Chapter 2
Kinematics in One Dimension
Kinematics deals with the concepts that
are needed to describe motion.
Dynamics deals with the effect that forces
have on motion.
Together, kinematics and dynamics form
the branch of physics known as Mechanics.
2.1 Displacement


x  final position
xo  initialposition
  
x  x  xo  displacement
2.1 Displacement

xo  2.0 m

x  5.0 m

x  7.0 m
  
x  x  xo  7.0 m  2.0 m  5.0 m
2.1 Displacement

x  2.0 m

x  5.0 m

xo  7.0 m
  
x  x  xo  2.0 m  7.0 m  5.0 m
2.1 Displacement

xo  2.0 m

x  5.0 m

x  7.0 m
  
x  x  xo  5.0 m   2.0 m  7.0 m
2.2.1. The branch of physics that deals with motion is called
mechanics. Kinematics is the portion of mechanics that
describes motion without any reference to which of the
following concepts?
a) forces
b) accelerations
c) velocities
d) displacements
e) time
2.2.1. The branch of physics that deals with motion is called
mechanics. Kinematics is the portion of mechanics that
describes motion without any reference to which of the
following concepts?
a) forces
b) accelerations
c) velocities
d) displacements
e) time
2.1.2. A particle travels along a curved path between two points
A and B as shown. Complete the following statement: The
displacement of the particle does not depend on
a) the location of A.
b) the location of B.
c) the direction of A from B.
d) the distance traveled from A to B.
e) the shortest distance between A and B.
2.1.2. A particle travels along a curved path between two points
A and B as shown. Complete the following statement: The
displacement of the particle does not depend on
a) the location of A.
b) the location of B.
c) the direction of A from B.
d) the distance traveled from A to B.
e) the shortest distance between A and B.
2.1.3. For which one of the following situations will the path
length equal the magnitude of the displacement?
a) An Olympic athlete is running around an oval track.
b) A roller coaster car travels up and down two hills.
c) A truck travels 4 miles west; and then, it stops and travels 2
miles west.
d) A ball rises and falls after being thrown straight up from the
earth's surface.
e) A ball on the end of a string is moving in a vertical circle.
2.1.3. For which one of the following situations will the path
length equal the magnitude of the displacement?
a) An Olympic athlete is running around an oval track.
b) A roller coaster car travels up and down two hills.
c) A truck travels 4 miles west; and then, it stops and travels 2
miles west.
d) A ball rises and falls after being thrown straight up from the
earth's surface.
e) A ball on the end of a string is moving in a vertical circle.
2.1.4. Complete the following statement: A displacement vector
a) is directed from an object’s final position toward its initial
position.
b) is always directed along a tangent to the object’s path.
c) has a magnitude that always equals the distance the object
traveled from its initial position to its final position.
d) has SI units of meter per second.
e) is directed from an object’s initial position toward its final
position.
2.1.4. Complete the following statement: A displacement vector
a) is directed from an object’s final position toward its initial
position.
b) is always directed along a tangent to the object’s path.
c) has a magnitude that always equals the distance the object
traveled from its initial position to its final position.
d) has SI units of meter per second.
e) is directed from an object’s initial position toward its final
position.
2.1.1. In the morning, a bird is in Tampa, Florida. In the afternoon, the bird is
near Orlando, Florida. Given this information, which one of the following
statements best describes the relationship between the magnitude of the
bird’s displacement and the distance the bird traveled?
a) The distance traveled is either greater than or equal to the magnitude of
bird’s displacement.
b) The distance traveled is either less than or equal to the magnitude of
bird’s displacement.
c) The distance traveled is equal to the magnitude of bird’s displacement.
d) The distance traveled is either less than or greater than the magnitude of
bird’s displacement.
e) The distance traveled is greater than the magnitude of bird’s
displacement.
2.1.1. In the morning, a bird is in Tampa, Florida. In the afternoon, the bird is
near Orlando, Florida. Given this information, which one of the following
statements best describes the relationship between the magnitude of the
bird’s displacement and the distance the bird traveled?
a) The distance traveled is either greater than or equal to the magnitude of
bird’s displacement.
b) The distance traveled is either less than or equal to the magnitude of
bird’s displacement.
c) The distance traveled is equal to the magnitude of bird’s displacement.
d) The distance traveled is either less than or greater than the magnitude of
bird’s displacement.
e) The distance traveled is greater than the magnitude of bird’s
displacement.
2.1.2. A race car, traveling at constant speed, makes one lap
around a circular track of radius r in a time t. The
circumference of a circle is given by C = 2r. Which one of
the following statements concerning this car is true?
a) The displacement of the car does not change with time.
b) The instantaneous velocity of the car is constant.
c) The average speed of the car is the same over any time
interval.
d) The average velocity of the car is the same over any time
interval.
e) The average speed of the car over any time interval is equal
2.1.2. A race car, traveling at constant speed, makes one lap
around a circular track of radius r in a time t. The
circumference of a circle is given by C = 2r. Which one of
the following statements concerning this car is true?
a) The displacement of the car does not change with time.
b) The instantaneous velocity of the car is constant.
c) The average speed of the car is the same over any time
interval.
d) The average velocity of the car is the same over any time
interval.
e) The average speed of the car over any time interval is equal
to the magnitude of the average velocity over the same time
interval.
2.2.1. A turtle and a rabbit are to have a race. The turtle’s average
speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The
distance from the starting line to the finish line is 1500 m. The
rabbit decides to let the turtle run before he starts running to give
the turtle a head start. What, approximately, is the maximum time
the rabbit can wait before starting to run and still win the race?
a) 15 minutes
b) 18 minutes
c) 20 minutes
d) 22 minutes
e) 25 minutes
2.2.1. A turtle and a rabbit are to have a race. The turtle’s average
speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The
distance from the starting line to the finish line is 1500 m. The
rabbit decides to let the turtle run before he starts running to give
the turtle a head start. What, approximately, is the maximum time
the rabbit can wait before starting to run and still win the race?
a) 15 minutes
b) 18 minutes
c) 20 minutes
d) 22 minutes
e) 25 minutes
2.2 Speed and Velocity
Average speed is the distance traveled divided by the time
required to cover the distance.
Distance
Average speed 
Elapsed time
SI units for speed: meters per second (m/s)
2.2 Speed and Velocity
Example 1 Distance Run by a Jogger
How far does a jogger run in 1.5 hours (5400 s) if his
average speed is 2.22 m/s?
Distance
Average speed 
Elapsed time
Distance  Average speedElapsed time
 2.22 m s 5400s   12000m
2.2 Speed and Velocity
Average velocity is the displacement divided by the elapsed
time.
Displacement
Average velocity
Elapsed time
 

