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. Thi s may include any or all of the elements below:
PREPARE
Set lip the problem: l\Iotion diagram
Pictorial representation
Draw a picture. y
Collect necessary information.
,
Do preliminary calculations. Do the necessary mathemati cs or reasoning.
SOLVE
,.,
YI' (1',)1' If
List or values Known Yi= Om (r,)i = 0 m/s
(I',)f
I; = Os
ta
ta
Check your an swer to see if it is complete in all detail s and makes physica l se nse.
~ ij
Graphicall't!presentation
(I ,
= !8 m/s l
I f = 150 s
ASSESS
Find
0
"
Yi' (v)",;
(V,)I andYr
IMPORTANT CONCEPTS A position-versus-time graph
Velocity is the rate of change of position: II
,
~x
= -
~f
x(m)
Acceleration is the rate of change of veloc ity:
V
6
~"' (/ = ,
A velocity-versus-time graph plots velocity on the vertical axis against time on the hori zontal axis.
plots position on the vertical ax is against time on the horizontal axis.
t
(m/s)
30
Acce)emliQn j, the slope or The velOCiTY graph.
4
~f
2
The units of acceleration are m/s 2 .
10
o +"-~~-~-~-~ I
(s)
0+--_-_-_-="- , (s) 2 3 4 o
0246810
An object is speeding up if V.I and ax have the same sign , slowing down if they have opposite sign s.
APPLICATIONS Uniform motion An object in uniform motion has a constant velocity. Its velocity graph is a horizontal line; its position graph is linear.
Motion with constant acceleration An object with constant acceleration has a co nstantly chang ing velocity. Its velocity graph is linear; its position graph is a parabola.
Free fall Free fall is a special case of constan tacceleratio n moti on; the acceleration has magnit ude g = 9.80 m/s 2 and is always directed vert ical ly downward whether an object is moving up or down .
, ",
Kinematic equatio n for uniform motion: Xf= Xi
+ VA' 61
Uniform motion is a special case of constantacceleration motion , with a.l , = O.
Kinematic eq uations for motion with constant accelerat ion: (V,)r = (V)i Xf =
Xi
(vJl =
+
(/x
61
+ (v,,), D.I + ta,( D.t )2 (v.T)? + 2ax D. X
.
,.
",
T he vc!oci ty grJ ph is a straight line with a slope of - 9.80 m/sl.
./
59
Question s
tMP)TM
!:!!/
INT integrate significant material from BID are of biological or medical interest.
For homework assigned on MasteringPhysics, go to
Prob lems labeled
www.masteringphysics.com
chapters;
earlier
Problem difficulty is labeled as I (straightforward) to 11111 (challenging),
QUESTIONS Conceptual Ouestions 1. A perso n gets in an elevator on the gro und floor and rides it to the lOp fl oor or a building. Sketch a veloc ity-ve rsus-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 lerati on. b. Give an example of a vertical motion with a negative ve locity and a negative acceleration. 3. S ketch a ve loc ity-versus-tim c graph for a roclnhat is thrown strai ght upward, from th e instant it leaves the hand until the instant it hits the ground. 4. You g? a. Just after leaving your hand ? b. At the very top (maximum height)? c. Just before hittin g the ground ? 8. A rock is (hrollln (not dropped) strai ght 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? Expl ain . b. Immediatel y before hitting the water, is the magn itude of the the roc k's acceleration grealer than g, less than g, or equal to g? Expla in. 9. Figure Q2.9 shows an object's positio n-verslis- time graph . The letters A to E correspo nd to various segments of the mot io n in which the graph has constant slo pe . x(rn)
B
2
c. In which segment(s) is the object mov ing to the ri ght ? d. Is the speed of the object during segment C greate r than , equal to, or less than its speed durin g segment E? Explain. 10. Figu re Q2.IO sho ws the positi o n graph for an object moving along the hori zontal ax is. a. Write a real istic moti o n short story for an objecl that would have thi s position graph. b. Draw the correspondin g velocity graph. x
\ FIGURE 02 . 10
\'-------~
B II. Figure 02.11 shows the pos iti o n- x versus-t ime graphs for two A obj ec ts, A an d B, that are mov in g along the same ax is. a. At the in stant ( = Ls, is the speed of A greater than , less t (s) 2 3 4 5 than , or equal to the speed of 0 B? Ex plain. FI GURE 02 .11 b. Do objects A and B ever have thesClme speed? If so, at what time or limes? Ex plain. 8 12. Figure Q2. 12 shows a position - x versus-time graph. At which lettered point or po int s is the object D a. Mov ing the fastest? E ' -_ _ b. Moving to the le ft ? F c. Speeding up? FIGURE 02 . 12 d. Slowing down? e. Turning around? 13. Fi g ure Q2.1 3 is the vel oc ity-versus-t ime graph for an objec t moving along the x-axis. a. Durin g which segment(s) is the veloc ity constant? b. During which segment( s) is the object speedin g up ? c. Durin g which seg ment(s) is the object slowin g down? d. During which segment(s) is the object standing still ? e . Durin g which segment(s) is the object mov ing to the right? I'~
A
(m/s)
3
2 - I
4
6
8
2
A
E
B
FIGURE 02 .9 - 2
0
F
2
3
- I
a. Write a reali st ic motion short story for an objec t that would have thi s positi on graph. b. In whi ch segmen t(s) is the object at rest?
