chapter 3 accelerated motion - mr....

Physics: Principles and Problems Solutions Manual 29

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3 Accelerated MotionCHAPTER

Practice Problems3.1 Acceleration

pages 57–64page 61

1. A dog runs into a room and sees a cat at theother end of the room. The dog instantlystops running but slides along the woodfloor until he stops, by slowing down with aconstant acceleration. Sketch a motion dia-gram for this situation, and use the velocityvectors to find the acceleration vector.

2. Figure 3-5 is a v-t graph for Steven as hewalks along the midway at the state fair.Sketch the corresponding motion diagram,complete with velocity vectors.

■ Figure 3-5

3. Refer to the v-t graph of the toy train in Figure 3-6 to answer the following questions.

■ Figure 3-6

a. When is the train’s speed constant?

5.0 to 15.0 sb. During which time interval is the train’s

acceleration positive?

0.0 to 5.0 sc. When is the train’s acceleration most

negative?

15.0 to 20.0 s

4. Refer to Figure 3-6 to find the average acceleration of the train during the follow-ing time intervals.

a. 0.0 s to 5.0 s

a! !

!

! 2.0 m/s2

b. 15.0 s to 20.0 s

a! !

!

! "1.2 m/s2

c. 0.0 s to 40.0 s

a! !v2 " v1!t2 " t1

4.0 m/s " 10.0 m/s!!!20.0 s " 15.0 s

v2 " v1!t2 " t1

10.0 m/s " 0.0 m/s!!!5.0 s " 0.0 s

v2 " v1!t2 " t1

30.020.010.0 40.00.0

6.04.02.0

12.010.08.0

Vel

ocit

y (m

/s)

Time (s)

Time (s) 0 1 2 3 4 567

8 109

Vel

ocit

y (m

/s)

Time (s)

5 6 7 8 943210 10

StopStart

1

v1

2

v2

3

v3

TimeintervalVelocity

Positionv2

!v1a

!

! 0.0 m/s2

5. Plot a v-t graph representing the followingmotion. An elevator starts at rest from theground floor of a three-story shoppingmall. It accelerates upward for 2.0 s at arate of 0.5 m/s2, continues up at a con-stant velocity of 1.0 m/s for 12.0 s, andthen experiences a constant downwardacceleration of 0.25 m/s2 for 4.0 s as itreaches the third floor.

page 646. A race car’s velocity increases from 4.0 m/s

to 36 m/s over a 4.0-s time interval. What isits average acceleration?

a! ! !""vt! ! ! 8.0 m/s2

7. The race car in the previous problem slowsfrom 36 m/s to 15 m/s over 3.0 s. What isits average acceleration?

a! ! !""vt! !!15 m/s

3."0 s

36 m/s! ! "7.0 m/s2

8. A car is coasting backwards downhill at aspeed of 3.0 m/s when the driver gets theengine started. After 2.5 s, the car is movinguphill at 4.5 m/s. If uphill is chosen as thepositive direction, what is the car’s averageacceleration?

a! ! !""vt! ! ! 3.0 m/s2

9. A bus is moving at 25 m/s when the driversteps on the brakes and brings the bus to astop in 3.0 s.

a. What is the average acceleration of thebus while braking?

a! ! !""vt!

! ! "8.3 m/s2

b. If the bus took twice as long to stop,how would the acceleration comparewith what you found in part a?

half as great ("4.2 m/s2)

10. Rohith has been jogging to the bus stop for2.0 min at 3.5 m/s when he looks at hiswatch and sees that he has plenty of timebefore the bus arrives. Over the next 10.0 s,he slows his pace to a leisurely 0.75 m/s.What was his average acceleration duringthis 10.0 s?

a! ! !""vt!

!

! "0.28 m/s2

11. If the rate of continental drift were toabruptly slow from 1.0 cm/yr to 0.5 cm/yrover the time interval of a year, what wouldbe the average acceleration?

a! ! !""vt! !

! "0.5 cm/yr2

Section Review3.1 Acceleration

pages 57–64page 6412. Velocity-Time Graph What information

can you obtain from a velocity-time graph?

The velocity at any time, the time atwhich the object had a particular velocity, the sign of the velocity, and thedisplacement.

13. Position-Time and Velocity-Time GraphsTwo joggers run at a constant velocity of 7.5 m/s toward the east. At time t " 0, one

0.5 cm/yr " 1.0 cm/yr!!!1.0 yr

0.75 m/s " 3.5 m/s!!!10.0 s

0.0 m/s " 25 m/s!!!3.0 s

4.5 m/s " ("3.0 m/s)!!!2.5 s

36 m/s " 4.0 m/s!!!4.0 s

Vel

ocit

y (m

/s)

1.0

Time (s)

15.05.00.0 10.0 20.0

0.0 m/s " 0.0 m/s!!!40.0 s " 0.0 s

30 Solutions Manual Physics: Principles and Problems

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is 15 m east of the origin and the other is15 m west.

a. What would be the difference(s) in theposition-time graphs of their motion?

Both lines would have the sameslope, but they would rise from thed-axis at different points, #15 m,and "15 m.

b. What would be the difference(s) in theirvelocity-time graphs?

Their velocity-time graphs would beidentical.

14. Velocity Explain how you would use avelocity-time graph to find the time atwhich an object had a specified velocity.

Draw or imagine a horizontal line at thespecified velocity. Find the point wherethe graph intersects this line. Drop aline straight down to the t-axis. Thiswould be the required time.

15. Velocity-Time Graph Sketch a velocity-timegraph for a car that goes east at 25 m/s for100 s, then west at 25 m/s for another 100 s.

16. Average Velocity and Average AccelerationA canoeist paddles upstream at 2 m/s andthen turns around and floats downstream at4 m/s. The turnaround time is 8 s.

a. What is the average velocity of the canoe?

Choose a coordinate system withthe positive direction upstream.

v! !

!

! "1 m/sb. What is the average acceleration of the

canoe?

a! ! !""vt!

! !vf

"

"

tvi!

!

! 0.8 m/s2

17. Critical Thinking A police officer clockeda driver going 32 km/h over the speed limitjust as the driver passed a slower car. Bothdrivers were issued speeding tickets. Thejudge agreed with the officer that both wereguilty. The judgement was issued based onthe assumption that the cars must havebeen going the same speed because theywere observed next to each other. Are thejudge and the police officer correct? Explainwith a sketch, a motion diagram, and aposition-time graph.

No, they had the same position, notvelocity. To have the same velocity, theywould have had to have the same rela-tive position for a length of time.

("4 m/s) " (2 m/s)!!!8 s

2 m/s # ("4 m/s)!!!2

vi # vf!2

25

Vel

ocit

y (m

/s)

"25

200Time (s)

100

Chapter 3 continued


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Practice Problems3.2 Motion with Constant Acceleration

pages 65–71page 6518. A golf ball rolls up a hill toward a miniature-golf hole. Assume that the

direction toward the hole is positive.

a. If the golf ball starts with a speed of 2.0 m/s and slows at a constant rate of0.50 m/s2, what is its velocity after 2.0 s?

vf ! vi # at

! 2.0 m/s # ("0.50 m/s2)(2.0 s)! 1.0 m/s

b. What is the golf ball’s velocity if the constant acceleration continues for 6.0 s?

vf ! vi # at

! 2.0 m/s # ("0.50 m/s2)(6.0 s)

! "1.0 m/sc. Describe the motion of the golf ball in words and with a motion diagram.

The ball’s velocity simply decreased in the first case. In the secondcase, the ball slowed to a stop and then began rolling back down thehill.