 x  x o x
v

t  to
t
2.2 Speed and Velocity
Example 2 The World’s Fastest Jet-Engine Car
Andy Green in the car ThrustSSC set a world record of
341.1 m/s in 1997. To establish such a record, the driver
makes two runs through the course, one in each direction,
to nullify wind effects. From the data, determine the average
velocity for each run.
2.2 Speed and Velocity

 x  1609 m
v

 339 .5 m s
t
4.740 s

 x  1609 m
v

 342 .7 m s
t
4.695 s
2.2 Speed and Velocity
The instantaneous velocity indicates how fast
the car moves and the direction of motion at each
instant of time.


x
v  lim
t 0 t
2.2.1. A motorcycle travels due south covering a total distance of
80.0 kilometers in 60.0 minutes. Which one of the following
statements concerning this situation is necessarily true?
a) The velocity of the motorcycle is constant.
b) The acceleration of the motorcycle must be non-zero.
c) The motorcycle traveled 40.0 kilometers during the first 30.0
minutes.
d) The speed of the motorcycle must be 80.0 kilometers per hour
throughout the entire trip.
e) The average velocity of the motorcycle is 80.0 kilometers per
hour, due south.
2.2.1. A motorcycle travels due south covering a total distance of
80.0 kilometers in 60.0 minutes. Which one of the following
statements concerning this situation is necessarily true?
a) The velocity of the motorcycle is constant.
b) The acceleration of the motorcycle must be non-zero.
c) The motorcycle traveled 40.0 kilometers during the first 30.0
minutes.
d) The speed of the motorcycle must be 80.0 kilometers per hour
throughout the entire trip.
e) The average velocity of the motorcycle is 80.0 kilometers per
hour, due south.
2.2.2. Which one of the following quantities is defined as the
distance traveled divided by the elapsed time for the travel?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.2. Which one of the following quantities is defined as the
distance traveled divided by the elapsed time for the travel?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.3. Which one of the following quantities is defined as an
object’s displacement divided by the elapsed time for the
displacement?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.3. Which one of the following quantities is defined as an
object’s displacement divided by the elapsed time for the
displacement?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.4. A train leaves a station, starting from rest, at time t = 0.0
s. At time t = 3600.0 s, the train is traveling due west at 28
m/s. In this example, the 28 m/s is the train’s
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.4. A train leaves a station, starting from rest, at time t = 0.0
s. At time t = 3600.0 s, the train is traveling due west at 28
m/s. In this example, the 28 m/s is the train’s
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.5. The speedometer on a car’s dashboard measures which
of the following quantities?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.5. The speedometer on a car’s dashboard measures which
of the following quantities?
a) average speed
b) average velocity
c) average acceleration
d) instantaneous velocity
e) instantaneous acceleration
2.2.1. A turtle and a rabbit are to have a race. The turtle’s average
speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The
distance from the starting line to the finish line is 1500 m. The
rabbit decides to let the turtle run before he starts running to give
the turtle a head start. What, approximately, is the maximum time
the rabbit can wait before starting to run and still win the race?
a) 15 minutes
b) 18 minutes
c) 20 minutes
d) 22 minutes
e) 25 minutes
2.2.1. A turtle and a rabbit are to have a race. The turtle’s average
speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The
distance from the starting line to the finish line is 1500 m. The
rabbit decides to let the turtle run before he starts running to give
the turtle a head start. What, approximately, is the maximum time
the rabbit can wait before starting to run and still win the race?
a) 15 minutes
b) 18 minutes
c) 20 minutes
d) 22 minutes
e) 25 minutes
2.2.2. A turtle, moving at a constant velocity of 0.9 m/s due south, is in a race
with a rabbit, who runs at a moderate speed of 9 m/s. When the turtle is
45 m from the finish line, the rabbit begins taunting the turtle by running
from the turtle to the finish line (without crossing it) and back to the turtle.
The rabbit continues going back and forth between the turtle and the
finish line until the turtle crosses the finish line. About how many meters
does the rabbit travel as the turtle travels that last 45 m? Assume the
rabbit always runs at 9 m/s and doesn’t lose any time changing direction.
a) 180 m
b) 270 m
c) 360 m
d) 450 m
e) 540 m
2.2.2. A turtle, moving at a constant velocity of 0.9 m/s due south, is in a race
with a rabbit, who runs at a moderate speed of 9 m/s. When the turtle is
45 m from the finish line, the rabbit begins taunting the turtle by running
from the turtle to the finish line (without crossing it) and back to the turtle.
The rabbit continues going back and forth between the turtle and the
finish line until the turtle crosses the finish line. About how many meters
does the rabbit travel as the turtle travels that last 45 m? Assume the
rabbit always runs at 9 m/s and doesn’t lose any time changing direction.
a) 180 m
b) 270 m
c) 360 m
d) 450 m
e) 540 m
2.3 Acceleration
The notion of acceleration emerges when a change in
velocity is combined with the time during which the
change occurs.
2.3 Acceleration
DEFINITION OF AVERAGE ACCELERATION
 

 v  v o v
a

t  to
t
2.3 Acceleration
Example 3 Acceleration and Increasing Velocity
Determine the average acceleration of the plane.

vo  0 m s

v  260km h
to  0 s
t  29 s
 
 v  v o 260km h  0 km h
km h
a

 9.0
t  to
29 s  0 s
s
2.3 Acceleration
2.3 Acceleration
Example 3 Acceleration and Decreasing
Velocity
 
