of constan t­acceleration moti on; the acceleration has magnitude g = 9.80 m/s 2 and is always directed vert ical ly downward whether an object is moving up or down.

, . .

", , ", The vc!oci ty grJph is a straight line with a slope of - 9.80 m/sl.

./

tMP)TM For homework assigned on MasteringPhysics, go to

!:!!/ www.masteringphysics.com

Problem difficulty is labeled as I (straightforward) to 11111 (challenging),

QUESTIONS

Conceptual Ouestions

1. A person gets in an elevator on the ground floor and rides it to the lOp floor or a building. Sketch a veloc ity-versus-time graph for t.hi s motion.

2. a. Give an example of a ve rtical molion with a pos itive velocity and a negati ve acce leration.

b. Give an example of a vertical motion with a negative ve loc­ity and a negative acceleration.

3. Sketch a ve loc ity-versus-timc graph for a roclnhat is thrown straight upward, from the instant it leaves the hand until the instant it hits the ground.

4. You <rre dri ving down the road at a constant speed. Another car go ing a bit faster catches up with you and passes you. Draw a position graph for both vehicles on the same set of axes, and note the point on the graph where the other vehicle passes yo u.

S. A ca r is travel ing north . Can its acceleration vector ever point sOlllh? Ex plain.

6. Certain an imals are capable of run ning at great speeds; o ther BID animals are capable of tremendolls accelerations. Speculate on

which would be more benefic ial to a predator-large maximum speed or l<rrge acceleration .

7. A ball is thrown straight up into the air. At each of the following instants, is the baJJ 's acceleration Cly equa l to g, -g, 0, < g, or > g? a. Just after leaving your hand? b. At the very top (maximum height)? c. Just before hitting the ground?

8. A rock is (hrollln (not dropped) straight down from a bridge into the ri ver below. a. Immediately after being released, is the magnitude of the

rock's acceleration greater than g, less than g, or equal to g? Explain .

b. Immediately before hitting the water, is the magnitude of the the rock's acceleration grealer than g, less than g, or equal to g? Expla in .

9. Figure Q2.9 shows an object's position-verslis- time graph . The letters A to E correspond to various segments of the mot ion in which the graph has constant slope.

x(rn) B

2

A

2 4 6 8 - I

E

FIGURE 02.9 - 2

a. Write a reali st ic motion short story for an objec t that would have thi s position graph.

b. In which segment(s) is the object at rest?

Questions 59

Prob lems labeled INT integrate significant material from earlier

chapters; BID are of biological or medical interest.

c. In which segment(s) is the object moving to the ri ght? d. Is the speed of the object during segment C greate r than ,

equal to, or less than its speed during segment E? Explain. 10. Figure Q2.IO shows the position graph for an object moving

along the hori zontal ax is. a. Write a real istic motion short story for an objecl that would

have thi s position graph. b. Draw the corresponding velocity graph.

x

\ FIGURE 02.10

~ \'-------II. Figure 02.11 shows the pos ition­

versus-t ime graphs for two objec ts, A and B, that are moving along the same ax is. a. At the instant (= L s, is the

speed of A greater than , less than , or equal to the speed of B? Explain.

b. Do objects A and B ever have

x

0 2 3

FIGURE 02.11

B

4

thesClme speed? If so, at what time or limes? Explain. 12. Figure Q2. 12 shows a position- x 8

versus-time graph. At which lettered point or points is the object a. Moving the fastest?

D

A

t (s) 5

b. Moving to the le ft ? E '-__

c. Speeding up? F

d. Slowing down? FIGURE 02.12

e. Turning around? 13. Figure Q2.1 3 is the veloc ity-versus-t ime graph for an objec t

moving along the x-axis. a. During which segment(s) is the veloc ity constant? b. During which segment( s) is the object speeding up? c. During which segment(s) is the object slowing down? d. During which segment(s) is the object standing still ? e . During which segment(s) is the object moving to the right?

I'~ (m/s)

3

2 A

B

0 F

I (s) 2 3 4 10

- I

- 2 D

FIGURE 02.13 - 3

60 CHAPTER 2 Motion in One Dimension

14. A car traveling at veloc ity v takes distance d to stop after the brakes are appl ied. W hat is the stopping distance if the car is ini tiaUy trave ling at veloc ity 2v? Assume that the acce leration due to the braking is the same in both cases.

Multiple-Choice Questions

IS. 1 Figure Q2.L5 shows the pos ition graph of a car travel­ing on a straight road. At which labeled instant is the speed of the car greatest?

