Physics For Scientists And Engineers 6E - part 14

Problems

36.

A glider on an air track carries a flag of length ! through a
stationary photogate, which measures the time interval $t

during which the flag blocks a beam of infrared light pass-
ing across the photogate. The ratio v

/$t

is the aver-

age velocity of the glider over this part of its motion. Sup-
pose the glider moves with constant acceleration.
(a) Argue for or against the idea that v

is equal to the in-

stantaneous velocity of the glider when it is halfway
through the photogate in space. (b) Argue for or against
the idea that v

is equal to the instantaneous velocity of the

glider when it is halfway through the photogate in time.

37.

A ball starts from rest and accelerates at 0.500 m/s

while

moving down an inclined plane 9.00 m long. When it
reaches the bottom, the ball rolls up another plane, where,
after moving 15.0 m, it comes to rest. (a) What is the speed
of the ball at the bottom of the first plane? (b) How long
does it take to roll down the first plane? (c) What is the ac-
celeration along the second plane? (d) What is the ball’s
speed 8.00 m along the second plane?

38.

Speedy Sue, driving at 30.0 m/s, enters a one-lane tunnel.
She then observes a slow-moving van 155 m ahead travel-
ing at 5.00 m/s. Sue applies her brakes but can accelerate
only at # 2.00 m/s

because the road is wet. Will there be

a collision? If yes, determine how far into the tunnel and
at what time the collision occurs. If no, determine the dis-
tance of closest approach between Sue’s car and the van.

39. Solve Example 2.8, “Watch out for the Speed Limit!” by a

graphical method. On the same graph plot position versus
time for the car and the police officer. From the intersec-
tion of the two curves read the time at which the trooper
overtakes the car.

Section 2.6 Freely Falling Objects

Note: In all problems in this section, ignore the effects of
air resistance.

40. A golf ball is released from rest from the top of a very tall

building. Neglecting air resistance, calculate (a) the position
and (b) the velocity of the ball after 1.00, 2.00, and 3.00 s.

41.

Every morning at seven o’clock
There’s twenty terriers drilling on the rock.
The boss comes around and he says, “Keep still
And bear down heavy on the cast-iron drill
And drill, ye terriers, drill.” And drill, ye terriers, drill.
It’s work all day for sugar in your tea
Down beyond the railway. And drill, ye terriers, drill.
The foreman’s name was John McAnn.
By God, he was a blamed mean man.
One day a premature blast went off
And a mile in the air went big Jim Goff. And drill ...
Then when next payday came around
Jim Goff a dollar short was found.
When he asked what for, came this reply:
“You were docked for the time you were up in the sky.” And drill...

—American folksong

What was Goff’s hourly wage? State the assumptions you
make in computing it.

42.

A ball is thrown directly downward, with an initial speed of
8.00 m/s, from a height of 30.0 m. After what time interval
does the ball strike the ground?

A student throws a set of keys vertically upward to her

sorority  sister,  who  is  in  a  window  4.00 m  above.  The  keys
are  caught  1.50 s  later  by  the  sister’s  outstretched  hand.
(a) With  what  initial  velocity  were  the  keys  thrown?
(b) What was the velocity of the keys just before they were
caught?

44. Emily challenges her friend David to catch a dollar bill as

follows. She holds the bill vertically, as in Figure P2.44,
with the center of the bill between David’s index finger
and thumb. David must catch the bill after Emily releases it
without moving his hand downward. If his reaction time is
0.2 s, will he succeed? Explain your reasoning.

43.

45.

In  Mostar,  Bosnia,  the  ultimate  test  of  a  young  man’s
courage  once  was  to  jump  off  a  400-year-old  bridge  (now
destroyed)  into  the  River  Neretva,  23.0 m  below  the
bridge. (a) How long did the jump last? (b) How fast was
the diver traveling upon impact with the water? (c) If the
speed of sound in air is 340 m/s, how long after the diver
took off did a spectator on the bridge hear the splash?

46.

A ball is dropped from rest from a height h above the
ground. Another ball is thrown vertically upwards from the
ground at the instant the first ball is released. Determine
the speed of the second ball if the two balls are to meet at
a height h/2 above the ground.
A baseball is hit so that it travels straight upward after be-
ing struck by the bat. A fan observes that it takes 3.00 s for
the ball to reach its maximum height. Find (a) its initial
velocity and (b) the height it reaches.

48. It is possible to shoot an arrow at a speed as high as

100 m/s. (a) If friction is neglected, how high would an ar-
row launched at this speed rise if shot straight up?
(b) How long would the arrow be in the air?