 v  v o 13m s  28m s
a

 5.0 m s 2
t  to
12 s  9 s
2.3 Acceleration
2.3.1. Which one of the following situations does the object have no
acceleration?
a) A ball at the end of a string is whirled in a horizontal circle at a
constant speed.
b) Seeing a red traffic light ahead, the driver of a minivan steps on the
brake. As a result, the minivan slows from 15 m/s to stop before
reaching the light.
c) A boulder starts from rest and rolls down a mountain.
d) An elevator in a tall skyscraper moves upward at a constant speed of
3 m/s.
e) A ball is thrown upward from the surface of the earth, slows to a
temporary stop at a height of 4 m, and begins to fall back toward the
ground.
2.3.1. Which one of the following situations does the object have no
acceleration?
a) A ball at the end of a string is whirled in a horizontal circle at a
constant speed.
b) Seeing a red traffic light ahead, the driver of a minivan steps on the
brake. As a result, the minivan slows from 15 m/s to stop before
reaching the light.
c) A boulder starts from rest and rolls down a mountain.
d) An elevator in a tall skyscraper moves upward at a constant speed of
3 m/s.
e) A ball is thrown upward from the surface of the earth, slows to a
temporary stop at a height of 4 m, and begins to fall back toward the
ground.
2.3.2. In which one of the following situations does the car have
an acceleration that is directed due north?
a) A car travels northward with a constant speed of 24 m/s.
b) A car is traveling southward as its speed increases from 24
m/s to 33 m/s.
c) A car is traveling southward as its speed decreases from 24
m/s to 18 m/s.
d) A car is traveling northward as its speed decreases from 24
m/s to
18 m/s.
e) A car travels southward with a constant speed of 24 m/s.
2.3.2. In which one of the following situations does the car have
an acceleration that is directed due north?
a) A car travels northward with a constant speed of 24 m/s.
b) A car is traveling southward as its speed increases from 24
m/s to 33 m/s.
c) A car is traveling southward as its speed decreases from 24
m/s to 18 m/s.
d) A car is traveling northward as its speed decreases from 24
m/s to
18 m/s.
e) A car travels southward with a constant speed of 24 m/s.
2.3.1. A ball is thrown toward a wall, bounces, and returns to the thrower with
the same speed as it had before it bounced. Which one of the following
statements correctly describes this situation?
a) The ball was not accelerated during its contact with the wall because its
speed remained constant.
b) The instantaneous velocity of the ball from the time it left the thrower’s
hand was constant.
c) The only time that the ball had an acceleration was when the ball started
from rest and left the hand of the thrower and again when the ball
returned to the hand and was stopped.
d) During this situation, the ball was never accelerated.
e) The ball was accelerated during its contact with the wall because its
direction changed.
2.3.1. A ball is thrown toward a wall, bounces, and returns to the thrower with
the same speed as it had before it bounced. Which one of the following
statements correctly describes this situation?
a) The ball was not accelerated during its contact with the wall because its
speed remained constant.
b) The instantaneous velocity of the ball from the time it left the thrower’s
hand was constant.
c) The only time that the ball had an acceleration was when the ball started
from rest and left the hand of the thrower and again when the ball
returned to the hand and was stopped.
d) During this situation, the ball was never accelerated.
e) The ball was accelerated during its contact with the wall because its
direction changed.
2.3.2. In an air race, two planes are traveling due east. Plane One has a
larger acceleration than Plane Two. Both accelerations are in the same
direction. Which one of the following statements is true concerning this
situation?
a) In the same time interval, the change in the velocity of the Plane Two is
greater than that of Plane One.
b) In the same time interval, the change in the velocity of the Plane One is
greater than that of Plane Two.
c) Within the time interval, the velocity of the Plane Two remains greater
than that of Plane One.
d) Within the time interval, the velocity of the Plane One remains greater
than that of Plane Two.
e) Too little information is given to compare the velocities of the planes or
how the velocities are changing.
2.3.2. In an air race, two planes are traveling due east. Plane One has a
larger acceleration than Plane Two. Both accelerations are in the same
direction. Which one of the following statements is true concerning this
situation?
a) In the same time interval, the change in the velocity of the Plane Two is
greater than that of Plane One.
b) In the same time interval, the change in the velocity of the Plane One is
greater than that of Plane Two.
c) Within the time interval, the velocity of the Plane Two remains greater
than that of Plane One.
d) Within the time interval, the velocity of the Plane One remains greater
than that of Plane Two.
e) Too little information is given to compare the velocities of the planes or
how the velocities are changing.
2.3.3. Two cars travel along a level highway. An observer
notices that the distance between the cars is increasing.
Which one of the following statements concerning this
situation is necessarily true?
a) Both cars could be accelerating at the same rate.
b) The leading car has the greater acceleration.
c) The trailing car has the smaller acceleration.
d) The velocity of each car is increasing.
e) At least one of the cars has a non-zero acceleration.
2.3.3. Two cars travel along a level highway. An observer
notices that the distance between the cars is increasing.
Which one of the following statements concerning this
situation is necessarily true?
a) Both cars could be accelerating at the same rate.
b) The leading car has the greater acceleration.
c) The trailing car has the smaller acceleration.
d) The velocity of each car is increasing.
e) At least one of the cars has a non-zero acceleration.
2.3.4. A police cruiser is parked by the side of the road when a
speeding car passes. The cruiser follows the speeding car.
Consider the following diagrams where the dots represent
the cruiser’s position at 0.5-s intervals. Which diagram(s)
are possible representations of the cruiser’s motion?
a) A only
b) B, D, or E only
c) C only
d) E only
e) A or C only
2.3.4. A police cruiser is parked by the side of the road when a
speeding car passes. The cruiser follows the speeding car.
Consider the following diagrams where the dots represent
the cruiser’s position at 0.5-s intervals. Which diagram(s)
are possible representations of the cruiser’s motion?
a) A only
b) B, D, or E only
c) C only
d) E only
e) A or C only
2.4 Equations of Kinematics for Constant Acceleration
 