-2 FIGURE 02 .13
-3
I (s)
10
4
D
60
CHAPTER 2
Motion in One Dim ension
14. A car travelin g at veloc ity v takes d istance d to stop after th e brakes are appl ied. W hat is the stopping d istance if the car is ini tiaUy trave lin g at veloc ity 2v? Ass ume th at the acce lerati on due to the braking is the same in both cases.
Multiple-Choice Questions IS. 1 Fig ure Q2.L5 shows the E pos itio n graph of a car traveling on a straight road. At which labeled instant is the speed of the car greatest? 16. I Fi gure Q2. 16 shows the pos iti on graph of a car travel- FIGURE OZ .15 in g on a stra ight road. The x vel oc 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 x(m) 17. 1 Figure 02 .1 7 shows an 40 object 's pos ition-versus-time 30 graph. W hat is the ve loc ity of 20 the object at t = 6 s? A. 0.67 ml' 10 B. 0.83 ml, O +----_---~I (s) C. 3.3 m/s o 5 10 D. 4.2 mls FIGURE OZ .17 E. 25 mls 18. I The fo ll owin g option s desc ri be th e mot io n of fo ur cars A- D. Which car has the largest accelerat ion? A. Goes from 0 mls to 10 mls in 5.0 s 8 . Goes fro m 0 mls to 5.0 mls in 2.0 s C. Goes from 0 mls to 20 mls in 7.0 s D. Goes fro m 0 mls to 3.0 mls in 1.0 s 19. I A car is trave lin g at Vx = 20 m/s. T he dr iver appli es th e 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 show n in Fi gure Q2.20. At t = 5.0 s, whi ch car has traveled the furthest?
2 1. I Whi ch of the three drag race rs in Question 20 had the greatest acceleration at I = 0 s? A. A ndy B. Belly C. Carl D. All had the same acceleration 22. II Ball 1 is thrown straight up in the air and, at the same in sta nt. ball 2 is released fro m rest and allowed to fall. Whi ch veloc ity graph in Figure Q2.22 best represents th e motion of the two ball s?
~,
A. Andy B. Belly C. Carl D. All have traveled the same d istance
I'~ ( m /s)
1',
(m/s)
4: W ,(,) 4:k===L,) o
Andy
FIGURE Q2 .20
5.0
o
Betty
5.0
/
~or') O~I (S) o Carl 5.0
FIGURE 02 .22
23 . II Figure Q2.23 shows a motion d iagram with the clock read ing (in seconds) shown al each pos iti on. From I = 9 s to I = 15 s the object is at the same pos iti on . After that, it returns along the same trac k. The pos itions of the dots for f ;::= 16 s are offset for clarity. Whi ch graph best" re prese nts the object's velocity?
o
•
• • • • ~ : ! . /9to 15 • • 21• 21• 20• 19• 18• 17• 16• 23 2
C.
FIGURE 02 .23
I'x
3
4
5
D.