47.

Figure P2.44

George Semple

A daring ranch hand sitting on a tree limb wishes to

drop vertically onto a horse galloping under the tree.
The constant speed of the horse is 10.0 m/s, and the dis-
tance from the limb to the level of the saddle is 3.00 m.
(a) What must be the horizontal distance between the
saddle and limb when the ranch hand makes his move?
(b) How long is he in the air?

50.

A  woman  is  reported  to  have  fallen  144 ft  from  the  17th
floor  of  a  building,  landing  on  a  metal  ventilator  box,
which she crushed to a depth of 18.0 in. She suffered only
minor injuries. Neglecting air resistance, calculate (a) the
speed of the woman just before she collided with the venti-
lator,  (b)  her  average  acceleration  while  in  contact  with
the box, and (c) the time it took to crush the box.

51.

The height of a helicopter above the ground is given by
h ! 3.00t

, where h is in meters and t is in seconds. After

2.00 s, the helicopter releases a small mailbag. How long
after its release does the mailbag reach the ground?

52.

A freely falling object requires 1.50 s to travel the last
30.0 m before it hits the ground. From what height above
the ground did it fall?

Section 2.7 Kinematic Equations Derived from

Calculus

Automotive engineers refer to the time rate of change of
acceleration as the “jerk.” If an object moves in one dimen-
sion such that its jerk J is constant, (a) determine expres-
sions for its acceleration a

(t), velocity v

(t), and position

x(t), given that its initial acceleration, velocity, and posi-
tion are a

, v

, and x

, respectively. (b) Show that

54.

A student drives a moped along a straight road as de-
scribed by the velocity-versus-time graph in Figure P2.54.
Sketch this graph in the middle of a sheet of graph paper.
(a) Directly above your graph, sketch a graph of the posi-
tion versus time, aligning the time coordinates of the two
graphs. (b) Sketch a graph of the acceleration versus time
directly below the v

-t graph, again aligning the time coor-

dinates. On each graph, show the numerical values of x
and a

for all points of inflection. (c) What is the accelera-

tion at t ! 6 s? (d) Find the position (relative to the start-
ing point) at t ! 6 s. (e) What is the moped’s final position
at t ! 9 s?

2J(v

)

53.

49.

55.

The speed of a bullet as it travels down the barrel of a
rifle toward the opening is given by v ! (# 5.00 " 10

(3.00 " 10

)t, where v is in meters per second and t is in

seconds. The acceleration of the bullet just as it leaves the
barrel is zero. (a) Determine the acceleration and position
of the bullet as a function of time when the bullet is in the
barrel. (b) Determine the length of time the bullet is ac-
celerated. (c) Find the speed at which the bullet leaves the
barrel. (d) What is the length of the barrel?

56.

The acceleration of a marble in a certain fluid is propor-
tional to the speed of the marble squared, and is given (in
SI units) by a ! # 3.00v

for v & 0. If the marble enters

this fluid with a speed of 1.50 m/s, how long will it take be-
fore the marble’s speed is reduced to half of its initial
value?

Additional Problems

57.

A car has an initial velocity v

when the driver sees an ob-

stacle in the road in front of him. His reaction time is $t

and the braking acceleration of the car is a. Show that the
total stopping distance is

Remember that a is a negative number.

58.

The yellow caution light on a traffic signal should stay on
long  enough  to  allow  a  driver  to  either  pass  through  the
intersection  or  safely  stop  before  reaching  the  intersec-
tion. A car can stop if its distance from the intersection is
greater  than  the  stopping  distance  found  in  the  previous
problem. If the car is less than this stopping distance from
the  intersection,  the  yellow  light  should  stay  on  long
enough to allow the car to pass entirely through the inter-
section. (a) Show that the yellow light should stay on for a
time interval

where $t

is the driver’s reaction time, v

is the velocity of

the car approaching the light at the speed limit, a is the
braking acceleration, and s

is the width of the intersec-

tion.  (b)  As  city  traffic  planner,  you  expect  cars  to
approach  an  intersection  16.0 m  wide  with  a  speed  of
60.0 km/h.  Be  cautious  and  assume  a  reaction  time
of 1.10 s to allow for a driver’s indecision. Find the length
of time the yellow light should remain on. Use a braking
acceleration of # 2.00 m/s

59.

