Explain.
8. Give an example (but not one from the text) of a device used to measure time and identify what change in that device indicates a change in time.
78 CHAPTER 2 | KINEMATICS
9. There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these
two quantities.
10. Does a car’s odometer measure position or displacement? Does its speedometer measure speed or velocity?
11. If you divide the total distance traveled on a car trip (as determined by the odometer) by the time for the trip, are you calculating the average
speed or the magnitude of the average velocity? Under what circumstances are these two quantities the same?
12. How are instantaneous velocity and instantaneous speed related to one another? How do they differ?
13. Is it possible for speed to be constant while acceleration is not zero? Give an example of such a situation.
14. Is it possible for velocity to be constant while acceleration is not zero? Explain.
15. Give an example in which velocity is zero yet acceleration is not.
16. If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the
acceleration positive or negative?
17. Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of
a negative velocity? Of a positive velocity?
2.6 Problem-Solving Basics for One-Dimensional Kinematics
18. What information do you need in order to choose which equation or equations to use to solve a problem? Explain.
19. What is the last thing you should do when solving a problem? Explain.
20. What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down?
21. An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change
direction? (c) Does the acceleration due to gravity have the same sign on the way up as on the way down?
22. Suppose you throw a rock nearly straight up at a coconut in a palm tree, and the rock misses on the way up but hits the coconut on the way
down. Neglecting air resistance, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it
had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain.
23. If an object is thrown straight up and air resistance is negligible, then its speed when it returns to the starting point is the same as when it was
released. If air resistance were not negligible, how would its speed upon return compare with its initial speed? How would the maximum height to
which it rises be affected?
24. The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration due to gravity being the same, how many
times higher could a safe fall on the Moon be than on Earth (gravitational acceleration on the Moon is about 1/6 that of the Earth)?
25. How many times higher could an astronaut jump on the Moon than on Earth if his takeoff speed is the same in both locations (gravitational
acceleration on the Moon is about 1/6 of g on Earth)?
2.8 Graphical Analysis of One-Dimensional Motion
26. (a) Explain how you can use the graph of position versus time in Figure 2.54 to describe the change in velocity over time. Identify (b) the time ( t a
, t b , t c , t d , or t e ) at which the instantaneous velocity is greatest, (c) the time at which it is zero, and (d) the time at which it is negative.
Figure 2.54
27. (a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in Figure 2.55. (b) Identify the time or times ( t a , t b , t c , etc.) at which the instantaneous velocity is greatest. (c) At which times is it zero? (d) At which times is it negative?
CHAPTER 2 | KINEMATICS 79
Figure 2.55
28. (a) Explain how you can determine the acceleration over time from a velocity versus time graph such as the one in Figure 2.56. (b) Based on the graph, how does acceleration change over time?
Figure 2.56
29. (a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in Figure 2.57. (b) Identify the time or
times ( t a , t b , t c , etc.) at which the acceleration is greatest. (c) At which times is it zero? (d) At which times is it negative?
Figure 2.57
30. Consider the velocity vs. time graph of a person in an elevator shown in Figure 2.58. Suppose the elevator is initially at rest. It then accelerates for 3 seconds, maintains that velocity for 15 seconds, then decelerates for 5 seconds until it stops. The acceleration for the entire trip is not constant
so we cannot use the equations of motion from Motion Equations for Constant Acceleration in One Dimension for the complete trip. (We could,
however, use them in the three individual sections where acceleration is a constant.) Sketch graphs of (a) position vs. time and (b) acceleration vs.
time for this trip.
80 CHAPTER 2 | KINEMATICS
Figure 2.58
31. A cylinder is given a push and then rolls up an inclined plane. If the origin is the starting point, sketch the position, velocity, and acceleration of the
cylinder vs. time as it goes up and then down the plane.
CHAPTER 2 | KINEMATICS 81
Problems & Exercises
12. The speed of propagation of the action potential (an electrical signal)
in a nerve cell depends (inversely) on the diameter of the axon (nerve
fiber). If the nerve cell connecting the spinal cord to your feet is 1.1 m
long, and the nerve impulse speed is 18 m/s, how long does it take for
the nerve signal to travel this distance?
13. Conversations with astronauts on the lunar surface were
characterized by a kind of echo in which the earthbound person’s voice
was so loud in the astronaut’s space helmet that it was picked up by the
astronaut’s microphone and transmitted back to Earth. It is reasonable to
assume that the echo time equals the time necessary for the radio wave
to travel from the Earth to the Moon and back (that is, neglecting any time
delays in the electronic equipment). Calculate the distance from Earth to
the Moon given that the echo time was 2.56 s and that radio waves travel
at the speed of light (3.00×108 m/s) .
