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# Problems | 53

2-1 No. The distance between cars will not remain constant. 2-1 1.2 m/s
Instead, the distance will continuously decrease. When 2-2 (a) 65 km/h (b) 2.5 s
you first begin breaking, the speed of your car is greater 2-3 Only (d) has the same dimensions on both sides of the
than the speed of the car in front. That is because the equation. Although we cannot obtain the exact
car in front began breaking 0.3 s earlier. Because the equation from dimensional analysis, we can often
cars lose speed at the same rate, the speed of your car obtain the functional dependence.
will remain greater than the speed of the car in front
2-4 54 ms
throughout the braking period.
2-5 (a) and (b) y max ⫺ y0 ⫽ 11.0 m (c) ⫺14.7 m/s; notice that
the final speed is the same as the initial speed
2-6 (a) ⫹0.981 m/s (b) ⫺0.981 m/s
(c) [(⫺0.981 m/s) ⫺ (⫹0.981 m/s]/(0.200 s) ⫽ ⫺9.81 m/s2
2-7 250 m

PROBLEMS
In a few problems, you are given more data than you • Single-concept, single-step, relatively easy
actually need; in a few other problems, you are required to •• Intermediate-level, may require synthesis of concepts
supply data from your general knowledge, outside sources,
••• Challenging
or informed estimates.
SSM Solution is in the Student Solutions Manual
Interpret as significant all digits in numerical values that
have trailing zeroes and no decimal points. Consecutive problems that are shaded are paired
problems.
For all problems, use g ⴝ 9.81 m/s2 for the free-fall
acceleration due to gravity and neglect friction and air
resistance unless instructed to do otherwise.

CONCEPTUAL PROBLEMS passes by the origin at t ⫽ 0.0 s. It continues on at 5.0 m>s for 5.0 s,
after which it gains speed at the constant rate of 0.50 m>s each
1 • What is the average velocity over the “round trip” of an second for 10.0 s. After gaining speed for 10.0 s, the cart loses
object that is launched straight up from the ground and falls speed at the constant rate of 0.50 m>s for the next 15.0 s.
straight back down to the ground? 9 • True/false: Average velocity always equals one-half the
2 • An object thrown straight up falls back and is caught at the sum of the initial and final velocities. Explain your choice.
same place it is launched from. Its time of flight is T; its maximum 10 • Identical twin brothers standing on a horizontal bridge
height is H. Neglect air resistance. The correct expression for its aver- each throw a rock straight down into the water below. They throw
age speed for the entire flight is (a) H/T, (b) 0, (c) H/(2T), (d) 2H/T. rocks at exactly the same time, but one hits the water before the
3 • Using the information in the previous question, what is its other. How can this be? Explain what they did differently. Ignore
average speed just for the first half of the trip? What is its average any effects due to air resistance.
velocity for the second half of the trip? (Answer in terms of H and T.) 11 •• Dr. Josiah S. Carberry stands at the top of the Sears Tower
4 • Give an everyday example of one-dimensional motion in Chicago. Wanting to emulate Galileo, and ignoring the safety of the
where (a) the velocity is westward and the acceleration is eastward, pedestrians below, he drops a bowling ball from the top of the tower.
and (b) the velocity is northward and the acceleration is northward. One second later, he drops a second bowling ball. While the balls are
in the air, does their separation (a) increase over time, (b) decrease,
5 • Stand in the center of a large room. Call the direction to (c) stay the same? Ignore any effects due to air resistance. SSM
across the room along a straight line, using a constant acceleration to 12 •• Which of the position-versus-time curves in Figure 2-28
quickly reach a steady speed along a straight line in the negative di- best shows the motion of an object (a) with positive acceleration,
rection. After reaching this steady speed, keep your velocity negative (b) with constant positive velocity, (c) that is always at rest, and
varied as you walked. (b) Sketch a graph of x versus t for your mo-
tion. Assume you started at x ⫽ 0. (c) Directly under the graph of (b)
Part (b), sketch a graph of vx versus t. SSM
(a) (c)
Position, m

## 6 • True/false: The displacement always equals the product of

the average velocity and the time interval. Explain your choice.
(e)
7 • Is the statement “for an object’s velocity to remain constant,
its acceleration must remain zero” true or false? Explain your choice. (d)

## 8 •• M ULTISTEP Draw careful graphs of the position and

velocity and acceleration over the time period 0 ⱕ t ⱕ 30 s for a
cart that, in succession, has the following motion. The cart is FIGURE 2-28
moving at the constant speed of 5.0 m>s in the ⫹x direction. It Time, s Problem 12
54 | CHAPTER 2 Motion in One Dimension

(d) with negative acceleration? (There may be more than one correct 17 •• True/false:
answer for each part of the problem.) (a) If the acceleration of an object is always zero, then it cannot be
13 •• Which of the velocity-versus-time curves in Figure 2-29 moving.
best describes the motion of an object (a) with constant positive (b) If the acceleration of an object is always zero, then its x-versus-t
acceleration, (b) with positive acceleration that is decreasing with curve must be a straight line.
time, (c) with positive acceleration that is increasing with time, and (c) If the acceleration of an object is nonzero at an instant, it may be
(d) with no acceleration? (There may be more than one correct an- momentarily at rest at that instant.
swer for each part of the problem.) SSM Explain your reasoning for each answer. If an answer is true, give
an example.
18 •• A hard-thrown tennis ball is moving horizontally when it
(b) bangs into a vertical concrete wall at perpendicular incidence. The
(a) ball rebounds straight back off the wall. Neglect any effects due to
Velocity, m/s

(c) gravity for the small time interval described here. Assume that
toward the wall is the ⫹x direction. What are the directions of its
(e) velocity and acceleration (a) just before hitting the wall, (b) at
(d) maximum impact, and (c) just after leaving the wall?

## 19 •• A ball is thrown straight up. Neglect any effects due to air

resistance. (a) What is the velocity of the ball at the top of its flight?
FIGURE 2-29 (b) What is its acceleration at that point? (c) What is different about
Time, s Problem 13 the velocity and acceleration at the top of the flight if instead the
ball impacts a horizontal ceiling very hard and then returns. SSM
14 •• The diagram in Figure 2-30 tracks the location of an ob- 20 •• An object that is launched straight up from the ground,
ject moving in a straight line along the x axis. Assume that the reaches a maximum height H, and falls straight back down to the
object is at the origin at t ⫽ 0. Of the five times shown, which time ground, hitting it T seconds after launch. Neglect any effects due
(or times) represents when the object is (a) farthest from the origin, to air resistance. (a) Express the average speed for the entire trip
(b) at rest for an instant, (c) in the midst of being at rest for a while, as a function of H and T. (b) Express the average speed for the
and (d) moving away from the origin? same interval of time as a function of the initial launch speed v0 .