FIGURE OZ.15

x

E

16. I Figure Q2. 16 shows the pos ition graph of a car travel­ing on a stra ight road. The veloc ity at instant I is __ -,-_ and the veloc ity at instan t 2 is ___ _ A. positi ve, negat ive ~,

B. posi tive, posit ive FIGURE OZ.16 C. negative. negative D. negative, zero E. positive, zero

17. 1 Figure 02.1 7 shows an x(m)

40 object 's pos ition-versus-time

30 graph. W hat is the ve loc ity of the object at t = 6 s? 20

A. 0.67 ml' 10 B. 0.83 ml, C. 3.3 m/s

O+----_---~I (s) o 5 10

D. 4.2 mls FIGURE OZ.17 E. 25 mls

18. I The fo llowing options desc ri be the mot ion of four cars A- D. Which car has the largest accelerat ion? A. Goes from 0 mls to 10 mls in 5.0 s 8 . Goes from 0 mls to 5.0 mls in 2.0 s C. Goes from 0 mls to 20 mls in 7.0 s D. Goes from 0 mls to 3.0 mls in 1.0 s

19. I A car is trave ling at Vx = 20 m/s. The dr iver applies the brakes, and the car slows with a.~ = - 4.0 m/s 2

. What is the stop­ping di stance? A. S.Om B. 25 m C. 40 m D. 50 m

20. II Ve loc ity-ve rsus-t ime graphs for three drag racers are shown in Figure Q2.20. At t = 5.0 s, which car has traveled the fur­thest? A. Andy B. Belly C. Carl D. All have traveled the same distance

I'~ (m/s)

4:W ,(,) 1', (m/s)

4:k===L,) ~or') / O~I(S)

o Andy 5.0 o Betty 5.0 o Carl 5.0

FIGURE Q2.20

2 1. I Which of the three drag race rs in Question 20 had the great­est acceleration at I = 0 s?

22.

A. Andy B. Belly C. Carl D. All had the same acceleration II Ball 1 is thrown straight up in the air and, at the same instant. ball 2 is released fro m rest and allowed to fall. Which veloc ity graph in Figure Q2.22 best represents the motion of the two ball s?

FIGURE 02 .22

23 . II Figure Q2.23 shows a motion diagram with the clock reading (in seconds) shown al each pos ition. From I = 9 s to I = 15 s the object is at the same pos ition . After that, it returns along the same track. The pos itions of the dots for f ;::= 16 s are offset for clarity. Which graph best" represents the object's velocity?

o • •

• 23

FIGURE 02 .23

2 •

• • 21 21

3 •

4

• 5 ~ : ! . /9to 15 •

• • • • • 20 19 18 17 16

C. I'x D. I'x

h, ~, 24. II A car can go from 0 to 60 mph in 7.0 s. Assumi ng that it

could maintain the same acceleration at higher speeds, how long would it take the car to go from 0 to 120 mph ? A. lOs B. 14 s C. 2 1 , D. 28 s

25. 11 A car can go from 0 to 60 mph in 12 s. A second car is capa­ble of twice the accelera ti on of the fi rst car. Assuming that it could maintain the same acceleration at highe r speeds, how much time will thi s second car take to go from 0 to 120 mph? A. 12 s B. 9.0s C. 6.0 s D. 3.0s

PROBLEMS

Section 2. t Describing Motion

I . II I Figure P2. l shows a m Ol ion diagram of a car travel ing down a street. The camera took one frame every second. A distance scale is prov ided. a. Measure the x-value of the car at each dot. Place your data in

a table, simi lar to Table 2. 1, showing each pos ition and the instanroftime at which it occurred.

b. Make a graph of x versus I, using the data in your table. Because you have data only at certain instants of lime, your graph should consist of dots that arc not connected together.

I frame per second

o 2()() 4flO 600 800 I flO()

FIGURE P2.1

2. I For each motion diagram in Figure P2.2, determine the sign (positi ve or negat ive) of the pos ition and the velocity.

(a) (bJ ('J • ~ I •• 0

t 0

t FIGURE P2.2 0

3. I Write a short desc ription of the mot ion of a real object for which Figure P2.3 would be a realist ic pos ition-versus- time graph .

x(m)

300

200

100

o ~~~~-c~~-+:I (s) o 200 400 600

FIGURE P2 .3

x(mi)

40

20

O +-~+C-~~~ I (h)

- 20

FIGURE P2.4

4. I Write a short desc ription of the motion of a real object fo r which Figure P2.4 would be a reaJi sti c pos ition-versus- time graph .

5. II The pos ition graph of Figure P2.5 shows a dog slowly sneaki ng up on a squi rrel, then putting on a burst of speed. a. For how many seconds does the dog move at the slower

speed? b. Draw the dog's ve loc ily-ve rsus-ti me graph. Include a

numerical scale on both axes.

x(m)

~D"., o 2 4 6 8 10

FIGURE P2.5

Probl ems 61

6. II The pos ition graph of Figure P2.6 represents the mot ion of a ball being roUed back and forth by two children. a. At what positions are the two children sitti ng? b. Draw the ba ll' s veloc ity-versus- time graph . Include a

numerical scale on both axes.