14. A football quarterback runs 15.0 m straight down the playing field in
2.50 s. He is then hit and pushed 3.00 m straight backward in 1.75 s. He
breaks the tackle and runs straight forward another 21.0 m in 5.20 s.
Figure 2.59
Calculate his average velocity (a) for each of the three intervals and (b)
1. Find the following for path A in Figure 2.59: (a) The distance traveled.
for the entire motion.
(b) The magnitude of the displacement from start to finish. (c) The
15. The planetary model of the atom pictures electrons orbiting the
displacement from start to finish.
atomic nucleus much as planets orbit the Sun. In this model you can view
2. Find the following for path B in Figure 2.59: (a) The distance traveled.
hydrogen, the simplest atom, as having a single electron in a circular
(b) The magnitude of the displacement from start to finish. (c) The
orbit 1.06×10−10 m in diameter. (a) If the average speed of the
displacement from start to finish.
electron in this orbit is known to be 2.20×106 m/s , calculate the
3. Find the following for path C in Figure 2.59: (a) The distance traveled.
(b) The magnitude of the displacement from start to finish. (c) The
number of revolutions per second it makes about the nucleus. (b) What is
displacement from start to finish.
the electron’s average velocity?
4. Find the following for path D in Figure 2.59: (a) The distance traveled.
(b) The magnitude of the displacement from start to finish. (c) The
displacement from start to finish.
16. A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s.
What is its acceleration?
17. Professional Application
5. (a) Calculate Earth’s average speed relative to the Sun. (b) What is its
Dr. John Paul Stapp was U.S. Air Force officer who studied the effects of
average velocity over a period of one year?
extreme deceleration on the human body. On December 10, 1954, Stapp
6. A helicopter blade spins at exactly 100 revolutions per minute. Its tip is
rode a rocket sled, accelerating from rest to a top speed of 282 m/s (1015
5.00 m from the center of rotation. (a) Calculate the average speed of the
km/h) in 5.00 s, and was brought jarringly back to rest in only 1.40 s!
blade tip in the helicopter’s frame of reference. (b) What is its average
Calculate his (a) acceleration and (b) deceleration. Express each in
velocity over one revolution?
multiples of g (9.80 m/s2) by taking its ratio to the acceleration of
7. The North American and European continents are moving apart at a
gravity.
rate of about 3 cm/y. At this rate how long will it take them to drift 500 km
18. A commuter backs her car out of her garage with an acceleration of
farther apart than they are at present?
1.40 m/s2 . (a) How long does it take her to reach a speed of 2.00 m/s?
8. Land west of the San Andreas fault in southern California is moving at
an average velocity of about 6 cm/y northwest relative to land east of the
(b) If she then brakes to a stop in 0.800 s, what is her deceleration?
fault. Los Angeles is west of the fault and may thus someday be at the
19. Assume that an intercontinental ballistic missile goes from rest to a
same latitude as San Francisco, which is east of the fault. How far in the
suborbital speed of 6.50 km/s in 60.0 s (the actual speed and time are
future will this occur if the displacement to be made is 590 km northwest,
classified). What is its average acceleration in m/s2 and in multiples of
assuming the motion remains constant?
g
9. On May 26, 1934, a streamlined, stainless steel diesel train called the
(9.80 m/s2)?
Zephyr set the world’s nonstop long-distance speed record for trains. Its
run from Denver to Chicago took 13 hours, 4 minutes, 58 seconds, and
2.5 Motion Equations for Constant Acceleration in One
was witnessed by more than a million people along the route. The total
distance traveled was 1633.8 km. What was its average speed in km/h
and m/s?
20. An Olympic-class sprinter starts a race with an acceleration of
10. Tidal friction is slowing the rotation of the Earth. As a result, the orbit
4.50 m/s2 . (a) What is her speed 2.40 s later? (b) Sketch a graph of her
of the Moon is increasing in radius at a rate of approximately 4 cm/year.
position vs. time for this period.