 x  xo
v
t  to
 
 v  vo
a
t  to
It is customary to dispense with the use of boldface symbols
overdrawn with arrows for the displacement, velocity, and
acceleration vectors. We will, however, continue to convey
the directions with a plus or minus sign.
x  xo
v
t  to
v  vo
a
t  to
2.4 Equations of Kinematics for Constant Acceleration
Let the object be at the origin when the clock starts.
xo  0
x  xo
v
t  to
to  0
x
v
t
x  vt 
1
2
vo  v t
2.4 Equations of Kinematics for Constant Acceleration
v  vo
a
t  to
v  vo
a
t
at  v  vo
v  vo  at
2.4 Equations of Kinematics for Constant Acceleration
Five kinematic variables:
1. displacement, x
2. acceleration (constant), a
3. final velocity (at time t), v
4. initial velocity, vo
5. elapsed time, t
2.4 Equations of Kinematics for Constant Acceleration
v  vo  at
x
1
2
vo  v t  vo  vo  att
1
2
x  vot  at
1
2
2
2.4 Equations of Kinematics for Constant Acceleration
x  vot  at
1
2
2
 6.0 m s 8.0 s  
 110 m
1
2
2.0 m s 8.0 s
2
2
2.4 Equations of Kinematics for Constant Acceleration
Example 6 Catapulting a Jet
Find its displacement.
vo  0 m s
x  ??
a  31m s2
v  62 m s
2.4 Equations of Kinematics for Constant Acceleration
v  vo
a
t
x
1
2
v  vo
t
a
vo  v  t 
1
2

v  vo 
vo  v 
a
v v
x
2a
2
2
o
2.4 Equations of Kinematics for Constant Acceleration

v v
62 m s   0 m s
x



62
m
2
2a
2 31m s
2
2
o
2

2

2.4 Equations of Kinematics for Constant Acceleration
Equations of Kinematics for Constant Acceleration
v  vo  at
x
1
2
vo  v t
v  v  2ax
2
2
o
x  vot  at
1
2
2
2.5 Applications of the Equations of Kinematics
Reasoning Strategy
1. Make a drawing.
2. Decide which directions are to be called positive (+) and
negative (-).
3. Write down the values that are given for any of the five
kinematic variables.
4. Verify that the information contains values for at least three
of the five kinematic variables. Select the appropriate equation.
5. When the motion is divided into segments, remember that
the final velocity of one segment is the initial velocity for the next.
6. Keep in mind that there may be two possible answers to a
kinematics problem.
2.5 Applications of the Equations of Kinematics
Example 8 An Accelerating Spacecraft
A spacecraft is traveling with a velocity of +3250 m/s. Suddenly
the retrorockets are fired, and the spacecraft begins to slow down
with an acceleration whose magnitude is 10.0 m/s2. What is
the velocity of the spacecraft when the displacement of the craft
is +215 km, relative to the point where the retrorockets began
firing?
x
a
v
vo
+215000 m
-10.0 m/s2
?
+3250 m/s
t
2.5 Applications of the Equations of Kinematics
2.5 Applications of the Equations of Kinematics
x
a
v
vo
+215000 m
-10.0 m/s2
?
+3250 m/s
v  v  2ax
2
2
o
t
v  v  2ax
2
o