4

2

0 I (s) 16 20

- 2

- 4

FIGURE P2.6 - 6

7. II A rural ma il carri er is driving slowly, putting ma il in ma il ­boxes near the road. He overshoots one mailbox.s tops.shi fts into reverse, and then backs up until he is at the right spot. The veloc ity graph of Figure P2.7 represents hi s mot ion. a. Draw the mail carrier 's pos ition-versus-time graph. Assume

that x = 0 m at f = 0 s. b. What is the pos ition of the mailbox?

I', (Ill /s) 1'.< (Ill/S)

3 3 r--,

2 2 I I I I

I I I I

0 1(8) 0 I (s) 2 4 6 'f - I 2 4 6 8 10 " - I I 1

L-.J - 2 I - 2 I

-3 - 3

FIGURE P2 .7 FIGURE P2.8

8. For the ve loc ity-versus-time graph of Figure P2.8: a. Draw the corresponding position-versus-time graph. Assume

that x = 0 m at f = 0 s. b. What is the object's posit ion at f = 12 s? c. Desc ri be a moving objec t that could have these graphs.

9. II A bicycl ist has the pos ition-versus- time graph shown in Figure P2.9. What is the bicycl ist's veloc ity at 1 = 10 s, at t = 25 s, and at f = 35 s?

FIGURE P2.9

x(m)

100

50 -

o +--~---c~-:--+-t (s) o 10 20 30 4{)

Section 2.2 Uniform Motion

10. I In co ll ege softball , the di stance from the pitcher' s mound to the balle r is 43 fee l. If the ba ll leaves the bat at 100 mph, how muc h time elapses between the hi t and the ball reac hi ng the pitcher?

62 CHAPTER 2 Motion in One Dimension

11. II Alan leaves Los Ange les at 8:00 A .M. to d ri ve to San Fran­c isco, 400 mi away. He trave ls at a steady 50 mph. Beth leaves Los Angeles at 9:00 A.M. and dri ves a steady 60 mph. a. Who gets to San Francisco first? b. How long does the first to arri ve have to wa it for the sec­

ond? 12. II Richard is driving home to vis it his parents. 125 mi of the tri p

.u e on the interstate highway where the speed limit is 65 mph . Normally Richard drives a1 the speed li mit, but today he is run­ning late and decides to take hi s chances by dri ving at 70 mph. How many mi nutes doeS' he save?

13. III In a 5.00 km race, one ru nner ru ns at a steady 12.0 kmlh and another runs at 14.5 km/h. "How long does the faster runner have to wait at the fi nish li ne to see the slower runner cross?

14. llil In an S.OO km race, one runner runs at a steady 11 .0 kmlh and anot her runs at 14.0 kmfh. How far from the fini sh line is the slower ru nner when the faster runner fini shes the race?

15. Il A car moves with constant velocity along a straight road. Its posi ti on is XI = 0 m at '1 = 0 s and is X2 = 30 III at 12 = 3.0 s . Answer the fo llowing by considering rat ios, withou t computi ng the car 's veloc ity. a. What is the car 's pos ition at 1 = 1.5 s? b. What will be its pos ition at I = 9.0 s?

16. II While run ning a marathon, a long-d istance run ner uses a stopwatch to time herse lf over a distance of 1.00 Ill. She finds that she runs thi s distance in IS s. Answer the following by con­sider ing rati os, without computing her ve locity. a. If she mainta ins her speed, how much time will it take her to

ru n the nex t 400 m? b. How long will it take her to ru n a mile at thi s speed?

Section 2.3 Instantaneous Velocity

17. 1 Figure P2. 17 shows the x (m)

pos ition graph of a particle. 20 a. Draw the parti cle's veloc ity

graph for the interval 10 Oss:t S: 4s.

b. Does thi s particle have a o +---I--~-~--'\- I (s) 0 24

turni ng point or po ints? If FtGURE P2 .17

so, at what time or times? 18. II A somewhat ideali zed graph of the speed of the blood in the BID ascend ing aorta during one beat of the heart appears as in

Figure P2. 1S. a. Approximately how far, in cm, does the blood move duri ng

one beat? b. Assume similar data for the mot ion of the blood in yo ur

aorta. Esti mate how many beats of the heart it wil.l it take the blood to get from your heart to your bra in.

I' (m/s)

1.0

0.5

o +---+--~I--~- f (s) FIGURE P2 .18 0 0.20 0.40 0.60

19. III A car starts from Xi = \0 m at Ii = 0 sand moves with the veloc ity graph shown in Figure P2. 19.

I'~ (m/s)

" 8

4

o +--~-2'---"3"~4c- 1 (s) a. W hat is the object'S

posit ion at 1 = 2 s, 3 s, and 4 s?