Assuming this to be a constant rate, how many years will pass before the
21. A well-thrown ball is caught in a well-padded mitt. If the deceleration
radius of the Moon’s orbit increases by 3.84×106 m (1%)?
of the ball is 2.10×104 m/s2 , and 1.85 ms (1 ms = 10−3 s) elapses
11. A student drove to the university from her home and noted that the
from the time the ball first touches the mitt until it stops, what was the
odometer reading of her car increased by 12.0 km. The trip took 18.0
initial velocity of the ball?
min. (a) What was her average speed? (b) If the straight-line distance
22. A bullet in a gun is accelerated from the firing chamber to the end of
from her home to the university is 10.3 km in a direction 25.0º south of
the barrel at an average rate of 6.20×105 m/s2 for 8.10×10−4 s .
east, what was her average velocity? (c) If she returned home by the
same path 7 h 30 min after she left, what were her average speed and
What is its muzzle velocity (that is, its final velocity)?
velocity for the entire trip?
23. (a) A light-rail commuter train accelerates at a rate of 1.35 m/s2 .
How long does it take to reach its top speed of 80.0 km/h, starting from
82 CHAPTER 2 | KINEMATICS
rest? (b) The same train ordinarily decelerates at a rate of 1.65 m/s2 .
with few life-threatening injuries. For these lucky pilots, the tree branches
and snow drifts on the ground allowed their deceleration to be relatively
How long does it take to come to a stop from its top speed? (c) In
small. If we assume that a pilot’s speed upon impact was 123 mph (54
emergencies the train can decelerate more rapidly, coming to rest from
m/s), then what was his deceleration? Assume that the trees and snow
80.0 km/h in 8.30 s. What is its emergency deceleration in m/s2 ?
stopped him over a distance of 3.0 m.
24. While entering a freeway, a car accelerates from rest at a rate of
35. Consider a grey squirrel falling out of a tree to the ground. (a) If we
2.40 m/s2
ignore air resistance in this case (only for the sake of this problem),
for 12.0 s. (a) Draw a sketch of the situation. (b) List the
determine a squirrel’s velocity just before hitting the ground, assuming it
knowns in this problem. (c) How far does the car travel in those 12.0 s?
fell from a height of 3.0 m. (b) If the squirrel stops in a distance of 2.0 cm
To solve this part, first identify the unknown, and then discuss how you
through bending its limbs, compare its deceleration with that of the
chose the appropriate equation to solve for it. After choosing the
airman in the previous problem.
equation, show your steps in solving for the unknown, check your units,
and discuss whether the answer is reasonable. (d) What is the car’s final
36. An express train passes through a station. It enters with an initial
velocity? Solve for this unknown in the same manner as in part (c),
velocity of 22.0 m/s and decelerates at a rate of 0.150 m/s2 as it goes
showing all steps explicitly.
through. The station is 210 m long. (a) How long is the nose of the train in
25. At the end of a race, a runner decelerates from a velocity of 9.00 m/s
the station? (b) How fast is it going when the nose leaves the station? (c)
If the train is 130 m long, when does the end of the train leave the
at a rate of 2.00 m/s2 . (a) How far does she travel in the next 5.00 s?
station? (d) What is the velocity of the end of the train as it leaves?
(b) What is her final velocity? (c) Evaluate the result. Does it make
sense?
37. Dragsters can actually reach a top speed of 145 m/s in only 4.45
s—considerably less time than given in Example 2.10 and Example
26. Professional Application:
2.11. (a) Calculate the average acceleration for such a dragster. (b) Find
Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm by
the final velocity of this dragster starting from rest and accelerating at the
the left ventricle of the heart. (a) Make a sketch of the situation. (b) List
rate found in (a) for 402 m (a quarter mile) without using any information
the knowns in this problem. (c) How long does the acceleration take? To
on time. (c) Why is the final velocity greater than that used to find the
solve this part, first identify the unknown, and then discuss how you
average acceleration? Hint: Consider whether the assumption of constant
chose the appropriate equation to solve for it. After choosing the
acceleration is valid for a dragster. If not, discuss whether the
equation, show your steps in solving for the unknown, checking your
acceleration would be greater at the beginning or end of the run and what
units. (d) Is the answer reasonable when compared with the time for a
effect that would have on the final velocity.
heartbeat?