## Position 21 •• A small lead ball is thrown directly upward. True or

false: (Neglect any effects due to air resistance.) (a) The magnitude
+
of its acceleration decreases on the way up. (b) The direction of its
C acceleration on its way down is opposite to the direction of its ac-
A B E Time celeration on its way up. (c) The direction of its velocity on its way
− D down is opposite to the direction of its velocity on its way up.
22 •• At t ⫽ 0, object A is dropped from the roof of a building.
FIGURE 2-30 Problems 14 and 15 At the same instant, object B is dropped from a window 10 m below
the roof. Air resistance is negligible. During the descent of B to the
ground, the distance between the two objects (a) is proportional to t,
15 •• An object moves along a straight line. Its position-
(b) is proportional to t2, (c) decreases, (d) remains 10 m throughout.
versus-time graph is shown in Figure 2-30. At which time or
times is its (a) speed at a minimum, (b) acceleration positive, and 23 •• C ONTEXT-R ICH You are driving a Porsche that acceler-
(c) velocity negative? SSM ates uniformly from 80.5 km/h (50 mi/h) at t ⫽ 0.00 s to 113 km/h
16 •• For each of the four graphs of x versus t in Figure 2-31 (70 mi/h) at t ⫽ 9.00 s. (a) Which graph in Figure 2-32 best describes
answer the following questions. (a) Is the velocity at time t2 greater the velocity of your car? (b) Sketch a position-versus-time graph
than, less than, or equal to the velocity at time t1 ? (b) Is the speed showing the location of your car during these nine seconds, assum-
at time t2 greater than, less than, or equal to the speed at time t1 ? ing we let its position x be zero at t ⫽ 0.

v v v v v
x x

t t t t t

## t1 t2 t1 t2 (a) (b) (c) (d) (e)

t t
FIGURE 2-32 Problem 23
(a) (b)
24 •• A small heavy object is dropped from rest and falls a dis-
x x tance D in a time T. After it has fallen for a time 2T, what will be its
(a) fall distance from its initial location, (b) its speed, and (c) its ac-
celeration? (Neglect air resistance.)
t2 25 •• In a race, at an instant when two horses are running right
t1 t1 t2 next to each other and in the same direction (the ⫹x direction),
t t
Horse A’s instantaneous velocity and acceleration are ⫹10 m/s and
⫹2.0 m/s2, respectively, and horse B’s instantaneous velocity and
(c) (d)
acceleration are ⫹12 m/s and ⫺1.0 m/s2, respectively. Which horse
FIGURE 2-31 Problem 16 is passing the other at this instant? Explain.
Problems | 55

26 •• True or false: (a) The equation x ⫺ xo ⫽ voxt ⫹ 12 axt2 is al- during the time interval. Note: tf is NOT twice tmax , but represents
ways valid for particle motion in one dimension. (b) If the velocity an arbitrary time. What is the relation between vJ and vmax ?
at a given instant is zero, the acceleration at that instant must also 32 •• Which graph (or graphs), if any, of vx versus t in
be zero. (c) The equation ⌬x ⫽ vav ⌬t holds for all particle motion in Figure 2-36 best describes the motion of a particle with (a) positive
one dimension. velocity and increasing speed, (b) positive velocity and zero accel-
27 •• If an object is moving in a straight line at constant accel- eration, (c) constant nonzero acceleration, and (d) a speed decrease?
eration, its instantaneous velocity halfway through any time inter-
val is (a) greater than its average velocity, (b) less than its average vx vx vx
velocity, (c) equal to its average velocity, (d) half its average veloc- + + +
ity, (e) twice its average velocity.
28 •• A turtle, seeing his owner put some fresh lettuce on the
opposite side of his terrarium, begins to accelerate (at a constant t t t
rate) from rest at time t ⫽ 0, heading directly toward the food. Let
t1 be the time at which the turtle has covered half the distance to his – – –
lunch. Derive an expression for the ratio of t2 to t1 , where t2 is the
time at which the turtle reaches the lettuce. (a) (b) (c)

## 29 •• The positions of two cars in parallel lanes of a straight vx vx

stretch of highway are plotted as functions of time in the + +
Figure 2-33. Take positive values of x as being to the right of the
origin. Qualitatively answer the following: (a) Are the two cars
t t
ever side by side? If so, indicate that time (those times) on the
axis. (b) Are they always traveling in the same direction, or are
they moving in opposite directions for some of the time? If – –
so, when? (c) Are they ever traveling at the same velocity? If so, (d) (e)
when? (d) When are the two cars the farthest apart? (e) Sketch
(no numbers) the velocity versus time curve for each car. SSM FIGURE 2-36 Problems 32 and 33

x
Car A 33 •• Which graph (or graphs), if any, of vx versus t in
Figure 2-36 best describes the motion of a particle with (a) negative
Car B velocity and increasing speed, (b) negative velocity and zero accel-
eration, (c) variable acceleration, and (d) increasing speed?
34 •• Sketch a v-versus-t curve for each of the following condi-
t(s) tions: (a) Acceleration is zero and constant while velocity is not
0 1 2 3 4 5 6 7 8 9 10 zero. (b) Acceleration is constant but not zero. (c) Velocity and ac-
FIGURE 2-33 Problem 29 celeration are both positive. (d) Velocity and acceleration are both
negative. (e) Velocity is positive and acceleration is negative.
( f ) Velocity is negative and acceleration is positive. (g) Velocity is
30 •• A car driving at constant velocity passes the origin at
momentarily zero but the acceleration is not zero.
time t ⫽ 0. At that instant, a truck, at rest at the origin, begins to
accelerate uniformly from rest. Figure 2-34 shows a qualitative 35 •• Figure 2-37 shows nine graphs of position, velocity, and
plot of the velocities of truck and car as functions of time. acceleration for objects in motion along a straight line. Indicate the
Compare their displacements (from the origin), velocities, and graphs that meet the following conditions: (a) Velocity is constant,
accelerations at the instant that their curves intersect.

vx ck x x x
Tru
Car
t t t

## (a) (b) (c)

t
vx vx vx
FIGURE 2-34 Problem 30
31 •• Reginald is out for a morning jog, and during the course t t t
of his run on a straight track, he has a velocity that depends upon
time as shown in Figure 2-35. That is, he begins at rest, and ends at
rest, peaking at a maximum velocity vmax at an arbitrary time tmax . (d) (e) (f)
A second runner, Josie, runs throughout the time interval t ⫽ 0 to
t ⫽ tf at a constant speed vJ, so that each has the same displacement ax ax ax

vx
t t t
vmax
Reginald

## (g) (h) (i)

Josie FIGURE 2-35
tmax tf t Problem 31 FIGURE 2-37 Problem 35
56 | CHAPTER 2 Motion in One Dimension