I'x
h, ~,
24. II A car can go from 0 to 60 mph in 7.0 s. Assumi ng that it co ul d maintain the same accelerati o n at hi gher speeds, how long would it take the car to go from 0 to 120 mph ? A. l Os 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 o n of the fi rst car. Assumin g that it co ul d maintain the same accelerati o n at hi g he r speeds, how much time will thi s seco nd car take to go fro m 0 to 120 mph ? A. 12 s B. 9.0s C. 6.0 s
D. 3.0s
Probl em s
61
PROBLEMS Section 2. t Describing Motion I. III Figure P2. l shows a
m Ol ion
diagram o f a car travel ing down
a street. The camera took one fra me every second. A distance
scale is prov ided. a. Measure the x-val ue of the car at each dot. Place your data in a table, simi lar to Table 2. 1, show in g each pos itio n and the instanroftime at whi ch it occurred. b. Make a graph of x vers us I, using the data in your table. Because you have data only at certain in stants of lime, your graph should consist of dots that arc not connected together.
6. II The pos ition graph of Figure P2.6 represe nts the mot ion of a ball being ro Ued back and forth by two children. a. At what positi o ns 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 frame per second
o
4flO
2()()
600
FIGURE P2 .1
2. I For eac h moti on d iagram in Fi gure P2.2, determine the sign (positi ve or negat ive) of the pos iti on and the velocit y. (bJ
••
• ~ I
('J 0
0
t
FIGURE P2 .2
0
t
3. I Write a short desc ripti on of the mot ion of a real object for which Figure P2.3 would be a realist ic pos ition-versus- time graph . x(mi)
x(m)
300
40
200
20
- 6
o ~~~~-c~~-+:I (s) o 200 400 600
I', (Ill/s)
1'.< (Ill/S )
3 2
3 2
FIGURE P2 .4
4. I Write a short desc ripti on of the moti on of a real object fo r which Figure P2.4 would be a reaJi sti c pos itio n-versus- time graph . 5. II The pos ition g raph of Fi gure P2.5 shows a dog slow ly sneaki ng up on a squi rrel, then putting on a burst of speed. a. For how many seco nds does the dog move at the slowe r speed? b. Draw the dog' s ve loc il y-ve rsus-ti me graph. In clude a numerical scale on both axes.
r--, I I
I I I
4
6
I I
I
0 - I
1(8)
2
4
6
I
'f
L-.J
FIGURE P2 .7
- 20
FIGURE P2 .3
7. II A rural ma il carri er is driving slow ly, puttin g ma il in ma il boxes near the road. He overshoots o ne mailbox.s tops.s hi fts into reverse, and then bac ks 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 itio n-versus-time graph. Assume that x = 0 m at f = 0 s. b. What is the pos ition of the mail box?
-2 -3
O +-~+C-~~~ I (h)
100
(s)
20
- 4
I flO()
800
FIGURE P2 .6
(a)
I
16
- 2
0
- I
I
2
-2 -3
1 I I
8
10
(s)
"
FIGURE P2.8
For the ve loc ity-versus-time graph of Figure P2.8: a. Draw the correspond ing position-versus-time graph. Assume th at x = 0 m at f = 0 s. b. What is the object's posit ion at f = 12 s? c. Desc ri be a mov ing objec t that could have these graphs. 9. II A bicycl ist has th e pos iti o n-vers us- 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?
8.
x(m)
100
50 -
FIGURE P2 .9
o+--~---c~-:--+-t (s) o 10 20 30 4{)
x(m)
~D "., o
FIGURE P2 .5
2
4
6
8
10
Section 2.2 Uniform Motion 10. I In co ll ege softball , the di stance from the pitc her' s moun d to the balle r is 43 fee l. If the ba ll leaves the bat at 100 mph , how muc h tim e elapses between th e hi t and the ball reac hi ng the pitcher?
62
Motion in One Dimension
CHAPTER 2
11. II Ala n 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 d ri 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 seco nd? 12. II Richard is driving home to vis it his pare nts. 125 mi of the tri p .u e o n the interstate hi ghway where the speed limit is 65 mph . Normally Richard drives a1 the speed li mit, but today he is run ning late and decides to take hi s chances by d ri 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 run s at 14.5 km/h. "H ow 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 run s at a steady 11 .0 kmlh and anot her run s at 14.0 kmfh. How far from the fini sh line is th e slower ru nner whe n the faster runner fini shes the race? 15. Il A car moves with co nstant velocity along a straight road. Its posi ti o n is XI = 0 m at '1 = 0 s an d is X2 = 30 III at 12 = 3.0 s . Answer the fo ll owin g by considerin g 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 itio n at I = 9.0 s? 16. II Whil e run ning a marathon , a lo ng-d istance run ner uses a stopwatch to time herse lf over a distance of 1.00 Ill. She finds th at she run s thi s d istance in IS s. Answer the following by consider in g rati os, without computin g her ve locity. a. If she mainta in s 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?