19. A bus that is traveling at 30.0 km/h speeds up at a constant rate of 3.5 m/s2.What velocity does it reach 6.8 s later?

vf ! vi # at

! 30.0 km/h # (3.5 m/s2)(6.8 s)!!1100

k0mm!"!!36

100

hs

!"! 120 km/h

Time interval 1 2 3

6

4

5

VelocityPositionVelocity

Time interval

#d

Time

Posi

tion

Passingposition

Position-Time Graph

t1 t2 t3 t40

1234

t0 t4t3t2t1

t0 t4t3t2t1

Motion diagram

Faster car

Slower car

Sketch


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20. If a car accelerates from rest at a constant5.5 m/s2, how long will it take for the car toreach a velocity of 28 m/s?

vf ! vi # at

so t ! !vf "

avi!

!

! 5.1 s

21. A car slows from 22 m/s to 3.0 m/s at aconstant rate of 2.1 m/s2. How many seconds are required before the car is traveling at 3.0 m/s?

vf ! vi # at

so t ! !vf "

avi!

!

! 9.0 s

page 6722. Use Figure 3-11 to determine the velocity

of an airplane that is speeding up at each ofthe following times.

■ Figure 3-11

Graph B represents constant speed. Sograph A should be used for the follow-ing calculations.a. 1.0 s

At 1.0 s, v ! 74 m/s

b. 2.0 s

At 2.0 s, v ! 78 m/sc. 2.5 s

At 2.5 s, v ! 80 m/s

23. Use dimensional analysis to convert an airplane’s speed of 75 m/s to km/h.

(75 m/s)!!36100

hs

!"!!1100

k0mm!" ! 2.7$102 km/h

24. A position-time graph for a pony runningin a field is shown in Figure 3-12. Drawthe corresponding velocity-time graph usingthe same time scale.

■ Figure 3-12

25. A car is driven at a constant velocity of 25 m/s for 10.0 min. The car runs out of gasand the driver walks in the same directionat 1.5 m/s for 20.0 min to the nearest gasstation. The driver takes 2.0 min to fill agasoline can, then walks back to the car at1.2 m/s and eventually drives home at 25 m/s in the direction opposite that of the original trip.

a. Draw a v-t graph using seconds as yourtime unit. Calculate the distance the dri-ver walked to the gas station to find thetime it took him to walk back to the car.

Time (s)

Stops andturns around

Position-time graph

Slowsdown

Speedsup

Speedsup

#x

#y

Dis

plac

emen

t (m

)V

eloc

ity

(m/s

)

Dis

plac

emen

t (m

)

Time (s) #x

#y

0.0 3.01.0 2.0

82

80

78

76

74

72

70

Vel

ocit

y (m

/s)

Time (s)

3.0 m/s " 22 m/s!!!

"2.1 m/s2

28 m/s " 0.0 m/s!!!5.5 m/s2

Chapter 3 continued

distance the driver walked to thegas station:

d! vt! (1.5 m/s)(20.0 min)(60 s/min)

! 1800 m! 1.8 km

time to walk back to the car:

t ! !dv! ! !

118.200

mm/s! ! 1500 s ! 25 min

b. Draw a position-time graph for the situation using the areas under thevelocity-time graph.

page 6926. A skateboarder is moving at a constant

velocity of 1.75 m/s when she starts up anincline that causes her to slow down with aconstant acceleration of #0.20 m/s2. Howmuch time passes from when she begins toslow down until she begins to move backdown the incline?

vf ! vi # at

t ! !vf "

avi! ! ! 8.8 s

27. A race car travels on a racetrack at 44 m/sand slows at a constant rate to a velocity of 22 m/s over 11 s. How far does it moveduring this time?

v! ! !"2v! ! !

(vf "

2vi)!

"d ! v!"t

! !(vf "

2vi)"t!

!

! "1.2$102 m

28. A car accelerates at a constant rate from 15 m/s to 25 m/s while it travels a distanceof 125 m. How long does it take to achievethis speed?

v! ! !"2v! ! !

(vf "

2vi)!

"d ! v!"t

! !(vf "

2vi)"t!

"t !

!

! 25 s

29. A bike rider pedals with constant accelera-tion to reach a velocity of 7.5 m/s over atime of 4.5 s. During the period of accelera-tion, the bike’s displacement is 19 m. Whatwas the initial velocity of the bike?

v! ! !"2v! ! !

(vf "

2vi)!

"d ! v!"t ! !(vf "

2vi)"t!

so vi ! !2""td

! " vf

!!4.5(s2)

"(19

7.m5

)m/s!

! 0.94 m/s

(2)(125 m)!!25 m/s " 15 m/s

2"d!(vf " vi)

(22 m/s " 44 m/s)(11 s)!!!2

0.0 m/s " 1.75 m/s!!!

"0.20 m/s2

0

5000

10,000

15,000

20,000

Posi

tion

(m)

1000 2000 3000

Time (s)

4000 5000

Vel

ocit

y (m

/s)

Time (s)!5

!10

!15!20

!25

0

5

15

20

25

3500

4000

3000

2500

2000

1500

1000500

10


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page 7130. A man runs at a velocity of 4.5 m/s for 15.0 min. When going up an increasingly

steep hill, he slows down at a constant rate of 0.05 m/s2 for 90.0 s and comes toa stop. How far did he run?

d ! v1t1 # !12!(v2f # v2i)t2

! (4.5 m/s)(15.0 min)(60 s/min) # !12!(0.0 m/s # 4.5 m/s)(90.0 s)

! 4.3$103 m

31. Sekazi is learning to ride a bike without training wheels. His father pushes him with a constant acceleration of 0.50 m/s2 for 6.0 s, and then Sekazi continues at 3.0 m/s for another 6.0 s before falling. What is Sekazi’s displacement? Solve thisproblem by constructing a velocity-time graph for Sekazi’s motion and comput-ing the area underneath the graphed line.

Part 1: Constant acceleration:

d1 ! !12!(3.0 m/s)(6.0 s)

! 9.0 mPart 2: Constant velocity:

d2 ! (3.0 m/s)(12.0 s " 6.0 s)

! 18 mThus d! d1 # d2 ! 9.0 m # 18 m ! 27 m

32. You start your bicycle ride at the top of a hill. You coast down the hill at a con-stant acceleration of 2.00 m/s2. When you get to the bottom of the hill, you aremoving at 18.0 m/s, and you pedal to maintain that speed. If you continue at thisspeed for 1.00 min, how far will you have gone from the time you left the hilltop?

Part 1: Constant acceleration:

vf2 ! vi

2 # 2a(df " di) and di ! 0.00 m

so df ! !vf

2

2"

avi

2!

since vi ! 0.00 m/s

df ! !v2

fa2!

! !(2()1(82..000

mm/s

/)s22)!

! 81.0 m

Vel

ocit

y (m

/s)

3.0

Time (s)

6.0 12.0

1 2

Chapter 3 continued

Part 2: Constant velocity:

d2 ! vt! (18.0 m/s)(60.0 s) ! 1.08$103 m

Thus d ! d1 # d2

! 81.0 m # 1.08$103 m ! 1.16$103 m

33. Sunee is training for an upcoming 5.0-km race. She starts out her training run by moving at a constant pace of 4.3 m/s for 19 min. Then she accelerates at a constant rate until she crosses the finish line, 19.4 s later. What is her acceleration during the last portion of the training run?

Part 1: Constant velocity:d ! vt

! (4.3 m/s)(19 min)(60 s/min)

! 4902 mPart 2: Constant acceleration:

df ! di # vit # !12!at2

a ! !

! 0.077 m/s2

Section Review3.2 Motion with Constant Acceleration

pages 65–71page 7134. Acceleration A woman driving at a speed of 23 m/s sees a deer on the road

ahead and applies the brakes when she is 210 m from the deer. If the deer doesnot move and the car stops right before it hits the deer, what is the accelerationprovided by the car’s brakes?

vf2 ! vi

2 # 2a(df " di)

a ! !2v(fd2

f

"

"

vdi2

i)!

!