- 4

FIGURE P2 .19

b. Does thi s car ever change di.recti on? If so, at what time? 20. Figure P2.20 shows a graph of actual pos ition-verslls- time

data for a particular type of drag racer known as a "funny car. " a. Estimate the car 's veloc ity at 2.0 s. b. Estimate the car's veloc ity at 4.0 s.

r(m)

400

300

200

100

0 I (s) FIGURE P2 .20 0 2 4 6

Section 2.4 Acceleration

2 1. II Figure P2.2 1 Shows the veloc ity graph of a bicycle. Draw the bicycle's acceleration graph for the in terval 0 s :5 1 :5 4 s. Give both axes an appropriate numeri cal scale.

1' ... (m/s)

4 --v, (m/s)

2

2 1

0 I (s) 8 10

- I

0 I (s) - 2 0 2 4

FIGURE P2.21 FIGURE P2 .22

22.

23.

24. BID

25.

III Figure P2 .22 shows the veloc ity graph of a train that starts from the ori gin at I = 0 s. a. Draw pos ition and accelerat ion graphs for the train . b. Find the accelerat ion of the train at ( = 3.0 s. I For each moti on diagram shown earli er in Figure P2.2, deter­mi ne the sign (positive or negative) of the acce lerat ion. II Figure P2. 18 showed data for the speed of blood in the aorta . Determine the magn itude of the acceleration for both phases, speeding up and slowing down. II Figure P2.25 is a somewhat simplified veloci ty graph for O lympic spr inter Carl Lewis starting a 100 III dash. Est imate hi s acce lerat ion duri ng each of the intervals A, B, and C.

L' ... (m/s)

12 10

8 6

4 A 2

c

B

O¥--~-~-~-~I (s) FIGURE P2 .2S 0 2 3 4

Section 2.5 Motion with Constant Acceleration

26. I A Thomson's gazelJ e can reach a speed of 13 mls in 3.0 s. A BID lion can reach a speed of 9.5 mls in 1.0 s. A lrout can reach a speed

of 2.8 mls in 0 .1 2 s. Which animal has the largest acce leration?

27. III When striking, the pike. a BID predatory fish, can acce lerate

from rest to a speed of 4 .0 m/s in 0. 11 s. a. What is the acceleration of

the pike during thi s str ike? b. How far does the pike move

during this strike? 28. II a. What constant acce lerat ion, in SI units, must a car have to

go from ze ro to 60 mph in IDs? b. What fraction of g is thi s? c. How far has the car trave led when it reaches 60 mph ?

Give your answer both in SI units and in feet. 29. Light-rail passenger trains that provide transportatio n within

30.

31.

32.

and between ci ties are capable o f modes t accelerati o ns. T he magni tude of the maximum acceleration is typica lly 1.3 m/s2, but the driver will usually maintain a constant acce leration that is less than the maximum. A train travel s through a congested part of town at 5.0 m/s. Once free of thi s area, it speeds up to 12 m/s in 8.0 s. At the edge of town, the driver aga in acce lerates, with the same accelerat io n, for a nother 16 s to reach a higher crui sing speed. What is the final speed? III A speed skater moving across frictionless ice at 8.0 m/s hits a 5.0-m-wide patch of rough ice. She slows stead ily. then contin­ues on at 6.0 m/s. What is her acceleration on the rough ice? JI A small propeller ai rpl ane can comfortably achieve a hi gh enough speed to take off on a runway that is 1/4 mile long. A large, fully loaded passenger jet has about the same accelerati on from rest, but it needs to achieve twice the speed to take off. What is the minimum runway length that will serve? Hint: You can solve this problem using ratios without hav ing any additional infonnati on. II Figure P2.32 shows a veloc-ity-versus-t ime graph for a par­ticle moving along the x-aris. At t = 0 s. assume that x = 0 m. a. What are the particle's posi­

ti on , veloc ity, and accelera­tion at t = 1.0 s?

b. What are the parti cle's posi­tion , ve locity, and accelera­ti on at t = 3.0 s?

v~ (mJs)

4+---,

2

O+--~-~~--+-I (s) o 2 3 4

FIGURE P2 .32

Section 2.6 Solving One· Dimensional Motion Problems

33. II A driver has a reac tion time of 0.50 s, and the maximum deceleration o f her car is 6.0 m/s2

. She is driving at 20 m/s when suddenly she sees an obstacle in the road 50 m in front of her. Can she stop the car in time to avo id a colli s ion?