19. III A car starts from I'~ (m/s) Xi = \0 m at Ii = 0 sand moves with the veloc ity 8 graph shown in Figure 4 P2. 19. o +--~-2'---"3"~4c- 1 (s) a. W hat is the o bject'S - 4 posit io n at 1 = 2 s, 3 s, FIGURE P2 .19 and 4 s? b. Does thi s car ever change d i.recti on? If so, at what time? 20. Figure P2.20 shows a graph of act ual pos itio n-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
FIGURE P2 .20
0
I
2
0
4
(s)
6
Section 2.4 Acceleration 2 1. II Figure P2.2 1 Shows the veloc ity graph of a bicycle. Draw the bicycl e's accelerati on graph for the in terval 0 s :5 1 :5 4 s. Give both axes an appropri ate numeri cal scale. 1'...
(m/s)
4
--
v, (m/s) 2
17. 1 Fig ure 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. o +---I--~-~--'\- I (s) b. Does thi s particle have a 0 24 turni ng po int 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 asce nd ing aorta durin g one beat of the heart appea rs as in Figure P2. 1S. a. Approx imately how far, in c m, does the bl ood move duri ng o ne beat? b. Ass ume simil ar data for the mot io n 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'
1
2
Section 2.3 Instantaneous Velocity
(m/s)
0
I
8
- I
0
0 2 FIGURE P2.21
I
(s)
(s)
10
-2
4 FIGURE P2 .22
22. III Figure P2 .22 shows the veloc it y graph of a train th at starts from the ori gin at I = 0 s. a. Draw pos ition and accelerat ion graphs for the train. b. Find the accelerat io n of the train at ( = 3.0 s. 23. I For eac h moti o n d iagram shown earli er in Figure P2.2, determi ne the sign (positi ve or negative) of the acce lerat ion. 24. II Figure P2. 18 showed data for the speed of blood in the aorta . BID Determine the magn itude of the accelerati o n for both phases, speed in g up and slowing down. 25. II Figure P2.25 is a somew hat L'... (m/s) simp lifi ed veloci ty graph for 12 c O lympic spr inter Carl Lewis 10 8 B starting a 100 III dash. Est im ate hi s acce lerat io n d uri ng eac h of 6 4 A the intervals A, B, and C. 2
1.0
O¥--~-~-~-~I (s)
FIGURE P2 .2S
2
3
4
Section 2.5 Motion with Constant Acceleration
0.5
FIGURE P2 .18
0
o +---+--~I--~- f (s) 0
0.20
0.40
0.60
26. I A Thomson's gazelJ e can reach a speed of 13 mls in 3.0 s. A BID lio n 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. Whi ch anim al has the largest acce leratio n?
Problems 27. III When strikin g, 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. Li ght-rail passenger trains that provide tran sportati o n within 29. and between ci ties are capab le o f modes t accelerati o ns. T he magni tude of the maximum acceleration is typi ca 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, w ith the same accelerat io n, for a nother 16 s to reach a hi gher crui sin g speed. What is the final speed? 30. 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 co ntinues on at 6.0 m/s. What is her acceleration on the rough ice ? 31. 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 passe nger jet has about the same accelerati on from rest, but it needs to achieve tw ice the speed to take off. What is the minimum runway length that will serve? Hint: You can solve this problem us ing ratios w ithout hav ing any additional infonn ati on. 32. II Fi gure P2.32 shows a velocv~ (mJs) ity-vers us-t ime graph for a particle moving along the x-aris. At t = 0 s. assume that x = 0 m. 2 a. What are the particle's posiO +--~-~~--+-I (s) ti on , veloc ity, and accelera2 3 4 o tion at t = 1.0 s? FIGURE P2 .32 b. What are the parti cle's position , ve locity, and accelerati on at t = 3.0 s?
4+-- -,
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/s 2 . 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 in sects w ith 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 rem arkabl e 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 hi g hway late one ni g ht at 20 m/s when a deer ste ps on to the road 35 m in front o f you. Your reaction tim e before stepping on the brakes is 0.50 s, and the maximum deceleration of your car is 10 m/s 2 . a. How much di stance is be tween you and the deer whe n you co me to a stop? b. What is the max imum speed you could have and still not hit the deer?