38. A bicycle racer sprints at the end of a race to clinch a victory. The
27. In a slap shot, a hockey player accelerates the puck from a velocity of racer has an initial velocity of 11.5 m/s and accelerates at the rate of
8.00 m/s to 40.0 m/s in the same direction. If this shot takes
0.500 m/s2 for 7.00 s. (a) What is his final velocity? (b) The racer
3.33×10−2 s , calculate the distance over which the puck accelerates. continues at this velocity to the finish line. If he was 300 m from the finish
line when he started to accelerate, how much time did he save? (c) One
28. A powerful motorcycle can accelerate from rest to 26.8 m/s (100 km/
other racer was 5.00 m ahead when the winner started to accelerate, but
h) in only 3.90 s. (a) What is its average acceleration? (b) How far does it
he was unable to accelerate, and traveled at 11.8 m/s until the finish line.
travel in that time?
How far ahead of him (in meters and in seconds) did the winner finish?
29. Freight trains can produce only relatively small accelerations and
39. In 1967, New Zealander Burt Munro set the world record for an Indian
decelerations. (a) What is the final velocity of a freight train that
motorcycle, on the Bonneville Salt Flats in Utah, of 183.58 mi/h. The one-
accelerates at a rate of 0.0500 m/s2 for 8.00 min, starting with an initial way course was 5.00 mi long. Acceleration rates are often described by
velocity of 4.00 m/s? (b) If the train can slow down at a rate of
the time it takes to reach 60.0 mi/h from rest. If this time was 4.00 s, and
0.550 m/s2
Burt accelerated at this rate until he reached his maximum speed, how
, how long will it take to come to a stop from this velocity?
long did it take Burt to complete the course?
(c) How far will it travel in each case?
40. (a) A world record was set for the men’s 100-m dash in the 2008
30. A fireworks shell is accelerated from rest to a velocity of 65.0 m/s
Olympic Games in Beijing by Usain Bolt of Jamaica. Bolt “coasted”
over a distance of 0.250 m. (a) How long did the acceleration last? (b)
across the finish line with a time of 9.69 s. If we assume that Bolt
Calculate the acceleration.
accelerated for 3.00 s to reach his maximum speed, and maintained that
31. A swan on a lake gets airborne by flapping its wings and running on
speed for the rest of the race, calculate his maximum speed and his
top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take
acceleration. (b) During the same Olympics, Bolt also set the world
record in the 200-m dash with a time of 19.30 s. Using the same
off and it accelerates from rest at an average rate of 0.350 m/s2 , how
assumptions as for the 100-m dash, what was his maximum speed for
far will it travel before becoming airborne? (b) How long does this take?
this race?
32. Professional Application:
A woodpecker’s brain is specially protected from large decelerations by
tendon-like attachments inside the skull. While pecking on a tree, the
41. Assume air resistance is negligible unless otherwise stated.
woodpecker’s head comes to a stop from an initial velocity of 0.600 m/s
42. Calculate the displacement and velocity at times of (a) 0.500, (b)
in a distance of only 2.00 mm. (a) Find the acceleration in m/s2 and in
1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial
velocity of 15.0 m/s. Take the point of release to be y
multiples of g ⎛
0 = 0 .
⎝ g = 9.80 m/s2⎞⎠ . (b) Calculate the stopping time. (c) The
tendons cradling the brain stretch, making its stopping distance 4.50 mm
43. Calculate the displacement and velocity at times of (a) 0.500, (b)
(greater than the head and, hence, less deceleration of the brain). What
1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down
is the brain’s deceleration, expressed in multiples of g ?
with an initial velocity of 14.0 m/s from the Verrazano Narrows Bridge in
New York City. The roadway of this bridge is 70.0 m above the water.
33. An unwary football player collides with a padded goalpost while
44. A basketball referee tosses the ball straight up for the starting tip-off.
running at a velocity of 7.50 m/s and comes to a full stop after
At what velocity must a basketball player leave the ground to rise 1.25 m
compressing the padding and his body 0.350 m. (a) What is his
above the floor in an attempt to get the ball?
deceleration? (b) How long does the collision last?
45. A rescue helicopter is hovering over a person whose boat has sunk.
34. In World War II, there were several reported cases of airmen who
One of the rescuers throws a life preserver straight down to the victim
jumped from their flaming airplanes with no parachute to escape certain
with an initial velocity of 1.40 m/s and observes that it takes 1.8 s to
death. Some fell about 20,000 feet (6000 m), and some of them survived,
CHAPTER 2 | KINEMATICS 83
reach the water. (a) List the knowns in this problem. (b) How high above
height reached, (b) its position and velocity 4.00 s after being released,
the water was the preserver released? Note that the downdraft of the
and (c) the time before it hits the ground.
helicopter reduces the effects of air