34. II Chameleons catch insects with their tongues, whi ch they can BID rapidly ex tend to great lengths. In a typical strike, the chameleo n's

to ngue accelerates at a remarkable 250 m/s2 for 20 ms, then travels at constant speed for another 30 ms. During thi s total time of 50 ms, 1/20 o f a second, how far does the tongue reac h?

35. !II Yo u' re dr iving dow n the hig hway late one nig ht at 20 m/s when a deer steps on to the road 35 m in front o f you. Your reac­tion time before stepping on the brakes is 0.50 s , and the maxi­mum deceleration of your car is 10 m/s2

.

a. How much di stance is be tween you and the deer whe n you come to a stop?

b. What is the max imum speed you could have and still not hit the deer?

Problems 63

36. Jl I A light-ra il trai n go ing from one stati o n to the next o n a straight section of track accelerates from rest at 1.1 m/s2 for 20 S.

It the n proceeds at constant speed fo r 11 00 m before slowing dow n at 2.2 m/s2 until it stops at the station. a. What is the di stance between the stations? b. How much time does it take the train to go between the

stati ons? 37. III A simple model for a person runni ng the 100 m dash is to

assu me the spri nter runs with cons tant acce leration unt il reaching top speed, then mai ntains that speed throug h the fini sh line. If a sprinter reaches hi s top speed of I L. 2 m/s in 2. 14 s, what will be hi s total time?

Section 2.7 Free Fall

38. III BaLI bear ings can be made by letting spheri ca l drops of mol ten metal fall ins ide a tall tower--called a sliot tower-and so lidify as they fall. a. If a bearing needs 4.0 s to solidify eno ugh for impact, how

high must the tower be? b. What is the bearing's impact veloc ity?

39. III In the chapter, we saw that a person' s react ion time is gener­BID all y not quick enough to allow the person to catch a dollar bill

dropped be tween the fingers. If a typ ical reaction time in thi s case is 0.25 s, how long would a bill need to be for a person to have a good chance of catching it ?

40. II A ball is thrown verti cally upward with a speed of 19.6 m/s. a. What are the ball 's ve locity and he igh t after 1.00,2 .00,3.00,

and 4.00 s? b. Draw the ball 's ve loc il y-versus- time graph. G ive both axes

an appropriate numerical scale. 4 1. A student at the top of a building of heig ht Ii throws ball A

stra ight upward with speed Vo and throws ball B straight down­ward with the same initi al speed. a. Compare the balls' accelerat ions. both direction and mag ni ­

tude, immediately after they leave her hand. Is one accelera­tion larger than the other? Or are the magn itudes equal?

b. Compare the final speeds of the ball s as they reac h the ground. Is one larger than the other? Or are they equal?

42. II Excellent human jumpers can leap straight up to a height of 110 cm o ff the gro und. To reach thi s he ight , with what speed would a person need to leave the grou nd?

43. II A footba ll is kicked straight up into the air; it hits the ground 5.2 s lale r. a. What was the greatest height reached by the ball? Assume it

is kicked from grou nd level. b. With what speed d id it leave the ki cker 's foot?

44. 11111 In an acti o n movie, the viU a in is rescued from the ocean by grabb ing onto the ladder hanging from a helicopter. He is so intent on g rippi ng the ladder that he lets go of hi s briefcase of counterfe it money when he is 130 m above the water. If the briefcase hits the water 6.0 s later, what was the speed at which the heli copter was ascend ing?

45. 11 11 A rock cli mber stands on top o f a 50-m-high cliff overhang­ing a pool of wa ter. He throws two stones vert ically downward 1.0 s apart and observes that they cause a single splash. The ini ­ti al speed of the firsl stone was 2.0 m/s. a. How long after the release of the first stone docs the second

stone hit the water? b. What was the initial speed of the second stone? C . What is the speed of each stone as they hit the water?

64 CHAPTER 2 Motion in One Dimension

General Problems

46. III Actual ve loc ity data fo r a li on pu rs ui ng prey are shown in BIO Figure P2.46. Estimate:

a. The initial accelerat ion of the lion. b. The acce leration of the lion at 2 s and at 4 s. c. The di stance traveled by the lion between 0 sand 8 s.

I'~ (m/s)

15

O+---~----,~ I (s) 048

FIGURE P2 .46

Prob lems 47 and 48 concern Ilen'e Unmyelinated fiber A:

impulses, electrical signals propa­gated alo ng nerve fi bers cons ist ing of many axol/s (fiberlike extensions of nerve ceLis) connected end-to-end.