(b) velocity reverses its direction, (c) acceleration is constant, and Why is that? (b) Estimate the speed of the ball that he is just
(d) acceleration is not constant. (e) Which graphs of position, veloc- releasing from his right hand. (c) Determine how high the ball
ity, and acceleration are mutually consistent? should have gone above the launch point and compare it to an es-
timate from the picture. Hint: You have a built-in distance scale if you
assume some reasonable value for the height of the juggler.
ESTIMATION AND APPROXIMATION
41 •• A rough rule of thumb for determining the distance be-
tween you and a lightning strike is to start counting the seconds
36 • C ONTEXT-R ICH While engrossed in thought about the that elapse (“one-Mississippi, two-Mississippi, . . .”) until you hear
scintillating lecture just delivered by your physics professor you the thunder (sound emitted by the lightning as it rapidly heats the
mistakenly walk directly into the wall (rather than through the air around it). Assuming the speed of sound is about 750 mi/h,
open lecture hall door). Estimate the magnitude of your average ac- (a) estimate how far away is a lightning strike if you counted about
celeration as you rapidly come to a halt. 5 s until you heard the thunder. (b) Estimate the uncertainty in the
37 • B IOLOGICAL A PPLICATION Occasionally, people can sur- distance to the strike in Part (a). Be sure to explain your assump-
vive falling large distances if the surface they land on is soft tions and reasoning. Hint: The speed of sound depends on the air tem-
enough. During a traverse of the Eiger’s infamous Nordvand, perature, and your counting is far from exact!
mountaineer Carlos Ragone’s rock anchor gave way and he plum-
meted 500 feet to land in snow. Amazingly, he suffered only a few
bruises and a wrenched shoulder. Assuming that his impact left a
hole in the snow 4.0 ft deep, estimate his average acceleration as he SPEED, DISPLACEMENT,
slowed to a stop (that is, while he was impacting the snow). SSM AND VELOCITY
38 •• When we solve free-fall problems near Earth, it’s impor-
tant to remember that air resistance may play a significant role. If its 42 • E NGINEERING A PPLICATION (a) An electron in a televi-
effects are significant, we may get answers that are wrong by orders sion tube travels the 16-cm distance from the grid to the screen at
of magnitude if we ignore it. How can we tell when it is valid to ig- an average speed of 4.0 ⫻ 107 m/s. How long does the trip take?
nore the effects of air resistance? One way is to realize that air re- (b) An electron in a current-carrying wire travels at an average
sistance increases with increasing speed. Thus, as an object falls and speed of 4.0 ⫻ 10⫺5 m/s. How long does it take to travel 16 cm?
its speed increases, its downward acceleration decreases. Under these
circumstances, the object’s speed will approach, as a limit, a value 43 • A runner runs 2.5 km, in a straight line, in 9.0 min and
called its terminal speed. This terminal speed depends upon such then takes 30 min to walk back to the starting point. (a) What is
things as the mass and cross-sectional area of the body. Upon reach- the runner’s average velocity for the first 9.0 min? (b) What is the
ing its terminal speed, its acceleration is zero. For a “typical” sky- average velocity for the time spent walking? (c) What is the av-
diver falling through the air, a typical terminal speed is about erage velocity for the whole trip? (d) What is the average speed
50 m/s (roughly 120 mph). At half its terminal speed, the sky- for the whole trip? SSM
diver’s acceleration will be about 34 g. Let us take half the terminal 44 • A car travels in a straight line with an average veloc-
speed as a reasonable “upper bound” beyond which we should not ity of 80 km/h for 2.5 h and then with an average velocity of
use our constant acceleration free-fall relationships. Assuming the 40 km/h for 1.5 h. (a) What is the total displacement for the 4.0-h
skydiver started from rest, (a) estimate how far, and for how long, trip? (b) What is the average velocity for the total trip?
the skydiver falls before we can no longer neglect air resistance.
(b) Repeat the analysis for a Ping-Pong ball, which has a terminal
45 • One busy air route across the Atlantic Ocean is about
speed of about 5.0 m/s. (c) What can you conclude by comparing
5500 km. The now-retired Concord, a supersonic jet capable of fly-
ing at twice the speed of sound, was used for traveling such routes.
39 •• B I O LO G I C A L A P P L I C AT I O N On June 14, 2005, Asafa (a) Roughly how long did it take for a one-way flight? (Use 343 m/s
Powell of Jamaica set a world’s record for the 100-m dash with a for the speed of sound.) (b) Compare this time to the time taken by
time t ⫽ 9.77 s. Assuming he reached his maximum speed in 3.00 s, a subsonic jet flying at 0.90 times the speed of sound.
and then maintained that speed until the finish, estimate his accel- 46 • The speed of light, designated by the universally
eration during the first 3.00 s. recognized symbol c, has a value, to two significant figures, of
40 •• The photograph in Figure 2-38 is a short-time exposure 3.0 ⫻ 108 m/s. (a) How long does it take for light to travel from the
(1/30 s) of a juggler with two tennis balls in the air. (a) The tennis Sun to Earth, a distance of 1.5 ⫻ 1011 m? (b) How long does it take
ball near the top of its trajectory is less blurred than the lower one. light to travel from the moon to Earth, a distance of 3.8 ⫻ 108 m?
47 • Proxima Centauri, the closest star to us besides our own
Sun, is 4.1 ⫻ 1013 km from Earth. From Zorg, a planet orbiting this
star, a Gregor places an order at Tony’s Pizza in Hoboken, New
Jersey, communicating by light signals. Tony’s fastest delivery craft
travels at 1.00 ⫻ 10⫺4c (see Problem 46). (a) How long does it take
Gregor’s order to reach Tony’s Pizza? (b) How long does Gregor
wait between sending the signal and receiving the pizza? If Tony’s
has a “1000-years-or-it’s-free” delivery policy, does Gregor have to
pay for the pizza? SSM
48 • A car making a 100-km journey travels 40 km/h for the
first 50 km. How fast must it go during the second 50 km to aver-
age 50 km/h?
FIGURE 2-38 49 •• C ONTEXT-R ICH Late in ice hockey games, the team that
Problem 40 is losing sometimes “pulls” their goalkeeper off the ice to add an
(Courtesy of Chuck Adler.) additional offensive player and increase their chances of scoring. In
Problems | 57