63
36. JlI A li ght-ra il trai n go in g 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 s peed 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 s imp le model for a person runni ng the 100 m dash is to assu me the spri nter run s with cons tant acce leration u nt 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 w ill 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 hi gh 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 eno ugh 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 catc hing 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 th e 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 w ith speed Vo and throws ball B straight downward with the same initi al speed. a. Compare the ball s' 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 heig ht of 110 cm o ff the gro und. To reac h thi s he ight , w ith what speed would a person need to leave the grou nd ? 43. II A footba ll is kicked strai ght up into the air; it hits the ground 5.2 s laler. 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 m ov ie, the viU a in is rescued from the ocea n 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 th e water. If the briefcase hits the water 6.0 s later, what was the speed at whi ch the heli copter was ascend ing? 45. 1111 A rock cli mber stan ds on top o f a 50- m-hi gh cliff overh angin g 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 lon g 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 Ac tual ve loc ity data fo r a li on pu rs ui ng prey are show n in BIO Fig ure P2.46. Estimate: a. The initial accelerat ion of the li o n. b. The acce leration of the lio n at 2 s and at 4 s. c. The di stance traveled by the li on between 0 sand 8 s. I'~ ( m/s)
15
5 1. II Figure P2.5 1 shows the mo ti on d iagram. made at two fra mes 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=
Sticky section
of track ~
0s
0+
o
-4
- 2
- 3
o - !
0
DOOOQO 2
3
4
6
Meters
FIGURE P2 .51
O+---~----,~ I (s)
048 FIGURE P2 .46
Prob lems 47 and 4 8 concern Ilen'e Unmyelinated fiber A: impulses, electrical signals propaIndividual gated alo ng nerve fi bers cons ist ing Panly myelinated fi ber B:/" awns of many axol/s (fi berlike extensio ns == == of nerve ceLis) con nected end-to-end . Axo ns come in two varieties: insu- Fully myelinated fiber C: lated axons w ith a sheath made o f FIGURE P2 .41 mye lin , and unins ulated axons with no such sheath. Myel inated (sheathed) axons conduct nerve impu lses much faste r than unm yelinated (unsheathed) axons. The impulse speed depends on the diameter o f the axons and the sheath, but a typical mye lin ated axon transmi ts nerve impulses at a speed of about 25 mis , much faster than the typi cal 2.0 mls for an unmyeli nated axo n. Figure P2.47 shows small porti ons of three nerve fi bers consisti ng o f axons of equal size. Two-thirds of the axons in fi ber B are myel inated. 47. I S uppose nerve impu lses 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 vel oc ity graph s fo r th e nerve impulses in all th ree cases. A nerve fib er is made up o f many axons, but show the propagat ion of the im pu lses only over the six axons show n here. 48. I S uppose that the ner ve fib ers in Fi gure P2.4 7 con nect a finBIO ger to your brain , a di stance o f 1.2 m. a. W hat are th e travel tim es of a nerve im pul se from fi nger (0 brain alo ng fi bers A and C? b. Fo r fib er B, 2/3 o f Ihe le ngth is composed of myeli nated axons, 1/3 unm yelin ated axo ns. Compute the travel time for a nerve impul se on thi s fiber. c. When you touch a hot stove w ith your fin ger, the se nsat io n o f pain must reac h your bra in as a nerve s ignal along a nerve fibe r before your muscles ca n react. Whi ch of the three fibers g ives you th e bes t protec ti o n aga in st a bu rn? Are any o f th ese fi bers uns ui table for transmittin g urgent sensory infonn atio n? 49. II A truck d ri ve r has a shipment of apples to del iver to a destinati o n 440 mi les away. The tr ip us uaLl y tak es him 8 ho urs. Today he fi nds him se lf daydreami ng and reali zes 120 mil es in to hi s trip th at he is runnin g 15 mi nutes la te r th an hi s usual pace at thi s po int. At what speed mu st he dri ve for the remain der o f the trip to co mplete the trip in the us ual amoun t of tim e? 50. II When you sneeze, the air in your lun gs accelerates from rest BIO to approx imately ISO kmlh in abo ut 0 .50 seco nds . a. What is the accelerat ion of the air in m/s 2 ? b. What is thi s accelerati on, in uni ts of g?