The  Acela  is  the  Porsche  of  American  trains.  Shown  in
Figure P2.59a, the electric train whose name is pronounced
ah-SELL-ah  is  in  service  on  the  Washington-New  York-
Boston  run.  With  two  power  cars  and  six  coaches,  it  can
carry 304 passengers at 170 mi/h. The carriages tilt as much
as  6- from  the  vertical  to  prevent  passengers  from  feeling
pushed  to  the  side  as  they  go  around  curves.  Its  braking
mechanism uses electric generators to recover its energy of
motion. A velocity-time graph for the Acela is shown in Fig-
ure P2.59b. (a) Describe the motion of the train in each suc-
cessive time interval. (b) Find the peak positive acceleration
of the train in the motion graphed. (c) Find the train’s dis-
placement in miles between t ! 0 and t ! 200 s.

light

! $

/2a) % (s

)

stop

/2a.

C H A P T E R 2 • Motion in One Dimension

(m/s)

–2

–6

–4

1 2 3 4 5 6

t(s)

8 9 10

–8

Figure P2.54

Problems

60.

Liz rushes down onto a subway platform to find her train al-
ready departing. She stops and watches the cars go by. Each
car is 8.60 m long. The first moves past her in 1.50 s and the
second in 1.10 s. Find the constant acceleration of the train.

61.

A dog’s hair has been cut and is now getting 1.04 mm
longer each day. With winter coming on, this rate of hair
growth is steadily increasing, by 0.132 mm/day every week.
By how much will the dog’s hair grow during 5 weeks?

62.

A test rocket is fired vertically upward from a well. A cata-
pult gives it an initial speed of 80.0 m/s at ground level. Its
engines then fire and it accelerates upward at 4.00 m/s

until it reaches an altitude of 1 000 m. At that point its en-
gines fail and the rocket goes into free fall, with an acceler-
ation of # 9.80 m/s

. (a) How long is the rocket in motion

above the ground? (b) What is its maximum altitude?
(c) What is its velocity just before it collides with the
Earth? (You will need to consider the motion while the en-
gine is operating separate from the free-fall motion.)

63.

A motorist drives along a straight road at a constant speed
of 15.0 m/s. Just as she passes a parked motorcycle police
officer, the officer starts to accelerate at 2.00 m/s

to over-

take her. Assuming the officer maintains this acceleration,
(a) determine the time it takes the police officer to reach
the motorist. Find (b) the speed and (c) the total displace-
ment of the officer as he overtakes the motorist.

64. In Figure 2.10b, the area under the velocity versus time

curve  and  between  the  vertical  axis  and  time  t (vertical
dashed  line)  represents  the  displacement.  As  shown,  this
area consists of a rectangle and a triangle. Compute their
areas  and  compare  the  sum  of  the  two  areas  with  the  ex-
pression on the right-hand side of Equation 2.12.

65.

Setting a new world record in a 100-m race, Maggie and

Judy cross the finish line in a dead heat, both taking 10.2 s.
Accelerating uniformly, Maggie took 2.00 s and Judy 3.00 s
to attain maximum speed, which they maintained for the
rest of the race. (a) What was the acceleration of each
sprinter? (b) What were their respective maximum speeds?
(c) Which sprinter was ahead at the 6.00-s mark, and by
how much?

66.

A commuter train travels between two downtown stations.
Because the stations are only 1.00 km apart, the train
never reaches its maximum possible cruising speed. Dur-
ing rush hour the engineer minimizes the time interval $t
between two stations by accelerating for a time interval $t

at a rate a

0.100 m/s

and then immediately braking

with acceleration a

! #

0.500 m/s

for a time interval

. Find the minimum time interval of travel $t and the

time interval $t

67.

A  hard  rubber  ball,  released  at  chest  height,  falls  to  the
pavement  and  bounces  back  to  nearly  the  same  height.
When it is in contact with the pavement, the lower side of
the  ball  is  temporarily  flattened.  Suppose  that  the  maxi-
mum depth of the dent is on the order of 1 cm. Compute
an order-of-magnitude estimate for the maximum acceler-
ation of the ball while it is in contact with the pavement.
State  your  assumptions,  the  quantities  you  estimate,  and
the values you estimate for them.