such cases, the goalie on the opposing team might have an oppor- 55 •• M ULTISTEP A car traveling at a constant speed of
tunity to score into the unguarded net 55.0 m away. Suppose you 20 m/s passes an intersection at time t ⫽ 0. A second car travel-
are the goaltender for your university team and are in just such a ing at a constant speed of 30 m/s in the same direction passes the
situation. You launch a shot (in hopes of getting your first career same intersection 5.0 s later. (a) Sketch the position functions x1 (t)
goal) on the frictionless ice. You eventually hear a disappointing and x 2 (t) for the two cars for the interval 0 ⱕ t ⱕ 20 s .
“clang” as the puck strikes a goalpost (instead of going in!) exactly (b) Determine when the second car will overtake the first.
2.50 s later. In this case, how fast did the puck travel? You should (c) How far from the intersection will the two cars be when they
assume 343 m/s for the speed of sound. pull even? (d) Where is the first car when the second car passes
50 •• Cosmonaut Andrei, your co-worker at the International the intersection? SSM
Space Station, tosses a banana at you at a speed of 15 m/s. At ex- 56 •• B IOLOGICAL A PPLICATION Bats use echolocation to de-
actly the same instant, you fling a scoop of ice cream at Andrei termine their distance from objects they cannot easily see in the
along exactly the same path. The collision between banana and ice dark. The time between the emission of a high-frequency sound
cream produces a banana split 7.2 m from your location 1.2 s after pulse (a click) and the detection of its echo is used to determine
the banana and ice cream were launched. (a) How fast did you such distances. A bat, flying at a constant speed of 19.5 m/s in a
toss the ice cream? (b) How far were you from Andrei when straight line toward a vertical cave wall, makes a single clicking
you tossed the ice cream? (Neglect any effects due to gravity.) noise and hears the echo 0.15 s later. Assuming that she contin-
51 •• Figure 2-39 shows the position of a particle as a function ued flying at her original speed, how close was she to the wall
of time. Find the average velocities for the time intervals a, b, c, and when she received the echo? Assume a speed of 343 m/s for the
d indicated in the figure. speed of sound.
57 ••• E NGINEERING A PPLICATION A submarine can use sonar
(sound traveling through water) to determine its distance from
x, m
other objects. The time between the emission of a sound pulse (a
6 “ping”) and the detection of its echo can be used to determine such
distances. Alternatively, by measuring the time between successive
4 echo receptions of a regularly timed set of pings, the submarine’s
speed may be determined by comparing the time between echoes to
2 the time between pings. Assume you are the sonar operator in a
submarine traveling at a constant velocity underwater. Your boat is
in the eastern Mediterranean Sea, where the speed of sound is
2 4 6 8 10 12 14 t, s known to be 1522 m/s. If you send out pings every 2.00 s, and your
–2 apparatus receives echoes reflected from an undersea cliff every
1.98 s, how fast is your submarine traveling?
–4

–6 ACCELERATION
a b c d

## 58 • A sports car accelerates in third gear from 48.3 km/h

FIGURE 2-39 Problem 51 (about 30 mi/h) to 80.5 km/h (about 50 mi/h) in 3.70 s. (a) What is
the average acceleration of this car in m/s2? (b) If the car maintained
this acceleration, how fast would it be moving one second later?
52 •• ENGINEERING APPLICATION It has been found that, on aver- 59 • An object is moving along the x axis. At t ⫽ 5.0 s, the ob-
age, galaxies are moving away from Earth at a speed that is propor- ject is at x ⫽ ⫹3.0 m and has a velocity of ⫹5.0 m/s. At t ⫽ 8.0 s, it
tional to their distance from Earth. This discovery is known as is at x ⫽ ⫹9.0 m and its velocity is ⫺1.0 m/s. Find its average ac-
Hubble’s law, named for its discoverer, astrophysicist Sir Edwin celeration during the time interval 5.0 s ⬍ t ⬍ 8.0 s. SSM
Hubble. He found that the recessional speed v of a galaxy a dis-
tance r from Earth is given by v ⫽ Hr, where H ⫽ 1.58 ⫻ 10⫺18 s⫺1 is 60 •• A particle moves along the x axis with velocity vx ⫽
called the Hubble constant. What are the expected recessional (8.0 m/s2)t 2 7.0 m/s. (a) Find the average acceleration for two dif-
speeds of galaxies (a) 5.00 ⫻ 1022 m from Earth, and (b) 2.00 ⫻ 1025 m ferent one-second intervals, one beginning at t ⫽ 3.0 s and the other
from Earth? (c) If the galaxies at each of these distances had traveled beginning at t ⫽ 4.0 s. (b) Sketch vx versus t over the interval
at their expected recessional speeds, how long ago would they have 0 ⬍ t ⬍ 10 s. (c) How do the instantaneous accelerations at the mid-
been at our location? dle of each of the two time intervals specified in Part (a) compare to
the average accelerations found in Part (a)? Explain.
53 •• The cheetah can run as fast as 113 km/h, the falcon can
fly as fast as 161 km/h, and the sailfish can swim as fast as
61 •• M ULTISTEP The position of a certain particle depends
105 km/h. The three of them run a relay with each covering a dis-
on time according to the equation x(t) ⫽ t2 ⫺ 5.0t ⫹ 1.0 , where
tance L at maximum speed. What is the average speed of this relay
x is in meters if t is in seconds. (a) Find the displacement and
team for the entire relay? Compare this average speed with the nu-
average velocity for the interval 3.0 s ⱕ t ⱕ 4.0 s . (b) Find the
merical average of the three individual speeds. Explain carefully
general formula for the displacement for the time interval from
why the average speed of the relay team is not equal to the numer-
t to t ⫹ ⌬t. (c) Use the limiting process to obtain the instanta-
ical average of the three individual speeds. SSM
neous velocity for any time t. SSM
54 •• Two cars are traveling along a straight road. Car A
62 •• The position of an object as a function of time is given
maintains a constant speed of 80 km/h and car B maintains a con-
by x ⫽ At2 ⫺ Bt ⫹ C, where A ⫽ 8.0 m/s2, B ⫽ 6.0 m/s, and
stant speed of 110 km/h. At t ⫽ 0, car B is 45 km behind car A. (a)
C ⫽ 4.0 m. Find the instantaneous velocity and acceleration as
How much farther will car A travel before car B overtakes it?
functions of time.
(b) How much ahead of A will B be 30 s after it overtakes A?
58 | CHAPTER 2 Motion in One Dimension