! "1.3 m/s2

35. Displacement If you were given initial and final velocities and the constantacceleration of an object, and you were asked to find the displacement, whatequation would you use?

vf2 ! vi

2 # 2adf

0.0 m/s " (23 m/s)2!!!(2)(210 m)

(2)(5.0$103 m " 4902 m " (4.3 m/s)(19.4 s))!!!!!!(19.4 s)2

2(df " di " vit)!!t2


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36. Distance An in-line skater first accelerates from 0.0 m/s to 5.0 m/s in 4.5 s, then continues at this constant speed for another 4.5 s. What is the total distance trav-eled by the in-line skater?

Accelerating

df ! v!tf ! !vi #

2vf!(tf)

! ! "(4.5 s)

! 11.25 mConstant speeddf ! vftf

! (5.0 m/s)(4.5 s)

! 22.5 mtotal distance ! 11.25 m # 22.5 m

! 34 m

37. Final Velocity A plane travels a distance of 5.0$102 m while being accelerateduniformly from rest at the rate of 5.0 m/s2. What final velocity does it attain?

vf2 ! vf

2 # 2a(df " di) and di ! 0, so

vf2 ! vf

2 # 2adf

vf ! "(0.0 m!/s)2 #! 2(5.0!m/s2)(!5.0$1!02 m)!! 71 m/s

38. Final Velocity An airplane accelerated uniformly from rest at the rate of 5.0m/s2 for 14 s. What final velocity did it attain?

vf ! vi # atf! 0 # (5.0 m/s2)(14 s) ! 7.0$101 m/s

39. Distance An airplane starts from rest and accelerates at a constant 3.00 m/s2 for 30.0 s before leaving the ground.

a. How far did it move?

df ! vitf # !12!atf2

! (0.0 m/s)(30.0 s)2 # !!12!"(3.00 m/s2)(30.0 s)2

! 1.35$103 mb. How fast was the airplane going when it took off?

vf ! vi # atf

! 0.0 m/s # (3.00 m/s2)(30.0 s)! 90.0 m/s

0.0 m/s # 5.0 m/s!!!2

Chapter 3 continued

40. Graphs A sprinter walks up to the starting blocks at a constant speed andpositions herself for the start of the race. She waits until she hears the startingpistol go off, and then accelerates rapidly until she attains a constant velocity.She maintains this velocity until she crosses the finish line, and then she slowsdown to a walk, taking more time to slow down than she did to speed up atthe beginning of the race. Sketch a velocity-time and a position-time graph torepresent her motion. Draw them one above the other on the same time scale.Indicate on your p-t graph where the starting blocks and finish line are.

41. Critical Thinking Describe how you could calculate the acceleration of an auto-mobile. Specify the measuring instruments and the procedures that you would use.

One person reads a stopwatch and calls out time intervals. Another person reads the speedometer at each time and records it. Plot speedversus time and find the slope.

Practice Problems3.3 Free Fall

pages 72–75page 7442. A construction worker accidentally drops a brick from a high scaffold.

a. What is the velocity of the brick after 4.0 s?

Say upward is the positive direction.

vf ! vi # at, a ! "g ! "9.80 m/s2

vf ! 0.0 m/s # ("9.80 m/s2)(4.0 s)

! "39 m/s when the upward direction is positiveb. How far does the brick fall during this time?

d ! vit # !12!at2

! 0 # !!12!"("9.80 m/s2)(4.0 s)2

! "78 mThe brick falls 78 m.

43. Suppose for the previous problem you choose your coordinate system so that the opposite direction is positive.

Starting blocks

Finishline

Posi

tion

Time

TimeV

eloc

ity


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a. What is the brick’s velocity after 4.0 s?

Now the positive direction is downward.

vf ! vi # at, a ! g ! 9.80 m/s2

vf ! 0.0 m/s # (9.80 m/s2)(4.0 s)

! #39 m/s when the downward direction is positiveb. How far does the brick fall during this time?

d ! vit # !12!at2, a ! g ! 9.80 m/s2

! (0.0 m/s)(4.0 s) #

!!12!"(9.80 m/s2)(4.0 s)2

! #78 m The brick still falls 78 m.

44. A student drops a ball from a window 3.5 m above the sidewalk. How fast is itmoving when it hits the sidewalk?

vf2 ! vi

2 # 2ad, a ! g and vi ! 0

so vf ! "2gd!! "(2)(9.8!0 m/s!2)(3.5!m)!! 8.3 m/s

45. A tennis ball is thrown straight up with an initial speed of 22.5 m/s. It is caughtat the same distance above the ground.

a. How high does the ball rise?

a ! "g, and at the maximum height, vf ! 0

vf2 ! vi

2 # 2ad becomes

vi2 ! 2gd

d ! !v2

ig

2! ! !(2

()2(29..580

mm/s

/)s22)! ! 25.8 m

b. How long does the ball remain in the air? Hint: The time it takes the ball to riseequals the time it takes to fall.

Calculate time to rise using vf ! vi # at, with a ! "g and vf ! 0

t ! !vg

i! ! !9

2.28.05

mm

//ss2! ! 2.30 s

The time to fall equals the time to rise, so the time to remain in the air istair ! 2trise ! (2)(2.30 s) ! 4.60 s

46. You decide to flip a coin to determine whether to do your physics or Englishhomework first. The coin is flipped straight up.

a. If the coin reaches a high point of 0.25 m above where you released it, whatwas its initial speed?

Chapter 3 continued

vf2 ! vi

2 # 2a"d

vi ! "vf2 #!2g"d! where a ! "g

and vf ! 0 at the height of the toss, so

vi ! "(0.0 m!/s)2 #! (2)(9.!80 m/!s2)(0.2!5 m)!! 2.2 m/s

b. If you catch it at the same height as you released it, how much time did itspend in the air?

vf ! vi # at where a ! "gvi ! 2.2 m/s and

vf ! "2.2 m/s

t ! !vf

"

"

gvi

!

!

! 0.45 s

Section Review3.3 Free Fall

pages 72–75page 7547. Maximum Height and Flight Time Acceleration due to gravity on Mars is about

one-third that on Earth. Suppose you throw a ball upward with the same velocityon Mars as on Earth.

a. How would the ball’s maximum height compare to that on Earth?

At maximum height, vf ! 0,

so df ! !v2

ig

2!, or three times higher.

b. How would its flight time compare?

Time is found from df ! !12!gtf2, or

tf ! #!2gdf!$. Distance is multiplied by 3 and g is divided by 3,

so the flight time would be three times as long.

48. Velocity and Acceleration Suppose you throw a ball straight up into the air.Describe the changes in the velocity of the ball. Describe the changes in theacceleration of the ball.

"2.2 m/s " 2.2 m/s!!!

"9.80 m/s2


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Velocity is reduced at a constant rate as the ball travels upward. At itshighest point, velocity is zero. As the ball begins to drop, the velocitybegins to increase in the negative direction until it reaches the heightfrom which it was initially released. At that point, the ball has the samespeed it had upon release. The acceleration is constant throughout theball’s flight.

49. Final Velocity Your sister drops your house keys down to you from the secondfloor window. If you catch them 4.3 m from where your sister dropped them,what is the velocity of the keys when you catch them?

Upward is positive

v2 ! vi2 # 2a"d where a ! "g

v ! "vi2 "!2g"d!

! "(0.0 m!/s)2 "! (2)(9.!80 m/!s2)("4!.3 m)!! 9.2 m/s

50. Initial Velocity A student trying out for the football team kicks the footballstraight up in the air. The ball hits him on the way back down. If it took 3.0 sfrom the time when the student punted the ball until he gets hit by the ball,what was the football’s initial velocity?

Choose a coordinate system with up as the positive direction and the origin at the punter. Choose the initial time at the punt and the final timeat the top of the football’s flight.vf ! vi # atf where a ! "g

vi ! vf # gtf

! 0.0 m/s # (9.80 m/s2)(1.5 s)

! 15 m/s

51. Maximum Height When the student in the previous problem kicked the foot-ball, approximately how high did the football travel?

vf2 ! vi

2 # 2a("d) where a ! "g

"d ! !vf

2

"

"

2gvi

2!