v   3250m s   2 10.0 m s 2 215000m
2
 2500m s
2.6 Freely Falling Bodies
In the absence of air resistance, it is found that all bodies
at the same location above the Earth fall vertically with
the same acceleration. If the distance of the fall is small
compared to the radius of the Earth, then the acceleration
remains essentially constant throughout the descent.
This idealized motion is called free-fall and the acceleration
of a freely falling body is called the acceleration due to
gravity.
g  9.80m s
2
or
32.2 ft s
2
2.6 Freely Falling Bodies
g  9.80 m s
2
2.6 Freely Falling Bodies
Example 10 A Falling Stone
A stone is dropped from the top of a tall building. After 3.00s
of free fall, what is the displacement y of the stone?
2.6 Freely Falling Bodies
y
a
?
-9.80 m/s2
v
vo
t
0 m/s
3.00 s
2.6 Freely Falling Bodies
y
a
?
-9.80 m/s2
y  vot  at
1
2
vo
t
0 m/s
3.00 s
2
 0 m s 3.00 s  
 44.1 m
v
1
2
 9.80m s 3.00 s
2
2
2.6 Freely Falling Bodies
Example 12 How High Does it Go?
The referee tosses the coin up
with an initial speed of 5.00m/s.
In the absence if air resistance,
how high does the coin go above
its point of release?
2.6 Freely Falling Bodies
y
a
v
vo
?
-9.80 m/s2
0 m/s
+5.00
m/s
t
2.6 Freely Falling Bodies
y
a
v
vo
?
-9.80 m/s2
0 m/s
+5.00
m/s
v v
y
2a
2
v  v  2ay
2
t
2
o
2
o