63 ••• The one-dimensional motion of a particle is plotted in 71 •• A load of bricks is lifted by a crane at a steady velocity
Figure 2-40. (a) What is the average acceleration in each of the in- of 5.0 m/s when one brick falls off 6.0 m above the ground.
tervals AB, BC, and CE? (b) How far is the particle from its starting (a) Sketch the position of the brick y(t) versus time, from the
point after 10 s? (c) Sketch the displacement of the particle as a func- moment it leaves the pallet until it hits the ground. (b) What is
tion of time; label the instants A, B, C, D, and E on your graph. (d) At the greatest height the brick reaches above the ground? (c) How
what time is the particle traveling most slowly? long does it take to reach the ground? (d) What is its speed just
before it hits the ground? SSM
72 •• A bolt comes loose from underneath an elevator that
vx, m/s is moving upward at a constant speed of 6.0 m/s. The bolt
15 reaches the bottom of the elevator shaft in 3.0 s. (a) How high
above the bottom of the shaft was the elevator when the bolt
10 came loose? (b) What is the speed of the bolt when it hits the bot-
5 tom of the shaft?
A B C D E
0
2 4 6 8 10 t, s 73 •• An object is dropped from rest at a height of 120 m. Find
–5 the distance it falls during its final second in the air.
–10 74 •• An object is released from rest at a height h. During the
final second of its fall, it traverses a distance of 38 m. Determine h.
–15
75 •• A stone is thrown vertically downward from the top of a
200-m cliff. During the last half second of its flight, the stone trav-
FIGURE 2-40 Problem 63 els a distance of 45 m. Find the initial speed of the stone. SSM
76 •• An object is released from rest at a height h. It travels 0.4h
during the first second of its descent. Determine the average veloc-
ity of the object during its entire descent.
CONSTANT ACCELERATION 77 •• A bus accelerates from rest at 1.5 m/s2 for 12 s. It then
AND FREE-FALL travels at constant velocity for 25 s, after which it slows to a stop
with an acceleration of magnitude 1.5 m/s2. (a) What is the total
distance that the bus travels? (b) What is its average velocity?
64 • An object projected vertically upward with initial speed
v0 attains a maximum height h above its launch point. Another 78 •• Al and Bert are jogging side-by-side on a trail in the
object projected up with initial speed 2v0 from the same height will woods at a speed of 0.75 m/s. Suddenly Al sees the end of the trail
attain a maximum height of (a) 4h, (b) 3h, (c) 2h, (d) h. (Air resistance 35 m ahead and decides to speed up to reach it. He accelerates at a
is negligible.) constant rate of 0.50 m/s2 while Bert continues on at a constant
speed. (a) How long does it take Al to reach the end of the trail?
65 • A car traveling along the x axis starts from rest at
(b) Once he reaches the end of the trail, he immediately turns
x ⫽ 50 m and accelerates at a constant rate of 8.0 m/s2. (a) How fast
around and heads back along the trail with a constant speed of
is it going after 10 s? (b) How far has it gone after 10 s? (c) What is
0.85 m/s. How long does it take him to meet up with Bert? (c) How
its average velocity for the interval 0 ⱕ t ⱕ 10 s ?
far are they from the end of the trail when they meet?
66 • An object traveling along the x axis with an initial veloc-
ity of ⫹5.0 m/s has a constant acceleration of ⫹2.0 m/s2. When its 79 •• You have designed a rocket to be used to sample the local
speed is 15 m/s, how far has it traveled? atmosphere for pollution. It is fired vertically with a constant up-
ward acceleration of 20 m/s2. After 25 s, the engine shuts off and
67 • An object traveling along the x axis at constant accel- the rocket continues rising (in freefall) for a while. (Air resistance is
eration has a velocity of ⫹10 m/s when it is at x ⫽ 6.0 m and of negligible.) The rocket eventually stops rising and then falls back to
⫹15 m/s when it is at x ⫽ 10.0 m. What is its acceleration? SSM the ground. You want to get a sample of air that is 20 km above the
ground. (a) Did you reach your height goal? If not, what would you
68 • The speed of an object traveling along the x axis in- change so that the rocket raches 20 km? (b) Determine the total time
creases at the constant rate of ⫹4.0 m/s each second. At t ⫽ 0.0 s, the rocket is in the air. (c) Find the speed of the rocket just before it
its velocity is ⫹1.0 m/s and its position is ⫹7.0 m. How fast is it hits the ground.
moving when its position is ⫹8.0 m, and how much time has
elapsed from the start at t ⫽ 0.0 s? 80 •• A flowerpot falls from a windowsill of an apartment that
is on the tenth floor of an apartment building. A person in an apart-
69 •• A ball is launched directly upward from ground level ment below, coincidentally in possession of a high-speed high-
with an initial speed of 20 m/s. (Air resistance is negligible.) precision timing system, notices that it takes 0.20 s for the pot to fall
(a) How long is the ball in the air? (b) What is the greatest height past his window, which is 4.0-m from top to bottom. How far above
reached by the ball? (c) How many seconds after launch is the ball the top of the window is the windowsill from which the pot fell?
15 m above the release point? (Neglect any effects due to air resistance.)
70 •• In the Blackhawk landslide in California, a mass of rock 81 •• In a classroom demonstration, a glider moves along an
and mud fell 460 m down a mountain and then traveled 8.00 km inclined track with constant acceleration. It is projected from the
across a level plain. It has been theorized that the rock and mud low end of the track with an initial velocity. After 8.00 s have
moved on a cushion of water vapor. Assume that the mass elapsed, it is 100 cm from the low end and is moving along the track
dropped with the free-fall acceleration and then slid horizontally, at a velocity of ⫺15 cm/s. Find the initial velocity and the
losing speed at a constant rate. (a) How long did the mud take to acceleration. SSM
drop the 460 m? (b) How fast was it traveling when it reached the 82 •• A rock dropped from a cliff covers one-third of its total
bottom? (c) How long did the mud take to slide the 8.00 km distance to the ground in the last second of its fall. Air resistance is
horizontally? negligible. How high is the cliff?
Problems | 59