!

! 11 m

52. Critical Thinking When a ball is thrown vertically upward, it continues upwarduntil it reaches a certain position, and then it falls downward. At that highestpoint, its velocity is instantaneously zero. Is the ball accelerating at the highestpoint? Devise an experiment to prove or disprove your answer.

The ball is accelerating; its velocity is changing. Take a strobe photo tomeasure its position. From photos, calculate the ball’s velocity.

(0.0 m/s)2 " (15 m/s)2!!!("2)(9.80 m/s2)

Chapter 3 continued

Chapter AssessmentConcept Mappingpage 8053. Complete the following concept map using

the following symbols or terms: d, velocity,m/s2, v, m, acceleration.

Mastering Conceptspage 8054. How are velocity and acceleration related?

(3.1)

Acceleration is the change in velocitydivided by the time interval in which itoccurs: it is the rate of change ofvelocity.

55. Give an example of each of the following.(3.1)

a. an object that is slowing down, but hasa positive acceleration

if forward is the positive direction, acar moving backward at decreasingspeed

b. an object that is speeding up, but has anegative acceleration

in the same coordinate system, a carmoving backward at increasingspeed

56. Figure 3-16 shows the velocity-time graphfor an automobile on a test track. Describehow the velocity changes with time. (3.1)

■ Figure 3-16

The car starts from rest and increasesits speed. As the car’s speed increases,the driver shifts gears.

57. What does the slope of the tangent to thecurve on a velocity-time graph measure?(3.1)

instantaneous acceleration

58. Can a car traveling on an interstate highwayhave a negative velocity and a positive accel-eration at the same time? Explain. Can thecar’s velocity change signs while it is travelingwith constant acceleration? Explain. (3.1)

Yes, a car’s velocity is positive or nega-tive with respect to its direction ofmotion from some point of reference.One direction of motion is defined aspositive, and velocities in that directionare considered positive. The oppositedirection of motion is considered nega-tive; all velocities in that direction arenegative. An object undergoing positiveacceleration is either increasing itsvelocity in the positive direction orreducing its velocity in the negativedirection. A car’s velocity can changesigns when experiencing constantacceleration. For example, it can betraveling right, while the acceleration isto the left. The car slows down, stops,and then starts accelerating to the left.

59. Can the velocity of an object change whenits acceleration is constant? If so, give anexample. If not, explain. (3.1)

Yes, the velocity of an object canchange when its acceleration is con-stant. Example: dropping a book. The

5 10 15 20 25 30 350

51015202530

Vel

ocit

y (m

/s)

Time (s)

velocityposition acceleration

v ad

m/s m/s2m

Quantities of motion


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longer it drops, the faster it goes, butthe acceleration is constant at g.

60. If an object’s velocity-time graph is a straightline parallel to the t-axis, what can you con-clude about the object’s acceleration? (3.1)

When the velocity-time graph is a lineparallel to the t-axis, the acceleration iszero.

61. What quantity is represented by the areaunder a velocity-time graph? (3.2)

the change in displacement

62. Write a summary of the equations forposition, velocity, and time for an objectexperiencing motion with uniform acceleration. (3.2)

tf ! !(vf "

avi)

!

vf ! vi # atf

v! ! !"2v! ! !

vf "

2vi

!

"d ! v!"t

! !vf "

2vi

!"t

assuming ti ! 0, then

"t ! tf

"d ! !!vf "

2vi

!"tf

63. Explain why an aluminum ball and a steelball of similar size and shape, droppedfrom the same height, reach the ground atthe same time. (3.3)

All objects accelerate toward theground at the same rate.

64. Give some examples of falling objects forwhich air resistance cannot be ignored.(3.3)

Student answers will vary. Some examples are sheets of paper,parachutes, leaves, and feathers.

65. Give some examples of falling objects forwhich air resistance can be ignored. (3.3)

Student answers will vary. Some exam-ples are a steel ball, a rock, and a per-son falling through small distances.

Applying Conceptspages 80–8166. Does a car that is slowing down always have

a negative acceleration? Explain.

No, if the positive axis points in thedirection opposite the velocity, theacceleration will be positive.

67. Croquet A croquet ball, after being hit by a mallet, slows down and stops. Do thevelocity and acceleration of the ball havethe same signs?

No, they have opposite signs.

68. If an object has zero acceleration, does itmean its velocity is zero? Give an example.

No, a ! 0 when velocity is constant.

69. If an object has zero velocity at someinstant, is its acceleration zero? Give anexample.

No, a ball rolling uphill has zero velocityat the instant it changes direction, butits acceleration is nonzero.

70. If you were given a table of velocities of anobject at various times, how would you findout whether the acceleration was constant?

Draw a velocity-time graph and seewhether the curve is a straight line or calculate accelerations using a! ! !

""vt!

and compare the answers to see if theyare the same.

71. The three notches in the graph in Figure 3-16 occur where the driver changedgears. Describe the changes in velocity andacceleration of the car while in first gear. Isthe acceleration just before a gear changelarger or smaller than the acceleration justafter the change? Explain your answer.

Chapter 3 continued

Velocity increases rapidly at first, then more slowly. Acceleration is greatestat the beginning but is reduced as velocity increases. Eventually, it is nec-essary for the driver to shift into second gear. The acceleration is smallerjust before the gear change because the slope is less at that point on thegraph. Once the driver shifts and the gears engage, acceleration and theslope of the curve increase.

72. Use the graph in Figure 3-16 and determine the time interval during which theacceleration is largest and the time interval during which the acceleration issmallest.

The acceleration is largest during an interval starting at t ! 0 and lasting about !

12! s. It is smallest beyond 33 s.

73. Explain how you would walk to produce each of the position-time graphs in Figure 3-17.

■ Figure 3-17

Walk in the positive direction at a constant speed. Walk in the positivedirection at an increasing speed for a short time; keep walking at a moderate speed for twice that amount of time; slow down over a shorttime and stop; remain stopped; and turn around and repeat the procedureuntil the original position is reached.

74. Draw a velocity-time graph for each of the graphs in Figure 3-18.

■ Figure 3-18

Time

Vel

ocit

y

Vel

ocit

y

Time

Vel

ocit

y

Time

Dis

plac

emen

t

Dis

plac

emen

t

Dis

plac

emen

t

Time Time Time

Dis

plac

emen

t

Dis

plac

emen

t

Time Time

A B

CD E

F

G H


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75. An object shot straight up rises for 7.0 sbefore it reaches its maximum height. A sec-ond object falling from rest takes 7.0 s toreach the ground. Compare the displacements of the two objects during thistime interval.

Both objects traveled the same dis-tance. The object that is shot straightupward rises to the same height fromwhich the other object fell.

76. The Moon The value of g on the Moon isone-sixth of its value on Earth.

a. Would a ball that is dropped by anastronaut hit the surface of the Moonwith a greater, equal, or lesser speedthan that of a ball dropped from thesame height to Earth?

The ball will hit the Moon with alesser speed because the accelera-tion due to gravity is less on theMoon.

b. Would it take the ball more, less, orequal time to fall?

The ball will take more time to fall.

77. Jupiter The planet Jupiter has about threetimes the gravitational acceleration of Earth.Suppose a ball is thrown vertically upwardwith the same initial velocity on Earth andon Jupiter. Neglect the effects of Jupiter’satmospheric resistance and assume thatgravity is the only force on the ball.

a. How does the maximum height reachedby the ball on Jupiter compare to themaximum height reached on Earth?

The relationship between d and g is

an inverse one: df ! !(vf

2

2"

gvi

2)!.

If g increases by three times, or

df ! !(vf

2

2

("

3gv)i2)

!, df changes by !13!.

Therefore, a ball on Jupiter would rise to a height of !

13! that on Earth.

b. If the ball on Jupiter were thrown withan initial velocity that is three timesgreater, how would this affect youranswer to part a?