v v
0 m s   5.00 m s 
y

 1.28 m
2
2a
2  9.80 m s
2
2
o
2

2

2.6.1. Two identical ping-pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the ball is
filled with water. The hole is sealed so that no water can escape. The
two balls are then dropped from rest at the exact same time from the roof
of a building. Assuming there is no wind, which one of the following
statements is true?
a) The two balls reach the ground at the same time.
b) The heavier ball reaches the ground a long time before the lighter ball.
c) The heavier ball reaches the ground just before the lighter ball.
d) The heavier ball has a much larger velocity when it strikes the ground
than the
light ball.
e) The heavier ball has a slightly larger velocity when it strikes the ground
than the light ball.
2.6.1. Two identical ping-pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the ball is
filled with water. The hole is sealed so that no water can escape. The
two balls are then dropped from rest at the exact same time from the roof
of a building. Assuming there is no wind, which one of the following
statements is true?
a) The two balls reach the ground at the same time.
b) The heavier ball reaches the ground a long time before the lighter ball.
c) The heavier ball reaches the ground just before the lighter ball.
d) The heavier ball has a much larger velocity when it strikes the ground
than the
light ball.
e) The heavier ball has a slightly larger velocity when it strikes the ground
than the light ball.
2.6.2. Two identical ping-pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the ball is
filled with water. The hole is sealed so that no water can escape. Each
ball is shot horizontally from a gun with an initial velocity v0 from the top
of a building. The following drawing shows several trajectories
numbered 1 through 5. Which of the following statements is true?
a) Both balls would follow
trajectory 5.
b) Both balls would follow
trajectory 3.
c) The lighter ball would follow 4 and the heavier ball would follow 2.
d) The lighter ball would follow 4 and the heavier ball would follow 3.
e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2 or
1, depending on the magnitude of v0.
2.6.2. Two identical ping-pong balls are selected for a physics
demonstration. A tiny hole is drilled in one of the balls; and the ball is
filled with water. The hole is sealed so that no water can escape. Each
ball is shot horizontally from a gun with an initial velocity v0 from the top
of a building. The following drawing shows several trajectories
numbered 1 through 5. Which of the following statements is true?
a) Both balls would follow
trajectory 5.
b) Both balls would follow
trajectory 3.
c) The lighter ball would follow 4 and the heavier ball would follow 2.
d) The lighter ball would follow 4 and the heavier ball would follow 3.
e) The lighter ball would follow 4 or 3 and the heavier ball would follow 2 or
1, depending on the magnitude of v0.
2.6.3. A cannon directed straight upward launches a ball with an
initial speed v. The ball reaches a maximum height h in a
time t. Then, the same cannon is used to launch a second
ball straight upward at a speed 2v. In terms of h and t, what
is the maximum height the second ball reaches and how
long does it take to reach that height?
a) 2h, t
b) 4h, 2t
c) 2h, 4t
d) 2h, 2t
e) h, t
2.6.3. A cannon directed straight upward launches a ball with an
initial speed v. The ball reaches a maximum height h in a
time t. Then, the same cannon is used to launch a second
ball straight upward at a speed 2v. In terms of h and t, what
is the maximum height the second ball reaches and how
long does it take to reach that height?
a) 2h, t
b) 4h, 2t
c) 2h, 4t
d) 2h, 2t
e) h, t
2.6 Freely Falling Bodies
Conceptual Example 14 Acceleration Versus Velocity
There are three parts to the motion of the coin. On the way
up, the coin has a vector velocity that is directed upward and
has decreasing magnitude. At the top of its path, the coin