83 •• A typical automobile under hard braking loses speed at 92 •• Starting from rest, a particle travels along the x axis with
a rate of about 7.0 m/s2; the typical reaction time to engage the a constant acceleration of ⫹3.0 m>s2 . At a time 4.0 s following its
brakes is 0.50 s. A local school board sets the speed limit in a school start, it is at x ⫽ ⫺100 m. At a time 6.0 s later it has a velocity of
zone such that all cars should be able to stop in 4.0 m. (a) What max- ⫹15 m>s. Find its position at this later time.
imum speed does this imply for an automobile in this zone?
(b) What fraction of the 4.0 m is due to the reaction time? SSM 93 •• If it were possible for a spacecraft to maintain a constant
acceleration indefinitely, trips to the planets of the Solar System
84 •• Two trains face each other on adjacent tracks. They are could be undertaken in days or weeks, while voyages to the nearer
initially at rest, and their front ends are 40 m apart. The train on the stars would only take a few years. (a) Using data from the tables at
left accelerates rightward at 1.0 m/s2. The train on the right accel- the back of the book, find the time it would take for a one-way trip
erates leftward at 1.3 m/s2. (a) How far does the train on the left from Earth to Mars (at Mars’ closest approach to Earth). Assume
travel before the front ends of the trains pass? (b) If the trains are that the spacecraft starts from rest, travels along a straight line, ac-
each 150 m in length, how long after the start are they completely celerates halfway at 1 g, flips around, and decelerates at 1 g for the
past one another, assuming their accelerations are constant? rest of the trip. (b) Repeat the calculation for a 4.1 ⫻ 1013-km trip to
85 •• Two stones are dropped from the edge of a 60-m cliff, the Proxima Centauri, our nearest stellar neighbor outside of the Sun.
second stone 1.6 s after the first. How far below the top of the cliff (See Problem 47.) SSM
is the second stone when the separation between the two stones 94 •• The Stratosphere Tower in Las Vegas is 1137 ft high. It
is 36 m? takes 1 min, 20 s to ascend from the ground floor to the top of the
86 •• A motorcycle officer hidden at an intersection observes a tower using the high-speed elevator. The elevator starts and ends at
car driven by an oblivious driver who ignores a stop sign and con- rest. Assume that it maintains a constant upward acceleration until
tinues through the intersection at constant speed. The police officer it reaches its maximum speed, and then maintains a constant accel-
takes off in pursuit 2.0 s after the car has passed the stop sign. She eration of equal magnitude until it comes to a stop. Find the mag-
accelerates at 4.2 m/s2 until her speed is 110 km/h, and then con- nitude of the acceleration of the elevator. Express this acceleration
tinues at this speed until she catches the car. At that instant, the car magnitude as a multiple of g (the acceleration due to gravity).
is 1.4 km from the intersection. (a) How long did it take for the of-
95 •• A train pulls away from a station with a constant accel-
ficer to catch up to the car? (b) How fast was the car traveling?
eration of 0.40 m/s2. A passenger arrives at a point next to the track
87 •• At t ⫽ 0, a stone is dropped from the top of a cliff above 6.0 s after the end of the train has passed the very same point. What
a lake. Another stone is thrown downward 1.6 s later from the same is the slowest constant speed at which she can run and still catch the
point with an initial speed of 32 m/s. Both stones hit the water at train? On a single graph, plot the position versus time curves for
the same instant. Find the height of the cliff. both the train and the passenger.
88 •• A passenger train is traveling at 29 m/s when the engi- 96 ••• Ball A is dropped from the top of a building of height h
neer sees a freight train 360 m ahead of his train traveling in the at the same instant that ball B is thrown vertically upward from the
same direction on the same track. The freight train is moving at a ground. When the balls collide, they are moving in opposite direc-
speed of 6.0 m/s. (a) If the reaction time of the engineer is 0.40 s, tions, and the speed of A is twice the speed of B. At what height
what is the minimum (constant) rate at which the passenger train does the collision occur?
must lose speed if a collision is to be avoided? (b) If the engineer’s
reaction time is 0.80 s and the train loses speed at the minimum rate 97 ••• Solve Problem 96 if the collision occurs when the balls
described in Part (a), at what rate is the passenger train approach- are moving in the same direction and the speed of A is 4 times
ing the freight train when the two collide? (c) For both reaction that of B.
times, how far will the passenger train have traveled in the time be- 98 ••• Starting at one station, a subway train accelerates from
tween the sighting of the freight train and the collision? rest at a constant rate of 1.00 m/s2 for half the distance to the next
89 •• B IOLOGICAL A PPLICATION The click beetle can project it- station, then slows down at the same rate for the second half of the
self vertically with an acceleration of about 400g (an order of mag- journey. The total distance between stations is 900 m. (a) Sketch a
nitude more than a human could survive!). The beetle jumps by graph of the velocity vx as a function of time over the full journey.
“unfolding” its 0.60-cm long legs. (a) How high can the click beetle (b) Sketch a graph of the position as a function of time over the full
jump? (b) How long is the beetle in the air? (Assume constant ac- journey. Place appropriate numerical values on both axes.
celeration while in contact with the ground and neglect air
resistance.)
99 ••• A speeder traveling at a constant speed of 125 km/h
90 •• An automobile accelerates from rest at 2.0 m/s2 for 20 s. races past a billboard. A patrol car pursues from rest with con-
The speed is then held constant for 20 s, after which there is an ac- stant acceleration of (8.0 km/h)/s until it reaches its maximum
celeration of ⫺3.0 m/s2 until the automobile stops. What is the total speed of 190 km/h, which it maintains until it catches up with
distance traveled? the speeder. (a) How long does it take the patrol car to catch the
91 •• Consider measuring the free-fall motion of a particle (ne- speeder if it starts moving just as the speeder passes? (b) How
glect air resistance). Before the advent of computer-driven data-log- far does each car travel? (c) Sketch x(t) for each car. SSM
ging software, these experiments typically employed a wax-coated
100 ••• When the patrol car in Problem 99 (traveling at
tape placed vertically next to the path of a dropped electrically con-
190 km/h) is 100 m behind the speeder (traveling at 125 km/h),
ductive object. A spark generator would cause an arc to jump be-
the speeder sees the police car and slams on his brakes, locking
tween two vertical wires through the falling object and through the
the wheels. (a) Assuming that each car can brake at 6.0 m/s2 and
tape, thereby marking the tape at fixed time intervals ⌬t. Show that
that the driver of the police car brakes instantly as she sees the
the change in height during successive time intervals for an object
brake lights of the speeder (reaction time ⫽ 0.0 s), show that the
falling from rest follows Galileo’s Rule of Odd Numbers: ⌬y21 ⫽ 3⌬y10 ,
cars collide. (b) At what time after the speeder applies his brakes
⌬y32 ⫽ 5⌬y10 , . . . , where ⌬y10 is the change in y during the first in-
do the two cars collide? (c) Discuss how reaction time would af-
terval of duration ⌬t, ⌬y21 is the change in y during the second
fect this problem.
interval of duration ⌬t, etc. SSM
60 | CHAPTER 2 Motion in One Dimension