With vf ! 0, the value df is directly proportional to the square of initial

velocity, vi. That is, df ! vf2 " !

(32vgi)2!.

On Earth, an initial velocity threetimes greater results in a ball risingnine times higher. On Jupiter, how-ever, the height of nine times higherwould be reduced to only threetimes higher because of df’s inverserelationship to a g that is threetimes greater.

78. Rock A is dropped from a cliff and rock B isthrown upward from the same position.

a. When they reach the ground at the bottom of the cliff, which rock has agreater velocity?

Rock B hits the ground with agreater velocity.

b. Which has a greater acceleration?

They have the same acceleration,the acceleration due to gravity.

c. Which arrives first?

rock A

Mastering Problems3.1 Accelerationpages 81–82Level 179. A car is driven for 2.0 h at 40.0 km/h, then

for another 2.0 h at 60.0 km/h in the samedirection.

a. What is the car’s average velocity?

Total distance:80.0 km # 120.0 km ! 200.0 kmtotal time is 4.0 hours, so,v! ! !

""dt! ! !

2040..00

hkm

! ! 5.0$101 km/h

b. What is the car’s average velocity if it isdriven 1.0$102 km at each of the twospeeds?

Chapter 3 continued

Total distance is 2.03102 km;

total time ! !14.00$.0

1k0m2 k

/hm

! # !16.00$.0

1k0m2 k

/hm

!

! 4.2 h

so v! ! !""dt! ! !

2.0$4.

120h2 km!

! 48 km/h

80. Find the uniform acceleration that causes a car’s velocity to change from 32 m/sto 96 m/s in an 8.0-s period.

a! ! !""vt!

! !v2

"

"

tv1

!

!!96 m/s8.

"0 s

32 m/s! ! 8.0 m/s2

81. A car with a velocity of 22 m/s is accelerated uniformly at the rate of 1.6 m/s2 for6.8 s. What is its final velocity?

vf ! vi # atf

! 22 m/s # (1.6 m/s2)(6.8 s)! 33 m/s

82. Refer to Figure 3-19 to find the acceleration of the moving object at each of thefollowing times.

■ Figure 3-19

a. during the first 5.0 s of travel

a! ! !""vt!

!

! 6.0 m/s2

b. between 5.0 s and 10.0 s

a! ! !""vt!

!

! 0.0 m/s2

30.0 m/s " 30.0 m/s!!!5.0 s

30.0 m/s " 0.0 m/s!!!5.0 s

Vel

ocit

y (m

/s)

Time (s)

5.0 10.0 15.0 20.0 25.0 30.00.0

10.0

20.0

30.0


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c. between 10.0 s and 15.0 s

a! ! !""vt!

!

! "2.0 m/s2

d. between 20.0 s and 25.0 s

a! ! !""vt!

!

! "4.0 m/s2

Level 283. Plot a velocity-time graph using the information

in Table 3-4, and answer the following questions.

a. During what time interval is the object speeding up? Slowing down?

speeding up from 0.0 s to 4.0 s; slowingdown from 5.0 s to 10.0 s

b. At what time does the object reverse direction?

at 10.0 sc. How does the average acceleration of the object in the interval between 0.0 s and

2.0 s differ from the average acceleration in the interval between 7.0 s and 12.0 s?

a! ! !""vt!

between 0.0 s and 2.0 s:

a! ! ! 4.0 m/s2

between 7.0 s and 12.0 s:

a! ! ! "4.0 m/s2"8.0 m/s " 12.0 m/s!!!12.0 s " 7.0 s

12.0 m/s " 4.0 m/s!!!2.0 s " 0.0 s

2.00 4.00 6.00 8.00 10.0

Time (s)

#8.00

#4.00

0.00

4.00

8.00

12.0

16.0

Vel

ocit

y (m

/s)

12.0

Table 3-4Velocity v. Time

Time (s) Velocity (m/s)0.00 4.001.00 8.002.00 12.03.00 14.04.00 16.05.00 16.06.00 14.07.00 12.08.00 8.009.00 4.00

10.0 0.0011.0 #4.0012.0 #8.00

0.0 m/s " 20.0 m/s!!!5.0 s

20.0 m/s " 30.0 m/s!!!5.0 s

Chapter 3 continued

84. Determine the final velocity of a protonthat has an initial velocity of 2.35$105 m/sand then is accelerated uniformly in anelectric field at the rate of #1.10$1012 m/s2

for 1.50$10#7 s.

vf ! vi # atf! 2.35$105 m/s #

("1.10$1012 m/s2)(1.50$10"7 s)! 7.0$104 m/s

Level 385. Sports Cars Marco is looking for a used

sports car. He wants to buy the one with thegreatest acceleration. Car A can go from 0m/s to 17.9 m/s in 4.0 s; car B can acceleratefrom 0 m/s to 22.4 m/s in 3.5 s; and car Ccan go from 0 to 26.8 m/s in 6.0 s. Rank thethree cars from greatest acceleration to least,specifically indicating any ties.

Car A:

a! ! ! ! 4.5 m/s2

Car B:

a! ! ! ! 6.4 m/s2

Car C:

a! ! ! ! 4.5 m/s2

Car B has the greatest acceleration of6.4 m/s2. Using significant digits, car Aand car C tied at 4.5 m/s2.

86. Supersonic Jet A supersonic jet flying at145 m/s experiences uniform acceleration atthe rate of 23.1 m/s2 for 20.0 s.

a. What is its final velocity?

vf ! vi # atf! 145 m/s # (23.1 m/s2)(20.0 s)! 607 m/s

b. The speed of sound in air is 331 m/s.What is the plane’s speed in terms ofthe speed of sound?

N ! !630371

mm

//ss!

! 1.83 times the speed of sound

3.2 Motion with Constant Accelerationpage 82Level 187. Refer to Figure 3-19 to find the distance

traveled during the following time intervals.

a. t " 0.0 s and t " 5.0 s

Area I ! !12!bh

! !!12!"(5.0 s)(30.0 m/s)

! 75 mb. t " 5.0 s and t " 10.0 s

Area II ! bh! (10.0 s " 5.0 s)(30.0 m/s)! 150 m

c. t " 10.0 s and t " 15.0 s

Area III # Area IV ! bh # !12!bh

! (15.0 s " 10.0 s)(20.0 m/s) #

!!12!"(15.0 s " 10.0 s)(10.0 m/s)

! 125 md. t " 0.0 s and t " 25.0 s

Area I # Area II #

(Area III # Area IV) # Area V # IV! 75 m # 150 m # 125 m #

bh # !12!bh

! 75 m # 150 m # 125 m #

(20.0 s " 15.0 s)(20.0 m/s) #

!!12!"(25.0 s " 20.0 s)

! 5.0$102 m

Level 288. A dragster starting from rest accelerates at

49 m/s2. How fast is it going when it hastraveled 325 m?

vf2 ! vi

2 # 2a(df " di)

vf ! "vi2 #!2a(df!" di)!

26.8 m/s " 0 m/s!!!6.0 s " 0.0 s

"v!"t

22.4 m/s " 0 m/s!!!3.5 s " 0.0 s

"v!"t

17.9 m/s " 0 m/s!!!4.0 s " 0.0 s

"v!"t


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! "(0.0 m!/s)2 #! (2)(49! m/s2)!(325 m!" 0.0! m)!! 180 m/s

89. A car moves at 12 m/s and coasts up a hill with a uniform acceleration of #1.6 m/s2.

a. What is its displacement after 6.0 s?


! (12 m/s)(6.0 s) #

!!12!"("1.6 m/s2)(6.0 s)2

! 43 mb. What is its displacement after 9.0 s?


! (12 m/s)(9.0 s) #

!!12!"("1.6 m/s2)(9.0 s)2

! 43 mThe car is on the way back down the hill. The odometer will show that the car traveled 45 m up the hill # 2 m back down ! 47 m.