momentarily has zero velocity. On the way down, the coin
has downward-pointing velocity with an increasing magnitude.
In the absence of air resistance, does the acceleration of the
coin, like the velocity, change from one part to another?
2.6 Freely Falling Bodies
Conceptual Example 15 Taking Advantage of Symmetry
Does the pellet in part b strike the ground beneath the cliff
with a smaller, greater, or the same speed as the pellet
in part a?
2.6.1. A rock is released from rest from a hot air balloon that is
at rest with respect to the ground a few meters below. If we
ignore air resistance as the rock falls, which one of the
following statements is true?
a) The rock will take longer than one second to reach the
ground.
b) The instantaneous speed of the rock just before it reaches
the ground will be 9.8 m/s.
c) The rock is considered a freely falling body after it is
released.
d) As the rock falls, its acceleration is 9.8 m/s2, directed
upward.
2.6.1. A rock is released from rest from a hot air balloon that is at rest
with respect to the ground a few meters below. If we ignore air
resistance as the rock falls, which one of the following statements
is true?
a) The rock will take longer than one second to reach the ground.
b) The instantaneous speed of the rock just before it reaches the
ground will be 9.8 m/s.
c) The rock is considered a freely falling body after it is released.
d) As the rock falls, its acceleration is 9.8 m/s2, directed upward.
e) After the ball is released it falls at a constant speed of 9.8 m/s.
2.6.2. Ping-pong ball A is filled with sand. Ping-pong ball B is
identical to A, except that it is empty inside. Ball A is
somewhat heavier than ball B because of the sand inside.
Both balls are simultaneously dropped from rest from the top
of a building. Which of these two balls has the greater
acceleration due to gravity, if any, as they fall?
a) ball A
b) ball B
c) Both ball A and ball B have zero acceleration.
d) Both ball A and ball B have the same acceleration.
2.6.2. Ping-pong ball A is filled with sand. Ping-pong ball B is
identical to A, except that it is empty inside. Ball A is
somewhat heavier than ball B because of the sand inside.
Both balls are simultaneously dropped from rest from the top
of a building. Which of these two balls has the greater
acceleration due to gravity, if any, as they fall?
a) ball A
b) ball B
c) Both ball A and ball B have zero acceleration.
d) Both ball A and ball B have the same acceleration.
2.6.3. A ball is thrown vertically upward from the surface of the earth.
The ball rises to some maximum height and falls back toward the
surface of the earth. Which one of the following statements
concerning this situation is true if air resistance is neglected?
a) As the ball rises, its acceleration vector points upward.
b) The ball is a freely falling body for the duration of its flight.
c) The acceleration of the ball is zero when the ball is at its highest
point.
d) The speed of the ball is negative while the ball falls back toward
the earth.
e) The velocity and acceleration of the ball always point in the same
direction.
2.6.3. A ball is thrown vertically upward from the surface of the earth.
The ball rises to some maximum height and falls back toward the
surface of the earth. Which one of the following statements
concerning this situation is true if air resistance is neglected?
a) As the ball rises, its acceleration vector points upward.
b) The ball is a freely falling body for the duration of its flight.
c) The acceleration of the ball is zero when the ball is at its highest
point.
d) The speed of the ball is negative while the ball falls back toward
the earth.
e) The velocity and acceleration of the ball always point in the same
direction.
2.7 Graphical Analysis of Velocity and Acceleration
x  8 m
Slope 

 4 m s
t
2s
2.7 Graphical Analysis of Velocity and Acceleration
2.7 Graphical Analysis of Velocity and Acceleration
2.7 Graphical Analysis of Velocity and Acceleration
v  12 m s
Slope 