a. Use th e sca le to dete rmin e th e pos iti o ns o f th e ce nter o f the bal l. Pl ace your data in a tab le, simi lar to Tab le 2. 1, show ing eac h pos iti o n an d the in s ta nt o f tim e at w hi c 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 co nsist o f dots that are not connec ted together. c. What is the cha ll ge in the ball 's pos itio n from 1 = 0 s to 1 = 1.0 s? d. What is the change in the ball's pos iti on fro m 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 accelerati on on the sticky secti on o f the track. Juli e drives LOO mi to Grand mother's house. On the way to 52. Grandm other' s, Ju lie dri ycs half th e distance at 40 mph and half th e di stan ce at 60 mph. O n her return trip, she dri ves half the time at 40 mph and half the time at 60 mph. a. How lo ng does it take Juli e to co mplete the trip to Gra ndmother 's house? b. How long does the return trip take? 53. II The takeoff speed for an A irbus A3 20 jet- I (s) liner is 80 m/s. Veloc ity data measured duro o ing takeoff are as shown in the table. 23 a. What is the takeo ff speed in miles per 10 hour? 20 46 b. What is the jet liner's acce leration durin g 30 69 takeoff? c. A t w hat time do th e whee ls leave the ground? d. For safety reasons, in case of an aborted takeo ff, the runway m ust be three times the takeoff di stance. Can an A320 take off sa fely o n a 2.5- mi -long runway? 54. 1111 Does a real automob ile have cons tant I (s) vA mph ) accelerat ion? Measured data fo r a Porsc he 0 0 944 Turbo at max imum acce lerati on are as 2 28 show n in the table. 4 46 a. Convert the veloc ities to mis, then make 6 60 a graph of ve loc ity ve rsus time. Based on your graph, is the accelerati o n con8 70 stan t? Ex plain. 10 78 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 an d 8.0 s. Gi ve yo ur an swer in S I unit s. Hint: Remem ber that accelerat ion is the slope of the veloc ity graph .
Problem s
55. II People hopin g to travel to other worlds are faced with hu ge chall enges. 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 cou ld accelerate at 1.0g for half a year, then continue at a constant speed. (Thi s is far beyond what can be ac hi eved with any known tec hnology.) How long wou ld it take you to reach the nearest star to earth ? 56. III You are dri ving to the grocery store at 20 m/s. You are 11 0 m from an inte rsection when thc traffi c li ght turns red. Assume th at your react ion time is 0.70 s and th at your car brakes with constant acce leration. a. Ho w Far are you from the inte rsec ti on whe n you beg in to appl y the brakes? b. What acceleration will bri ng you to rest right at the in tersection? 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 eyeli d? Assume it to be constant. c. Wh at 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 ca pablc of leaping vert ica ll y to the re markable hc ight of 2.3 m. To jump thi s hi gh, th e bush baby accelerates over a distance of 0. 16 m whi le rapi dly ex te nding it s legs. The accelerati on during the jump is approx imatel y consta nt. Wh at is the acce lera ti on in m/s2 and in g's? 59. 1111 Whe n jumping, a flea reaches a takeo ff 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 hi gh will it go? (Ignore air res istance for thi s proble m; in real ity, air res istance plays a large ro le , and the fl ca will not reac h thi s he ight.) 60. 1II Certain insects ca n achieve BID seemi ngly imposs ible accelerati ons while j umpin g. T he c li ck bee tlc accelerates at an aston ishin g 400g over a distance of 0.60 em as it rapi dly be nds its thorax, makin g the "cl ic k" th at gives it its name. a. Assumi ng the beetle j umps 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 hi gh would it go? 61 . 1111 Di vers compete by diving in to a 3.0-m-deep pool from a pl atform 10 m above the water. What is the magnitude of the mini mum acce lerat ion in th e wate r needed to keep a diver from hitti ng the bottom of the pool? Ass ume the acceleration is constant.