90. Race Car A race car can be slowed with a constant acceleration of #11 m/s2.

a. If the car is going 55 m/s, how many meters will it travel before it stops?

vf2 ! vi

2 # 2adf

df !

!

! 1.4$102 mb. How many meters will it take to stop a car going twice as fast?

vf2 ! vi

2 # 2adf

df !

! ! 550 m,

which is about 4 times longer than when going half the speed.

91. A car is traveling 20.0 m/s when the driver sees a child standing on the road. Shetakes 0.80 s to react, then steps on the brakes and slows at 7.0 m/s2. How fardoes the car go before it stops?

reaction displacement dr ! (20.0 m/s)(0.80 s) ! 16 m

(0.0 m/s)2 " (110 m/s)2!!!(2)("11 m/s2)

vf2 " vi

2!!2a

(0.0 m/s)2 " (#55 m/s)2!!!(2)("11 m/s2)

vf2 " vi

2!!2a

Chapter 3 continued

df !

braking displacement

db !

! 29 mtotal displacement is

dr # db ! 16 m # 29 m ! 45 m

Level 392. Airplane Determine the displacement of a

plane that experiences uniform accelerationfrom 66 m/s to 88 m/s in 12 s.

df ! v!t ! !(vf #

2vi)t!

!

! 9.2$102 m

93. How far does a plane fly in 15 s while itsvelocity is changing from 145 m/s to 75 m/s at a uniform rate of acceleration?

d ! v!t ! !(vf #

2vi)t!

!

! 1.6$103 m

94. Police Car A speeding car is traveling at aconstant speed of 30.0 m/s when it passes astopped police car. The police car acceler-ates at 7.0 m/s2. How fast will it be goingwhen it catches up with the speeding car?

dspeeder ! vspeedert

dpolice ! vi policet # !12!apolicet2

vspeedert ! vi policet # !12!apolicet2

since vi police ! 0 then

vspeedert ! !12!apolicet2

0 ! !12!apolicet2 " vspeedert

0 ! t!!12!apolicet " vspeeder"

therefore

t ! 0 and !12!apolicet " vspeeder ! 0

t !

! !(2)

7(3.00.

m0

/ms2

/s)!

! 8.6 s

After t ! 8.6 s, the police car’s velocitywasvf ! vi # at

! 0.0 m/s # (7.0 m/s2)(8.6 s)

! 6.0$101 m/s

95. Road Barrier The driver of a car going90.0 km/h suddenly sees the lights of a bar-rier 40.0 m ahead. It takes the driver 0.75 sto apply the brakes, and the average acceler-ation during braking is #10.0 m/s2.

a. Determine whether the car hits the barrier.

The car will traveld ! vt ! (25.0 m/s)(0.75 s)! 18.8 m (Round off at the end.)before the driver applies the brakes.Convert km/h to m/s.

vi !

! 25.0 m/s

vf2 ! vi

2 # 2a(df " di)

df ! # di

! # 18.8 m

! 5.0$101 m, yes it hits the barrier

(0.0 m/s)2 " (25.0 m/s)2!!!(2)("10.0 m/s2)

vf2 " vi

2!!2a

(90.0 km/h)(1000 m/km)!!!3600 s/h

2vspeeder!!apolice

(75 m/s # 145 m/s)(15 m/s)!!!!2

(88 m/s # 66 m/s)(12 s)!!!2

(0.0 m/s)2 " (20.0 m/s)2!!!(2)("7.0 m/s2)

vf2 " vi

2!!2a


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b. What is the maximum speed at whichthe car could be moving and not hit thebarrier 40.0 m ahead? Assume that theacceleration doesn’t change.

dtotal ! dconstant # ddecelerating

! 40.0 mdc ! vt ! (0.75 s)v

dd ! !!2("1"0.v0

2

m/s2)!

!

40 m ! (0.75 s)v #

v2 # (15 m/s)v " 800 m2/s2 ! 0Using the quadratic equation:v ! 22 m/s (The sense of the prob-lem excludes the negative value.)

3.3 Free Fallpage 82Level 196. A student drops a penny from the top of a

tower and decides that she will establish acoordinate system in which the direction ofthe penny’s motion is positive. What is thesign of the acceleration of the penny?

The direction of the velocity is positive,and velocity is increasing. Therefore,the acceleration is also positive.

97. Suppose an astronaut drops a feather from1.2 m above the surface of the Moon. If theacceleration due to gravity on the Moon is1.62 m/s2 downward, how long does it takethe feather to hit the Moon’s surface?

df ! vitf # atf2 ! (0 m/s)tf # atf2

tf ! #!2adf!$ ! #$ ! 1.2 s

98. A stone that starts at rest is in free fall for 8.0 s.

a. Calculate the stone’s velocity after 8.0 s.

vf ! vi # atf where a ! "g! vi " gtf! 0.0 m/s " (9.80 m/s2)(8.0 s)! "78 m/s (downward)

b. What is the stone’s displacement duringthis time?

Choose the coordinate system tohave the origin where the stone is at rest and positive to be upward.

df ! vit # !12!atf2 where a ! "g

! vit " !12!gtf2

! 0.0 m " !!12!"(9.80 m/s2)(8.0 s)2

! "3.1$102 m

99. A bag is dropped from a hovering helicopter.The bag has fallen for 2.0 s. What is the bag’svelocity? How far has the bag fallen?

Velocity:vf ! vi # atf where a ! "g

! vi " gtf

! 0.0 m/s " (9.80 m/s2)(2.0 s)

! "2.0$101 m/sDisplacement:df ! vitf # !

12!atf2 where a ! "g

! vitf " !12!gt2

! 0.0 m " !!12!"(9.80 m/s2)(2.0 s)2

! "2.0$101 m

Level 2100. You throw a ball downward from a window

at a speed of 2.0 m/s. How fast will it bemoving when it hits the sidewalk 2.5 mbelow?

Choose a coordinate system with thepositive direction downward and theorigin at the point where the ballleaves your hand.

(2)(1.2 m)!!(1.62 m/s2)

v2!!20.0 m/s2

v2!!20.0 m/s2

02 " v2!2a

Chapter 3 continued

vf2 ! vi

2 # 2adf where a ! g

vf ! "vi2 #!2gdf!

! "(2.0 m!/s)2 #! (2)(9.!80 m/!s2)(2.5! m)!! 7.3 m/s

101. If you throw the ball in the previous problem up instead of down, how fast willit be moving when it hits the sidewalk?

Choose the same coordinate system.

vf2 ! vi

2 # 2adf where a ! g

vf ! "vi2 #!2gdf!

! "(2.0 m!/s)2 #! (2)(9.!80 m/!s2)(2.5! m)!! 7.3 m/s

(df is the displacement, not the total distance traveled.)

Level 3102. Beanbag You throw a beanbag in the air and catch it 2.2 s later.

a. How high did it go?

Choose a coordinate system with the upward direction positive and theorigin at the point where the beanbag left your hand. Assume that youcatch the beanbag at the same place where you threw it.Therefore, thetime to reach the maximum height is half of the time in the air. Choose tito be the time when the beanbag left your hand and tf to be the time atthe maximum height. Each formula that you know includes vi, so youwill have to calculate that first.vf ! vi # atf where a ! "g

vi ! vf # gtf! 0.0 m/s # (9.80 m/s2)(1.1 s)! 11 m/s

Now you can use an equation that includes the displacement.df ! di # vitf # !

12!atf2 where a ! "g

! di # vitf " !12!gtf2

! 0.0 m # (11 m/s)(1.1 s) "

!!12!"(9.80 m/s2)(1.1 s)2

! 6.2 mb. What was its initial velocity?

vi ! 11 m/s


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Mixed Reviewpages 82–84Level 1103. A spaceship far from any star or planet

experiences uniform acceleration from65.0 m/s to 162.0 m/s in 10.0 s. How fardoes it move?