" Individual Panly myelinated fi ber B:/ awns

== == Axo ns come in two varieties: insu- Fully myelinated fiber C:

lated axons with a sheath made of mye lin , and uninsulated axons with

FIGURE P2 .41

no such sheath. Myel inated (sheathed) axons conduct nerve impulses much faste r than unmyelinated (unsheathed) axons. The impulse speed depends on the diameter o f the axons and the sheath, but a typical mye linated axon transmi ts nerve impulses at a speed of about 25 mis , much faster than the typical 2.0 mls for an unmyeli nated axo n. Figure P2.47 shows small porti ons of three nerve fibers consisti ng of axons of equal size. Two-thirds of the axons in fi ber B are myel inated.

47. I Suppose nerve impulses s imultaneously enter the left side of BIO the nerve fi bers sketched in Figure P2.47, then propagate to the

ri ght. Draw quali tati vely accurate pos ition and veloc ity graphs fo r the nerve impulses in all th ree cases. A nerve fiber is made up of many axons, but show the propagat ion of the impulses only over the six axons shown here.

48. I Suppose that the nerve fibers in Figure P2.47 con nect a fin­BIO ger to your brain , a di stance of 1.2 m.

a. What are the travel times of a nerve impul se from fi nger (0

brain alo ng fi bers A and C? b. Fo r fiber B, 2/3 of Ihe length is composed of myeli nated

axons, 1/3 unm yelinated axo ns. Compute the travel time for a nerve impulse on thi s fiber.

c. When you touch a hot stove with your finger, the sensat ion of pain must reac h your bra in as a nerve s ignal along a nerve fibe r before your muscles can react. Which of the three fibers g ives you the bes t protec ti on aga inst a bu rn? Are any of these fi bers unsui table for transmitting urgent sensory infonnation?

49. II A truck d ri ve r has a shipment of apples to del iver to a desti­nation 440 mi les away. The tr ip usuaLl y takes him 8 hours. Today he fi nds himself daydreami ng and reali zes 120 miles in to hi s trip that he is running 15 mi nutes la te r than hi s usual pace at thi s po int. At what speed must he dri ve for the remain­der o f the tr ip to complete the tr ip in the usual amoun t of time?

50. II When you sneeze, the air in your lungs accelerates from rest BIO to approx imately ISO kmlh in about 0 .50 seconds .

a. What is the accelerat ion of the air in m/s2?

b. What is thi s accelerati on, in uni ts of g?

51.

52.

53.

54.

II Figure P2.5 1 shows the mo ti on d iagram. made at two frames of film per second, of a ball rolli ng along a trac k. The track has a 3 .0-m-long sticky section .

2 fram es per second

1 = 0 s

Sticky section of track ~

0+ o o DOOOQO

- 4 - 3 - 2 - ! 0 2 3 4 6

Meters

FIGURE P2 .51

a. Use the sca le to dete rmine the pos itions o f the center o f the bal l. Place your data in a tab le, s imi lar to Table 2. 1, show ing each pos ition and the ins ta nt o f time at whic h it occurred.

b. Make a graph of x versus t fo r the bal l. Because you have data only at certain instants of time, yo ur graph should con­sist o f dots that are not connec ted together.

c. What is the cha llge in the ball 's pos ition from 1 = 0 s to 1 = 1.0 s?

d. What is the change in the ball's pos ition from t = 2.0 s to ( = 4.0 s7

e. What is the ball's velocity before reaching the sticky sec ti on? f. What is the ball 's veloc ity after passing the stick"), sec ti on? g . Detemline the ball 's acceleration on the sticky secti on of the

track. Juli e drives LOO mi to Grand mother 's house. On the way to

Grandmother' s, Ju lie dri ycs half the distance at 40 mph and half the di stance at 60 mph. On her return trip, she dri ves half the time at 40 mph and half the time at 60 mph. a. How long does it take Juli e to complete the trip to Grand-

mother 's house? b. How long does the return trip take? II The takeoff speed for an Airbus A320 jet­liner is 80 m/s. Veloc ity data measured dur­ing takeoff are as shown in the table. a. What is the takeoff speed in miles per

hour? b. What is the jet liner's acce leration during

takeoff? c. At what time do the whee ls leave the

ground?

I (s)

o o 10 23

20 46 30 69

d. For safety reasons, in case of an aborted takeoff, the runway must be three times the takeoff di stance. Can an A320 take off sa fely on a 2.5-mi -long runway?

1111 Does a real automobile have cons tant accelerat ion? Measured data fo r a Porsche 944 Turbo at max imum acce lerati on are as shown in the table. a. Convert the veloc ities to mis, then make

a graph of ve loc ity ve rsus time. Based on your graph, is the accelerati on con-stan t? Ex plain.

b. Draw a smooth curve through the points on your graph, the n use your graph to esrinw(e the car 's accele rati on at 2.0 s

I (s) vA mph)

0 0

2 28

4 46 6 60 8 70

10 78

and 8.0 s. Give yo ur answer in SI unit s. Hint: Remember that accelerat ion is the slope of the veloc ity graph .