 6 m s 2
t
2s
2.7.1. A dog is initially walking due east. He stops, noticing a cat behind
him. He runs due west and stops when the cat disappears into some
bushes. He starts walking due east again. Then, a motorcycle passes
him and he runs due east after it. The dog gets tired and stops running.
Which of the following graphs correctly represent the position versus
time of the dog?
2.7.1. A dog is initially walking due east. He stops, noticing a cat behind
him. He runs due west and stops when the cat disappears into some
bushes. He starts walking due east again. Then, a motorcycle passes
him and he runs due east after it. The dog gets tired and stops running.
Which of the following graphs correctly represent the position versus
time of the dog?
2.7.2. The graph below represents the speed of a car traveling due east for a portion
of its travel along a horizontal road. Which of the following statements
concerning this graph is true?
a) The car initially increases its speed, but then the
speed decreases at a constant rate until the car stops.
b) The speed of the car is initially constant, but then
it has a variable positive acceleration before it stops.
c) The car initially has a positive acceleration, but then it has a variable negative
acceleration before it stops.
d) The car initially has a positive acceleration, but then it has a variable positive
acceleration before it stops.
e) No information about the acceleration of the car can be determined from this
graph.
2.7.2. The graph below represents the speed of a car traveling due east for a portion
of its travel along a horizontal road. Which of the following statements
concerning this graph is true?
a) The car initially increases its speed, but then the
speed decreases at a constant rate until the car stops.
b) The speed of the car is initially constant, but then
it has a variable positive acceleration before it stops.
c) The car initially has a positive acceleration, but then it has a variable negative
acceleration before it stops.
d) The car initially has a positive acceleration, but then it has a variable positive
acceleration before it stops.
e) No information about the acceleration of the car can be determined from this
graph.
2.7.1. A dog is walking along a street. As the dog moves, a
graph is made of its position on the vertical axis with the
elapsed time on the horizontal axis. The slope of the curve
is determined at some point on the graph. The slope of this
curve is a measurement of which of the following
parameters?
a) the dog’s instantaneous velocity
b) the dog’s acceleration
c) the dog’s speed
d) the dog’s average velocity
e) the elapsed time for the dog’s walk
2.7.1. A dog is walking along a street. As the dog moves, a
graph is made of its position on the vertical axis with the
elapsed time on the horizontal axis. The slope of the curve
is determined at some point on the graph. The slope of this
curve is a measurement of which of the following
parameters?
a) the dog’s instantaneous velocity
b) the dog’s acceleration
c) the dog’s speed
d) the dog’s average velocity
e) the elapsed time for the dog’s walk
2.7.2. Starting from rest, a particle that is confined to move
along a straight line is accelerated at a rate of 5.0 m/s2.
Which one of the following statements concerning the slope
of the position versus time graph for this particle is true?
a) The slope has a constant value of 5.0 m/s.
b) The slope has a constant value of 5.0 m/s2.
c) The slope is both constant and negative.
d) The slope is not constant and increases with increasing
time.
e) The slope is not constant and decreases with increasing
time.
2.7.2. Starting from rest, a particle that is confined to move
along a straight line is accelerated at a rate of 5.0 m/s2.
Which one of the following statements concerning the slope
of the position versus time graph for this particle is true?
a) The slope has a constant value of 5.0 m/s.
b) The slope has a constant value of 5.0 m/s2.
c) The slope is both constant and negative.
d) The slope is not constant and increases with increasing
time.
e) The slope is not constant and decreases with increasing
time.
2.7.3. The graph shows the velocity of an object versus the
elapsed time. During which interval on the graph does the
object’s acceleration decrease with time?
a) A
b) B
c) C
d) D
e) E
2.7.3. The graph shows the velocity of an object versus the
elapsed time. During which interval on the graph does the
object’s acceleration decrease with time?
a) A
b) B
c) C
d) D
e) E
2.7.4. Complete the following statement: the instantaneous
acceleration of an object can be determined by determining
the slope of
a) the object’s velocity versus elapsed time graph.
b) the object’s displacement versus elapsed time graph.
c) the object’s distance versus elapsed time graph.
d) the object’s acceleration versus elapsed time graph.
2.7.4. Complete the following statement: the instantaneous
acceleration of an object can be determined by determining
the slope of
a) the object’s velocity versus elapsed time graph.
b) the object’s displacement versus elapsed time graph.
c) the object’s distance versus elapsed time graph.
d) the object’s acceleration versus elapsed time graph.