Choosing a coordinate system with the origin at the point where the speedis 65.0 m/s and given vi ! 65.0 m/s,vf ! 162.0 m/s, and tf ! 10.0 s and needing df, we use the formula with the average velocity.

df ! di # !12!(vi # vf)tf

df ! 0 # !12!(65.0 m/s # 162.0 m/s)(10.0 s)

! 1.14$103 m

104. Figure 3-20 is a strobe photo of a horizontally moving ball. What informationabout the photo would you need and whatmeasurements would you make to estimatethe acceleration?

■ Figure 3-20

You need to know the time betweenflashes and the distance between thefirst two images and the distancebetween the last two. From these, youget two velocities. Between these twovelocities, a time interval of t secondsoccurred. Divide the differencebetween the two velocities by t.

105. Bicycle A bicycle accelerates from 0.0 m/sto 4.0 m/s in 4.0 s. What distance does ittravel?

df ! v!tf ! !vi #

2vf!tf

! ! "(4.0 s)

! 8.0 m

106. A weather balloon is floating at a constantheight above Earth when it releases a packof instruments. a. If the pack hits the ground with a

velocity of #73.5 m/s, how far did thepack fall?

vf2 ! vi

2 # 2adf

df ! !vf

2

2"

avi

2!

!

! "276 m

b. How long did it take for the pack to fall?

vf ! vi # atf where a ! "g

tf ! !vf

"

"

gvi

!

!

! 7.50 s

Level 2107. Baseball A baseball pitcher throws a fast-

ball at a speed of 44 m/s. The accelerationoccurs as the pitcher holds the ball in hishand and moves it through an almoststraight-line distance of 3.5 m. Calculatethe acceleration, assuming that it is constant and uniform. Compare this accel-eration to the acceleration due to gravity.

vf2 ! vi

2 # 2adf

a !

! !(4

(42)

m(3

/s.5)2

m")

0! ! 2.8$102 m/s2

! 29, or 29 times g2.8$102 m/s2!!9.80 m/s2

vf2 " vi

2!!2df

"73.5 m/s " 0.00 m/s!!!

"9.80 m/s2

("73.5 m/s)2 " (0.00 m/s)2!!!!(2)("9.80 m/s2)

0.0 m/s # 4.0 m/s!!!2

Chapter 3 continued

108. The total distance a steel ball rolls downan incline at various times is given inTable 3-5.

a. Draw a position-time graph of themotion of the ball. When setting upthe axes, use five divisions for each 10 m of travel on the d-axis. Use fivedivisions for 1 s of time on the t-axis.

b. Calculate the distance the ball hasrolled at the end of 2.2 s.

After 2.2 seconds the ball hasrolled approximately 10 m.

109. Engineers are developing new types ofguns that might someday be used tolaunch satellites as if they were bullets.One such gun can give a small object avelocity of 3.5 km/s while moving itthrough a distance of only 2.0 cm.

a. What acceleration does the gun givethis object?

vf2 ! vi

2 # 2adf

or vf2 ! 2adf

a ! !!(3

(.25)$(0

1.0023

0m

m/s

))2

!

! 3.1$108 m/s2

b. Over what time interval does the acceleration take place?

d !

t ! !vf

2#

dfvi

! !

! 11$10"6 s! 11 microseconds

110. Sleds Rocket-powered sleds are used totest the responses of humans to accelera-tion. Starting from rest, one sled can reacha speed of 444 m/s in 1.80 s and can bebrought to a stop again in 2.15 s.

a. Calculate the acceleration of the sledwhen starting, and compare it to themagnitude of the acceleration due togravity, 9.80 m/s2.

a! ! !""vt! ! !

v2"

"

tv1

!

!

! 247 m/s2

!92.4870

mm

//ss22! ! 25 times g

b. Find the acceleration of the sled as it isbraking and compare it to the magni-tude of the acceleration due to gravity.

a ! !""vt! ! !

v2"

"

tv1

!

!

! "207 m/s2

!92.0870

mm

//ss22! ! 21 times g

0.00 m/s " 444 m/s!!!2.15 s

444 m/s " 0.00 m/s!!!1.80 s

(2)(0.020 m)!!!3.5$103 m/s # 0.0 m/s

(vf # vi)t!!2

vf2

!2df

1.00.0 2.0 3.0 4.0 5.0

Time (s)

Top ofincline

10.0

20.0

30.0

40.0

50.0

Posi

tion

(m)

Table 3-5Distance v. Time

Time (s) Distance (m)

0.0 0.0

1.0 2.0

2.0 8.0

3.0 18.0

4.0 32.0

5.0 50.0


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111. The velocity of a car changes over an 8.0-stime period, as shown in Table 3-6.

a. Plot the velocity-time graph of themotion.

b. Determine the displacement of the carduring the first 2.0 s.

Find the area under the v-t curve.

d! !12!bh

! !!12!"(2.0 s)(8.0 m/s " 0.0 m/s)

! 8.0 mc. What displacement does the car have

during the first 4.0 s?


d ! !12!bh

! !!12!"(4.0 s)(16.0 m/s " 0.0 m/s)

! 32 md. What is the displacement of the car

during the entire 8.0 s?


d ! !12!bh # bh

! !!12!"(5.0 s)(20.0 m/s " 0.0 m/s) #

(8.0 s " 5.0 s)(20.0 m/s)! 110 m

e. Find the slope of the line between t " 0.0 s and t " 4.0 s. What does thisslope represent?

a ! !""vt! !

! 4.0 m/s2, accelerationf. Find the slope of the line between

t " 5.0 s and t " 7.0 s. What does thisslope indicate?

a ! !""vt! !

! 0.0 m/s2, constant velocity

Level 3112. A truck is stopped at a stoplight. When the

light turns green, the truck accelerates at2.5 m/s2. At the same instant, a car passesthe truck going 15 m/s. Where and whendoes the truck catch up with the car?

Car:df ! di # vtfdcar ! di # vcartf ! vcartf

! 0 # (15 m/s)tfTruck:

df ! di # vitf # !12!atf2

dtruck ! !12!atrucktf2

! 0 # 0 # !!12!"(2.5 m/s2)tf2

When the truck catches up, the displacements are equal.

vcartf ! !12!atrucktf2

0 ! !12!atrucktf2 " vcartf

0 ! tf!!12!atrucktf " vcar"

20.0 m/s " 20.0 m/s!!!7.0 s " 5.0 s

16.0 m/s " 0.00 m/s!!!4.0 s " 0.0 s

1.00.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Time (s)

4.0

8.0

12.0

16.0

20.0

Vel

ocit

y (m

/s)

Table 3-6Velocity v. Time

Time (s) Velocity (m/s)0.0 0.01.0 4.02.0 8.03.0 12.04.0 16.05.0 20.06.0 20.07.0 20.08.0 20.0

Chapter 3 continued

therefore

tf ! 0 and !12!atrucktf " vcar ! 0

tf ! !a2

t

v

ru

c

c

a

k

r!

! !(2

2)(.155mm/s/s)

!

! 12 s

df ! (15 m/s)tf! (15 m/s)(12 s) ! 180 m

113. Safety Barriers Highway safety engineersbuild soft barriers, such as the one shownin Figure 3-21, so that cars hitting themwill slow down at a safe rate. A personwearing a safety belt can withstand anacceleration of #3.0$102 m/s2. How thickshould barriers be to safely stop a car thathits a barrier at 110 km/h?

■ Figure 3-21

vi ! ! 31 m/s

vf2 ! vi

2 # 2adf

with vf ! 0 m/s, vi2 ! "2adf, or

df ! !"

2vai2

! !

! 1.6 m thick

114. Karate The position-time and velocity-time graphs of George’s fist breaking awooden board during karate practice areshown in Figure 3-22.

■ Figure 3-22

a. Use the velocity-time graph to describethe motion of George’s fist during thefirst 10 ms.