55. II People hoping to travel to other worlds are faced with huge challenges. One of thc biggest is the time requi rcd for a journcy. The nearest star is 4. 1 X 10Ib m away. Suppose you had a spacec rart that could accelerate at 1.0g for half a year, then con­tinue at a constant speed. (This is far beyond what can be ac hieved with any known technology.) How long wou ld it take you to reach the nearest star to earth?

56. III You are driving to the grocery store at 20 m/s. You are 11 0 m from an inte rsection when thc traffi c light turns red. Assume that your react ion time is 0.70 s and that your car brakes with constant acce leration. a. How Far are you from the inte rsec tion when you begin to

apply the brakes? b. What acceleration will bri ng you to rest right at the in tersec­

tion? c. How long does it take you to stop?

57. J When you blink your eye, the upper lid goes From rest with your BID cye open to completely cove ri ng your eye in a time of 0.024 s.

a. Estimate the di stance that the top lid of your eye moves dur­ing a bli nk.

b. What is the acceleration of your eyelid? Assume it to be con­stant.

c. What is your upper eyel id 's fin al speed as it hi ts the bottom eyel id?

58. III A bush baby, an African BID primate, is capablc of leap­

ing vert ica lly to the remark­able hcight of 2.3 m. To jump thi s high, the bush baby accelerates over a distance of 0. 16 m whi le rapidly ex tend­ing it s legs. The acceleration during the jump is approx i­mately constant. What is the acce lera ti on in m/s2 and in g's?

59. 1111 When jumping, a flea reaches a takeoff speed of 1.0 mls over BID a di stance of 0.50 mm .

a. What is the flea's accelerat ion during the jump phase? b. How long does the acceleration phase last? c. If the flea jumps strajght up, how high will it go? (Ignore air

res istance for thi s problem; in real ity, air res istance plays a large ro le , and the fl ca will not reach thi s height.)

60. 1II Certain insects can achieve BID seemi ngly imposs ible accel­

erations while j umping. The c li ck bee tlc accelerates at an aston ishing 400g over a dis­tance of 0.60 em as it rapidly bends its thorax, making the "cl ick" that gives it its name. a. Assumi ng the beetle jumps straight up, at what speed does it

leave the ground? b. How much time is required for the beetle to reach thi s speed? c. Ignori ng air res istance, how high would it go?

61 . 1111 Di vers compete by diving in to a 3.0-m-deep pool from a plat­form 10 m above the water. What is the magnitude of the mini ­mum acce lerat ion in the wate r needed to keep a diver from hitting the bottom of the pool? Assume the acceleration is con­stant.

Problems 65

62. 11 11 A studen t stand ing on the ground throws a ball straight up. The ball leaves the studen t's hand with a speed of 15 mls when the hand is 2.0 m above the grou nd. How long is the baU in the air before it hit s the ground? (The student moves her hand out of the way.)

63. 1111 A rock is tossed stra ight up with a speed of 20 m/s. When it returns, it fall s into a hole 10 m deep. a. What is the rock's velocity as it hits the bottom of the hole? b. How long is the rock in the air, from the instant it is released

until it hilS the bottom of the hole? 64 . 11111 A 200 kg weather rocket is loaded with 100 kg of fue l and

fi red stra ight up. It accelerates upward at 30.0 m/s2 for 30.0 s, then runs out of fuel. Ignore any air resistance effects. a. What is thc rocket's maximum altitude? b. How long is the rocket in the air? c. Draw a veloc ity-ve rsus-time graph for the rocket from

liftoff un til it hi ts the ground. 65. 11111 Ajuggler throws a baU straight up into the air with a speed of

10 m/s. With what speed would she need to throw a second ball half a second later, starting from the same pos ition as the first, in order to hi t the fi rst ball at the top of its trajectory?

66. 1111 A hotel e levator ascends 200 m with a max imum speed of 5.0 m/s. Its acceleration and deceleration both have a magni ­tude of 1.0 m/s2. a. How far does the elevator move while accelerating to full

speed from rest? b. How long does it take to make the complete trip from bot­

tom to top? 67. 1111 A car starts from rest at a stop sign. It accelerates at 2.0 m/s2

for 6.0 seconds, coasts For 2.0 s, and then slows down at a rate of 1.5 m/s2 for the next stop sign. How far apart are the stop signs?

68. 11 11 A toy train is pushed Forward and released at Xi = 2.0 m with a speed of 2.0 m/s. It ro ll s at a steady speed for 2.0 s, then one wheel beg ins to stick. The tra in comes to a stop 6.0 m from the point at which it was released. Whal is the tra in 's acce leration after its wheel begins to stick?