68. At NASA’s John H. Glenn research center in Cleveland,

Ohio,  free-fall  research  is  performed  by  dropping  experi-
ment  packages  from  the  top  of  an  evacuated  shaft  145 m
high.  Free  fall  imitates  the  so-called  microgravity  environ-
ment of a satellite in orbit. (a) What is the maximum time
interval  for  free  fall  if  an  experiment  package  were  to  fall
the entire 145 m? (b) Actual NASA specifications allow for a
5.18-s drop time interval. How far do the packages drop and
(c) what is their speed at 5.18 s? (d) What constant accelera-
tion  would  be  required  to  stop  an  experiment  package  in
the distance remaining in the shaft after its 5.18-s fall?
An  inquisitive  physics  student  and  mountain  climber
climbs a 50.0-m cliff that overhangs a calm pool of water.
He  throws  two  stones  vertically  downward,  1.00 s  apart,
and observes that they cause a single splash. The first stone
has an initial speed of 2.00 m/s. (a) How long after release
of the first stone do the two stones hit the water? (b) What
initial velocity must the second stone have if they are to hit
simultaneously? (c) What is the speed of each stone at the
instant the two hit the water?

70.

A rock is dropped from rest into a well. The well is not re-
ally 16 seconds deep, as in Figure P2.70. (a) The sound of
the splash is actually heard 2.40 s after the rock is released
from rest. How far below the top of the well is the surface
of the water? The speed of sound in air (at the ambient
temperature) is 336 m/s. (b)

What If? If the travel time

for the sound is neglected, what percentage error is intro-
duced when the depth of the well is calculated?

71.

To protect his food from hungry bears, a boy scout raises
his food pack with a rope that is thrown over a tree limb at
height h above his hands. He walks away from the vertical
rope with constant velocity v

boy

, holding the free end of

the rope in his hands (Fig. P2.71). (a) Show that the speed
v of the food pack is given by x(x

)

1/2

boy

where x

69.

(a)

–50

100

150

200

–100

50 100 150 200 250 300 350 400

–50

(b)

v(mi/h)

t(s)

Figure P2.59 (a) The Acela—1 171 000 lb of cold steel

thundering along at 150 mi/h. (b) Velocity-versus-time graph

for the Acela.

Courtesy Amtrak Nec Media Relations

Time (s)

Height (m)

Time (s)

Height (m)

0.00

5.00

2.75

7.62

0.25

5.75

3.00

7.25

0.50

6.40

3.25

6.77

0.75

6.94

3.50

6.20

1.00

7.38

3.75

5.52

1.25

7.72

4.00

4.73

1.50

7.96

4.25

3.85

1.75

8.10

4.50

2.86

2.00

8.13

4.75

1.77

2.25

8.07

5.00

0.58

2.50

7.90

Height of a Rock versus Time

Table P2.74

is the distance he has walked away from the vertical rope.
(b) Show that the acceleration a of the food pack is
h

)

3/2

boy

. (c) What values do the acceleration a

and velocity v have shortly after he leaves the point under
the pack (x ! 0)? (d) What values do the pack’s velocity
and acceleration approach as the distance x continues to
increase?

72.

In Problem 71, let the height h equal 6.00 m and the speed
v

boy

equal 2.00 m/s. Assume that the food pack starts from

rest. (a) Tabulate and graph the speed–time graph.
(b) Tabulate and graph the acceleration-time graph. Let
the range of time be from 0 s to 5.00 s and the time inter-
vals be 0.500 s.
Kathy Kool buys a sports car that can accelerate at the rate
of 4.90 m/s

. She decides to test the car by racing with an-

other speedster, Stan Speedy. Both start from rest, but ex-
perienced Stan leaves the starting line 1.00 s before Kathy.
If Stan moves with a constant acceleration of 3.50 m/s

and Kathy maintains an acceleration of 4.90 m/s

, find

(a) the time at which Kathy overtakes Stan, (b) the dis-
tance she travels before she catches him, and (c) the
speeds of both cars at the instant she overtakes him.

73.

74.

Astronauts on a distant planet toss a rock into the air.

With  the  aid  of  a  camera  that  takes  pictures  at  a  steady
rate,  they  record  the  height  of  the  rock  as  a  function  of
time as given in Table P2.74. (a) Find the average velocity
of  the  rock  in  the  time  interval  between  each  measure-
ment  and  the  next.  (b)  Using  these  average  velocities  to
approximate  instantaneous  velocities  at  the  midpoints  of
the time intervals, make a graph of velocity as a function of
time. Does the rock move with constant acceleration? If so,
plot a straight line of best fit on the graph and calculate its
slope to find the acceleration.

Two objects, A and B, are connected by a rigid rod that has
a length L. The objects slide along perpendicular guide
rails, as shown in Figure P2.75. If A slides to the left with a
constant speed v, find the velocity of B when . ! 60.0°.

75.

C H A P T E R 2 • Motion in One Dimension

v a

boy

Figure P2.70

Figure P2.71 Problems 71 and 72.

Figure P2.75

By permission of John Hart and Creators Syndicate, Inc.

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