101 ••• Leadfoot Lou enters the “Rest-to-Rest” auto competition, 106 •• Consider the velocity graph in Figure 2-42. Assuming
in which each contestant’s car begins and ends at rest, covering a x ⫽ 0 at t ⫽ 0, write correct algebraic expressions for x(t), vx (t), and
fixed distance L in as short a time as possible. The intention is to ax (t) with appropriate numerical values inserted for all constants.
demonstrate driving skills, and to find which car is the best at the
total combination of speeding up and slowing down. The course is de- vx, m/s
signed so that maximum speeds of the cars are never reached. (a) If
Lou’s car maintains an acceleration (magnitude) of a during 50
speedup, and maintains a deceleration (magnitude) of 2a during
braking, at what fraction of L should Lou move his foot from the gas
pedal to the brake? (b) What fraction of the total time for the trip has 0
elapsed at that point? (c) What is the fastest speed Lou’s car ever 10 t, s
reaches? Neglect Lou’s reaction time, and answer in terms of a and L.
102 ••• A physics professor, equipped with a rocket backpack, –50
steps out of a helicopter at an altitude of 575 m with zero initial ve-
locity. (Neglect air resistance.) For 8.0 s, she falls freely. At that time,
she fires her rockets and slows her rate of descent at 15 m/s2 until FIGURE 2-42 Problem 106
her rate of descent reaches 5.0 m/s. At this point, she adjusts her
rocket engine controls to maintain that rate of descent until she
107 ••• Figure 2-43 shows the acceleration of a particle versus
reaches the ground. (a) On a single graph, sketch her acceleration
time. (a) What is the magnitude, in m/s, of the area of the shaded
and velocity as functions of time. (Take upward to be positive.)
box? (b) The particle starts from rest at t ⫽ 0. Estimate the velocity
(b) What is her speed at the end of the first 8.0 s? (c) What is the du-
at t ⫽ 1.0 s, 2.0 s, and 3.0 s by counting the boxes under the curve.
ration of her slowing-down period? (d) How far does she travel
(c) Sketch the curve vx versus t from your results for Part (b); then
while slowing down? (e) How much time is required for the entire
estimate how far the particle travels in the interval t ⫽ 0 to t ⫽ 3.0 s.
trip from the helicopter to the ground? ( f ) What is her average ve-
locity for the entire trip?
ax, m/s2
INTEGRATION 4
OF THE EQUATIONS OF MOTION
3
103 • The velocity of a particle is given by vx(t) ⫽ (6.0 m/s )t 2

⫹ (3.0 m/s). (a) Sketch v versus t and find the area under the 2
curve for the interval t ⫽ 0 to t ⫽ 5.0 s. (b) Find the position func-
tion x(t). Use it to calculate the displacement during the interval 1
t ⫽ 0 to t ⫽ 5.0 s. SSM
104 • Figure 2-41 shows the velocity of a particle versus 0
time. (a) What is the magnitude, in meters, represented by the 0 1 2 3 4 t, s
area of the shaded box? (b) Estimate the displacement of the
particle for the two 1-s intervals, one beginning at t ⫽ 1.0 s and FIGURE 2-43 Problem 107
the other at t ⫽ 2.0 s. (c) Estimate the average velocity for the
interval 1.0 s ⱕ t ⱕ 3.0 s . (d) The equation of the curve is vx ⫽ 108 ••• Figure 2-44 is a graph of vx versus t for a particle moving
(0.50 m/s3)t2. Find the displacement of the particle for the inter- along a straight line. The position of the particle at time t ⫽ 0 is
val 1.0 s ⱕ t ⱕ 3.0 s by integration and compare this answer with x0 ⫽ 5.0 m. (a) Find x for various times t by counting boxes, and
your answer for Part (b). Is the average velocity equal to the sketch x as a function of t. (b) Sketch a graph of the acceleration ax
mean of the initial and final velocities for this case? as a function of the time t. (c) Determine the displacement of the
particle between t ⫽ 3.0 s and 7.0 s.
v, m/s
vx, m/s
8
8
7
6 6
5
4
4
3 2
2
1 0

1 2 3 4 t, s −2

−4
FIGURE 2-41 Problem 104
−6
105 •• The velocity of a particle is given by vx ⫽ (7.0 m/s3)t2 ⫺ 0 1 2 3 4 5 6 7 8 9 10 t, s
5.0 m/s. If the particle is at the origin at t0 ⫽ 0, find the position
function x(t). FIGURE 2-44 Problem 108
Problems | 61