69. III Heather and Jerry are standing on a bridge 50 m above a river. Heather throws a rock stra ight down with a speed of 20 mls. Jerry, at exactly the same instant of time, throws a rock straight up with the same speed. Ignore air resistance. a. How much time elapses between the first splash and the sec­

ond splash? b. Which rock has the faster speed as it hi ts the water?

70. 1111 A motori st is dri ving at 20 mls when she sees that a traffic light 200 m ahead has j ust turned red. She knows that thi s li ght stays red for 15 s, and she wants to reach the light j ust as it turns green aga in. It takcs her 1.0 s to step on the brakes and begin slowing at a constant deceleration . What is her speed as she reaches the light at the instant it turns green?

7 1. 11 111 A "rocket car" is launched along a long straight track at ( = 0 s. It moves with cons tant acceleration GI = 2.0 mls2

. At ,= 2.0 s, a second car is launched along a paralle l track, from the same start ing point, with constan t accelerati on G2 = 8.0 I11/s2.

a. At what time does the second car catch up with the first one? b. How far have the cars traveled when the second passes the fi rst?

72. III A Parse he challenges a Honda to a 400 m race. Because the Porsche 's acceleration of 3.5 m/s 2 is larger than the Honda's 3.0 m/s2

, the Honda gets a 50-m head start. Assume, somewhat unreal isticall y, that bo th cars can maintai n these accelerations the enti re di stance. Who wins, and by how much time?

66 CHAPTER 2 Motion in One Dimension

73. 1111 1 The minimum stopping distance for a car traveling at a speed of 30 m/s is 60 m, including the distance traveled during the dri ver's react ion time of 0.50 s. a. What is the mini mum slOpping distance for the same car

travel ing at a speed of 40 m1s? b. Draw a pos ition-versus-time graph for the mot ion of the car

in part a. Assume the car is at Xj = 0 m when the driver first sees the emergency situation ahead that call s for a rapid halt.

74. 1111 1 AJocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt faUs off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket's acceleration?

Passage Problems

Free Fall on Different Worlds

Objects ill free fall on the earth have acceleration ay = -9.8 m/s2. On the moon, free-fall acceleration is approx imately 116 of the accelera­tion on earth. This changes the scale of problems involv ing free fall. For instance, suppose you jump straight upward, leaving the ground with ve loc ity Vi and then stead il y slowing un til reaching zero ve locity

Stop to Think 2.1: D. The moti on consists of two constant-velocity phases; the second one has a greater ve locity. The correct graph has two straight-l ine segments, with the second one hav ing a steeper slope.

Stop to Think 2.2: B. The displacement is the area under a ve locity­ve rsus-t ime curve. In all four cases, the graph is a straight line, so the area under the curve is a rectangle. The area is the product of the length times the height, so the largest disp lacement belongs to the graph with the largest product of the leng th (the time interva l, in s) times the height (the velocity, in m/s).

Stop to Think 2.3: C. Consider the slope of the pos ition-versus-time graph ; it starts ou t positive and constant, then decreases to zero. Thus the veloc ity graph must start with a constan t pos itive value, then decrease to zero.

at your highest point. Because your initial velocity is determi ned mostly by the strength of your leg musctes, we can assume your ini ­t ial ve locity would be the same on the moon. But conside ri ng the final equation in Table 2.4 we can see that, with a smaller free- fall accelerat ion, your max imum height would be greater. The following quest ions ask you to think about how certa in athletic feats might be performed in thi s reduced-gravity environment.

75. I If an astronaut can jump straight up to a height of 0.50 m on earth, how high could he jump on the moon? A. 1.2 m B. 3.0 m C. 3.6 m D. 18 m

76. I On the earth, an astronaut can safely jump (0 the ground from a height of 1.0 m; her velocity when reaching the ground is slow enough to not cause inj ury. From what height could the astronaut safe ly jump to the ground on the moon? A. 2.4 m B.6.0 m C. 7.2 m D.36 m

77 . I On the earth , an astronaut throws a ball straight upward; it stays in the air for a total time of3.0 s before reachi ng the ground again. If a ball were to be thrown upward with the same initial speed on the moon, how much time would pass before it hi t the ground? A. 7.3s B.1 8s C. 44s D. 108s

Stop to Think 2.4: C. Acceleration is the slope of the veloc ity­versus- time graph. The largest magni tude of the slope is at poin t C.

Stop to Think 2.5: E. An accelerat ion of 1.2 m1s2 corresponds to an increase of 1.2 m/s every second. At thi s rate, the cruising speed of 6.0 m/s will be reached after 5.0 s.

Stop to Think 2.6: D. The fi nal veloc ity wi ll have the same magnitude as the ini tial veloc ity, but the veloc ity is negati ve because the rock will be moving downward.