The fist moves downward at about "13 m/s for about 4 ms.It then suddenly comes to a halt(accelerates).

b. Estimate the slope of the velocity-timegraph to determine the acceleration ofhis fist when it suddenly stops.

a ! !""vt! !

! 3.7$103 m/s2

c. Express the acceleration as a multipleof the gravitational acceleration, g " 9.80 m/s2.

! 3.8$102

The acceleration is about 380g.d. Determine the area under the velocity-

time curve to find the displacement ofthe fist in the first 6 ms. Compare thiswith the position-time graph.

The area can be approximated by arectangle:("13 m/s)(0.006 s) ! "8 cmThis is in agreement with the position-time graph where the handmoves from #8 cm to 0 cm, for anet displacement of "8 cm.

3.7$103 m/s2!!9.80 m/s2

0 m/s " ("13 m/s)!!!7.5 ms " 4.0 ms

Fist

5.0 10.0 15.0 20.0 25.0 30.0!15.0

!10.0

!5.0

0.0

!5.0

0.0

5.0

10.0

Disp

lace

men

t (cm

)V

eloc

ity

(m/s

)

Time (ms)

"(31 m/s)2!!!(2)("3.0$102 m/s2)

(110 km/h)(1000 m/km)!!!3600 s/h


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115. Cargo A helicopter is rising at 5.0 m/swhen a bag of its cargo is dropped. Thebag falls for 2.0 s.

a. What is the bag’s velocity?

vf ! vi # atf where a ! "g

! vi " gtf

! 5.0 m/s " (9.80 m/s2)(2.0 s)! "15 m/s

b. How far has the bag fallen?

df ! vitf # !12!atf2 where a ! "g

! vitf " !12!gtf2

! (5.0 m/s)(2.0 s) "

!!12!"(9.80 m/s2)(2.0 s)2

! "1.0$101 mThe bag has fallen 1.0$101 m

c. How far below the helicopter is thebag?

The helicopter has risen

df ! vitf ! (5.0 m/s2)(2.0 s)

! 1.0$101 mThe bag is 1.0$101 m below the origin and 2.0$101 m below thehelicopter.

Thinking Criticallypage 84116. Apply CBLs Design a lab to measure the

distance an accelerated object moves overtime. Use equal time intervals so that youcan plot velocity over time as well as dis-tance. A pulley at the edge of a table with amass attached is a good way to achieveuniform acceleration. Suggested materialsinclude a motion detector, CBL, lab cart,string, pulley, C-clamp, and masses.Generate distance-time and velocity-timegraphs using different masses on the pulley. How does the change in mass affectyour graphs?

Students’ labs will vary. Studentsshould find that a change in the massover the edge of the table will not

change the distance the cart moves,because the acceleration is alwaysthe same: g.

117. Analyze and Conclude Which has thegreater acceleration: a car that increases itsspeed from 50 km/h to 60 km/h, or a bikethat goes from 0 km/h to 10 km/h in thesame time? Explain.

a ! !vf

"

"

tvi

!

For car, a !

! !10

"kmt

/h!

For bike, a !

! !10

"kmt

/h!

The change in velocity is the same.

118. Analyze and Conclude An express train,traveling at 36.0 m/s, is accidentally side-tracked onto a local train track. Theexpress engineer spots a local train exactly1.00$102 m ahead on the same track andtraveling in the same direction. The localengineer is unaware of the situation. Theexpress engineer jams on the brakes andslows the express train at a constant rateof 3.00 m/s2. If the speed of the localtrain is 11.0 m/s, will the express train be able to stop in time, or will there be a collision? To solve this problem, takethe position of the express train when the engineer first sights the local train as a point of origin. Next, keeping inmind that the local train has exactly a1.00$102 m lead, calculate how far eachtrain is from the origin at the end of the12.0 s it would take the express train to stop (accelerate at #3.00 m/s2 from 36 m/s to 0 m/s).

a. On the basis of your calculations,would you conclude that a collisionwill occur?

Express:df ! vitf # !

12!atf2

! (36.0 m/s)(12.0 s) #

10 km/h " 0 km/h!!!

"t

60 km/h " 50 km/h!!!

"t

Chapter 3 continued

!!12!"("3.00 m/s2)(12.0 s)2

! 432 m " 216 m ! 216 mLocal:df ! di # vitf # atf2

! 100 m # (11.0 m/s)(12.0 s) # 0! 232 m

On this basis, no collision will occur.b. The calculations that you made do not allow for the possibility that a colli-

sion might take place before the end of the 12 s required for the expresstrain to come to a halt. To check this, take the position of the express trainwhen the engineer first sights the local train as the point of origin and calcu-late the position of each train at the end of each second after the sighting.Make a table showing the distance of each train from the origin at the endof each second. Plot these positions on the same graph and draw two lines.Use your graph to check your answer to part a.

They collide between 6 and 7 s.

Writing in Physicspage 84119. Research and describe Galileo’s contributions to physics.

Student answers will vary. Answers should include Galileo’s experimentsdemonstrating how objects accelerate as they fall. Answers mightinclude his use of a telescope to discover the moons of Jupiter and therings of Saturn, and his reliance on experimental results rather thanauthority.


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2 4 6 8 10 12t

Time (s)

50

100

150

200

250

Posi

tion

(m)

d

Express

Local

t (s) d (Local) d (Express)(m) (m)

1 111 35

2 122 66

3 133 95

4 144 120

5 155 145

6 166 162

7 177 179

8 188 192

9 199 203

10 210 210

11 221 215

12 232 216


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120. Research the maximum acceleration ahuman body can withstand without black-ing out. Discuss how this impacts thedesign of three common entertainment ortransportation devices.

Answers will vary. Because humanscan experience negative effects, likeblackouts, the designers of rollercoasters need to structure the down-ward slopes in such a way that thecoaster does not reach accelerationsthat cause blackouts. Likewise, engi-neers working on bullet trains, eleva-tors, or airplanes need to design thesystem in such a way that allows theobject to rapidly accelerate to highspeeds, without causing the passen-gers to black out.

Cumulative Reviewpage 84121. Solve the following problems. Express your

answers in scientific notation. (Chapter 1)

a. 6.2$10#4 m % 5.7$10#3 m

6.3$10"3 mb. 8.7$108 km # 3.4$107 m

8.4$108 kmc. (9.21$10#5 cm)(1.83$108 cm)

1.69$104 cm2

d. (2.63$10#6 m)/(4.08$106 s)

6.45$10"13 m/s

122. The equation below describes the motionof an object. Create the correspondingposition-time graph and motion diagram.Then write a physics problem that couldbe solved using that equation. Be creative.d " (35.0 m/s) t # 5.0 m (Chapter 2)

Graph and motion diagram indicateconstant velocity motion with a velocity of 35.0 m/s and initial positionof "5.0 m. Answers will vary for thecreate-a-problem part.

Challenge Problempage 75You notice a water balloon fall past your class-room window. You estimate that it took the balloon about t seconds to fall the length of thewindow and that the window is about y metershigh. Suppose the balloon started from rest,approximately how high above the top of thewindow was it released? Your answer should bein terms of t, y, g, and numerical constants.

Down is positive. Work this problem in twostages. Stage 1 is falling the distance D to the top of the window. Stage 2 is falling thedistance y from the top of the window to thebottom of the window.Stage 1: the origin is at the top of the fall.

vf12 ! vi1

2 # 2a(df1 " di1)

! 0 # 2g(D " 0)

vf1 ! "2gD!

Stage 2: the origin is at the top of the window.

df2 ! di1 # vi1tf2 # !12!atf22

y ! 0 # vf1t # !12!gt2

! 0 # !"2gD!"(t) # !12!gt2

"2gD! ! !yt! " !

g2t!

D ! !21g!!!yt! " !

g2t!"

2

#10042 86 10

400

0

100

200

300

Motion Diagram

Position-Time Graph

Time (s)

Posi

tion

(m)

Chapter 3 continued

chapter 3 accelerated motion - mr....

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