109 ••• C ONCEPTUAL Figure 2-45 shows a plot of x versus t for what value of ⌬t would you expect to measure, assuming gexp is the
an object moving along a straight line. For this motion, sketch standard value (9.81 m/s2)? (c) During the experiment, a slight
graphs (using the same t axis) of (a) vx as a function of t, and (b) ax error is made. Instead of locating the first photogate even with the
as a function of t. (c) Use your sketches to qualitatively compare the top of the table, your not-so-careful lab partner locates it 0.50 cm
time(s) when the object is at its largest distance from the origin to lower than the top of the table. However, she does manage to prop-
the time(s) when its speed is greatest. Explain why the times are not erly locate the second photogate at a height of 0.50 m above the
the same. (d) Use your sketches to qualitatively compare the time(s) floor. However, she releases the marble from the same height that it
when the object is moving fastest to the time(s) when its accelera- was released from when the photogate was 1.00 m above the floor.
tion is the largest. Explain why the times are not the same. SSM What value of gexp will you and your partner determine? What per-
centage difference does this represent from the standard value of g?
x 114 ••• M ULTISTEP The position of a body oscillating on a
spring is given by x ⫽ A sin vt, where A and v (lower case Greek
omega) are constants, A ⫽ 5.0 cm, and v ⫽ 0.175 s⫺1. (a) Plot x as a
function of t for 0 ⱕ t ⱕ 36 s . (b) Measure the slope of your graph at
t t ⫽ 0 to find the velocity at this time. (c) Calculate the average
velocity for a series of intervals, beginning at t ⫽ 0 and ending at
t ⫽ 6.0, 3.0, 2.0, 1.0, 0.50, and 0.25 s. (d) Compute dx/dt to find the
velocity at time t ⫽ 0. (e) Compare your results in Parts (c) and (d)
and explain why your Part (c) results approach your Part (d) result.
FIGURE 2-45 Problem 109
115 ••• C ONCEPTUAL Consider an object that is attached to a
110 ••• M ULTISTEP The acceleration of a certain rocket is given horizontally oscillating piston. The object moves with a velocity
by ax ⫽ bt, where b is a positive constant. (a) Find the position func- given by v ⫽ B sin(vt), where B and v (lower case Greek omega)
tion x(t) if x ⫽ x0 and vx ⫽ v0x at t ⫽ 0. (b) Find the position and ve- are constants and v is in s⫺1. (a) Explain why B is equal to the max-
locity at t ⫽ 5.0 s if x0 ⫽ 0, v0x ⫽ 0 and b ⫽ 3.0 m/s3. (c) Compute imum speed vmax . (b) Determine the acceleration of the object as a
the average velocity of the rocket between t ⫽ 4.5 s and 5.5 s at function of time. Is the acceleration constant? (c) What is the maxi-
t ⫽ 5.0 s if x0 ⫽ 0, v0x ⫽ 0 and b ⫽ 3.0 m/s3. Compare this average mum acceleration (magnitude) in terms of v and vmax . (d) At t ⫽ 0,
velocity with the instantaneous velocity at t ⫽ 5.0 s. the object’s position is known to be x0 . Determine the position as a
function of time in terms of t, v, x0 and vmax . SSM
111 ••• In the time interval from 0.0 s to 10.0 s, the accelera- 116 ••• Suppose the acceleration of a particle is a function of x,
tion of a particle traveling in a straight line is given by ax ⫽ where ax(x) ⫽ (2.0 s⫺2)x. (a) If the velocity is zero when x ⫽ 1.0 m,
(0.20 m/s3)t. Let to the right be the ⫹x direction. The particle ini- what is the speed when x ⫽ 3.0 m? (b) How long does it take the
tially has a velocity to the right of 9.5 m/s and is located 5.0 m particle to travel from x ⫽ 1.0 m to x ⫽ 3.0 m.
to the left of the origin. (a) Determine the velocity as a function
of time during the interval; (b) determine the position as a func-
tion of time during the interval; (c) determine the average veloc- 117 ••• A rock falls through water with a continuously de-
ity between t ⫽ 0.0 s and 10.0 s, and compare it to the average of creasing acceleration. Assume that the rock’s acceleration as a
the instantaneous velocities at the start and ending times. Are function of velocity has the form ay ⫽ g ⫺ bvy where b is a posi-
these two averages equal? Explain. SSM tive constant. (The ⫹y direction is directly downward.) (a) What
112 ••• Consider the motion of a particle that experiences a are the SI units of b? (b) Prove mathematically that if the rock is
variable acceleration given by ax ⫽ a0x ⫹ bt, where a0x and b are released from rest at time t ⫽ 0, the acceleration will depend
constants and x ⫽ x0 and vx ⫽ v0x at t ⫽ 0. (a) Find the instanta- exponentially on time according to ay (t) ⫽ ge2bt. (c) What is the
neous velocity as a function of time. (b) Find the position as a terminal speed for the rock in terms of g and b? (See Problem 38
function of time. (c) Find the average velocity for the time for an explanation of the phenomenon of terminal speed.) SSM
interval with an initial time of zero and arbitrary final time t. 118 ••• A small rock sinking through water (see Problem 117)
(d) Compare the average of the initial and final velocities to your experiences an exponentially decreasing acceleration given by
answer to Part (c). Are these two averages equal? Explain. ay (t) ⫽ ge2bt, where b is a positive constant that depends on the
shape and size of the rock and the physical properties of the
water. Based upon this, find expressions for the velocity and po-
GENERAL PROBLEMS sition of the rock as functions of time. Assume that its initial po-
sition and velocity are both zero and that the ⫹y direction is di-
rectly downward.
113 ••• C ONTEXT-R ICH You are a student in a science class that
is using the following apparatus to determine the value of g. Two
photogates are used. (Note: You may be familiar with photogates in 119 ••• S PREADSHEET The acceleration of a skydiver jumping
everyday living. You see them in the doorways of some stores. They from an airplane is given by ay ⫽ g ⫺ bv2y , where b is a positive
are designed to ring a bell when someone interrupts the beam while constant that depends on the skydiver’s cross-sectional area and
walking through the door.) One photogate is located at the edge of the density of the surrounding atmosphere she is diving through.
a table that is 1.00 m above the floor, and the second photogate is The ⫹y directions is directly downward. (a) If her initial speed is
located directly below the first, at a height 0.500 m above the floor. zero when stepping from a hovering helicopter, show that her
You are told to drop a marble through these gates, releasing it from speed as a function of time is given by vy(t) ⫽ vt tanh (t/T), where
rest a negligible distance above the upper gate. The upper gate vt is the terminal speed (see Problem 38) given by vt ⫽ 1g/b,
starts a timer as the ball passes through its beam. The second pho- and T ⫽ vt >g is a time-scale parameter. (b) What fraction of the ter-
togate stops the timer when the ball passes through its beam. (a) minal speed is the speed at t ⫽ T. (c) Use a spreadsheet program
Prove that the experimental magnitude of free-fall acceleration is to graph vy (t) as a function of time, using a terminal speed of
given by gexp ⫽ (2⌬y)/(⌬t)2, where ⌬y is the vertical distance be- 56 m/s (use this value to calculate b and T). Does the resulting
tween the photogates and ⌬t is the fall time. (b) For your setup, curve make sense?
62 | CHAPTER 2 Motion in One Dimension

120 ••• A PPROXIMATION Imagine that you are standing at a the light is red. Immediately after passing through the intersection,
wishing well, wishing that you knew how deep the surface of the you take your foot off the accelerator, relieved. However, down the
water was. Cleverly, you make your wish. Then you take a penny road you are pulled over for speeding. You assume that you were
from your pocket and drop it into the well. Exactly three seconds ticketed for the speed of your car as it exited the intersection.
after you dropped the penny, you hear the sound it made when it Determine this speed and decide whether you should fight this
struck the water. If the speed of sound is 343 m/s, how deep is the ticket in court. Explain.
well? Neglect any effects due to air resistance.
122 ••• For a spherical celestial object of radius R, the acceleration
121 ••• C ONTEXT-R ICH You are driving a car at the 25-mi/h due to gravity g at a distance x from the center of the object is
speed limit when you observe the light at the intersection 65 m in g ⫽ g0R 2>x 2, where g0 is the acceleration due to gravity at the ob-
front of you turn yellow. You know that at that particular intersec- ject’s surface and x ⬎ R. For the moon, take g0 ⫽ 1.63 m/s2 and
tion the light remains yellow for exactly 5.0 s before turning red. R ⫽ 3200 km. If a rock is released from rest at a height of 4R above
After you think for 1.0 s, you then accelerate the car at a constant the lunar surface, with what speed does the rock impact the moon?
rate. You somehow manage to pass your 4.5-m-long car completely Hint: Its acceleration is a function of position and increases as the object
through the 15.0-m-wide intersection just as the light turns red, falls. So do not use constant acceleration free-fall equations, but go back
thus narrowly avoiding a ticket for being in an intersection when to basics.
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