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1111 A studen t stand ing on the ground throws a ball strai ght 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 stude nt moves he r hand out of the way.) 1111 A roc k is tossed stra ight up with a speed of 20 m/s. Wh en it returns, it fall s into a hole 10 m deep. a. What is the rock's veloc ity as it hits the bottom of the hole? b. How long is the roc k in the air, from the in stant it is released until it hilS the bottom of the hole? 11111 A 200 kg weather rocket is loaded with 100 kg of fue l and fi red stra ight up . It accelerates up ward at 30.0 m/s 2 for 30. 0 s, then runs out of fuel. Ignore any air resistance effects. a. Wh at is thc roc ket's maximum altitude? b. How long is the roc ket in the air? c. Draw a veloc ity-ve rsus-time graph fo r the roc ket from liftoff un til it hi ts the ground. 11111 Ajuggler throws a baUstraight 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, startin g from the same pos ition as the first, in order to hi t the fi rst ball at the top of its trajectory? 1111 A hotel e levator ascends 200 m with a max imum speed of 5.0 m/s. Its acceleration and decelerati on both have a magni tude of 1.0 m/s 2. a. How far does the el evator move whil e acceleratin g to full speed from rest? b. How long does it take to make the complete trip from bottom to top? 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/s 2 for the next stop sign. How far apart are the stop signs? 1111 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 in s 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 le rati on after its wheel begins to stick? III Heather and Jerry are standing on a bri dge 50 m above a river. Heath er throws a rock stra ight dow n with a speed of 20 mls. Je rry, 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 spl ash and the second splash? b. Which rock has the faster speed as it hi ts the water? 1111 A motori st is dri ving at 20 mls when she sees that a traffi c li ght 200 m ahead has j ust turned red. She kn ows th at thi s li ght stays red for 15 s, and she wants to reach the light j ust as it turns gree n aga in. It takcs her 1.0 s to step on the brakes and beg in slowin g at a constant decelerati on . What is her speed as she reac hes the li ght at the in stant it turns green? 11111 A "roc ket car" is launched along a long strai ght trac k at 2 ( = 0 s. It moves with cons tant accelerati on GI = 2.0 mls . At 2.0 s, a secon d car is launched along a paralle l trac k, fro m the same start ing point, with constan t accelerati on G2 = 8.0 I11/s2. a. At what time does the seco nd car catch up with the first one? b. How far have the cars traveled when the second passes the fi rst? III A Parse he c hall enges a Honda to a 400 m race. Because the Porsche's accel eration of 3.5 m/s 2 is larger than the Hond a's 3.0 m/s 2 , the Honda gets a 50- m head start. Assume, somew hat unreal isticall y, that bo th cars can maintai n these accelerati ons the enti re di stance. Who wins, and by how much time?
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CHAPTER
2
Motion in One Dimension
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1111 1 The minimum stopping distance for a car traveling at a speed of 30 m/s is 60 m, in clud ing th e dista nce traveled durin g the dri ver's react ion time of 0.50 s. a. Wh at 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 fo r the mot ion of the car in part a. Ass ume 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/s 2 . On the moon, free- fall acceleration is approx imately 116 of the acceleration on earth. Thi s changes the scale of problems in volv ing free fall. For in stance, suppose you j ump straight upward, leav ing the ground with ve loc ity Vi and then stead il y slowin g un til reachin g zero ve loc ity
Stop to Think 2.1: D. The moti on consists of two constant-veloc ity phases; the seco nd one has a greater ve loc ity. The correct graph has two straight-l ine seg ments, with the seco nd one hav in g a steeper slope. Stop to Think 2.2: B. The displ acement is the area under a ve locityve rsus-t ime c urve . In all fo ur cases, the graph is a straight line, so the area under the curve is a rectangle. The area is th e product of the le ngth times th e he ight, so the largest disp laceme nt be longs to the graph with the largest product of the leng th (the time interva l, in s) times the height (the vel ocity, in m/s). Stop to Think 2.3: C. Consider the slope of the pos ition-vers us-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 val ue, then decrease to zero.
at yo ur hi ghest point. Because your initial vel oc ity is determi ned mostly by the strength of your leg mu sctes, we can assume your ini tial ve loc ity would be the same on the moo n. But conside ri ng the final eq uati on in Table 2.4 we ca n see that, with a smaller free- fall accelerat ion, your max imum height wou ld be greater. The follow ing quest ions ask you to thin k about how certa in athletic feats mi ght be performed in thi s reduced-gravity e nvironment. 75. I If an astronaut can jump straight up to a he ight of 0.50 m on earth, how hi gh 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 safel y j ump (0 the ground from a he ight of 1.0 m; her veloc ity when reac hing the groun d is slow eno ugh to not cause inj ury. From what height coul d the astro naut sa fe ly j ump 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 initi al 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 T hink 2.5: E. An accelerat ion of 1.2 m1s 2 corresponds to an increase of 1.2 m/s eve ry second. At thi s rate, the cru isin g 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 sa me magnitude as the ini tial veloc ity, but the veloc ity is negati ve because the rock will be mov ing downward.
