Chapter 2
One-Dimensional
Kinematics
Description of motion in one dimension
Copyright © 2010 Pearson Education, Inc.
Units of Chapter 2
• Position, Distance, and Displacement
• Average Speed and Velocity
• Instantaneous Velocity
• Acceleration
• Motion with Constant Acceleration
• Applications of the Equations of Motion
• Freely Falling Objects
Copyright © 2010 Pearson Education, Inc.
2-1 Position, Distance, and Displacement
Before describing motion, you must set up a
coordinate system – define an origin and a
positive direction.
Position: described by x
Copyright © 2010 Pearson Education, Inc.
2-1 Position, Distance, and Displacement
The distance is the total length of travel; if you
drive from your house to the grocery store and
back, you have covered a distance of 8.6 mi.
Copyright © 2010 Pearson Education, Inc.
2-1 Position, Distance, and Displacement
Displacement is the change in position. If you
drive from your house to the grocery store and
then to your friend’s house, your displacement
is 2.1 mi and the distance you have traveled is
10.7 mi.
displacement = change in position = final position − initial position
displacement = Δx = Δx f − xi
Copyright © 2010 Pearson Education, Inc.
Question 2.1
Walking the Dog
You and your dog go for a walk to the
park. On the way, your dog takes many
side trips to chase squirrels or examine
a) yes
fire hydrants. When you arrive at the
park, do you and your dog have the same
displacement?
b) no
Question 2.1
Walking the Dog
You and your dog go for a walk to the
park. On the way, your dog takes many
side trips to chase squirrels or examine
a) yes
fire hydrants. When you arrive at the
park, do you and your dog have the same
b) no
displacement?
Yes, you have the same displacement. Because you and your dog had
the same initial position and the same final position, then you have (by
definition) the same displacement.
Follow-up: have you and your dog traveled the same distance?
2-2 Average Speed and Velocity
The average speed is defined as the distance
traveled divided by the time the trip took:
Average speed = distance / elapsed time
Is the average speed of the red car 40.0 mi/h, more than
40.0 mi/h, or less than 40.0 mi/h?
Copyright © 2010 Pearson Education, Inc.
2-2 Average Speed and Velocity
The average speed is defined as the distance
traveled divided by the time the trip took:
Average speed = distance / elapsed time
Is the average speed of the red car 40.0 mi/h, more than
40.0 mi/h, or less than 40.0 mi/h?
t1 =
t2 =
4.00mi
= (4.00 / 30.0) h
30.0mi / h
4.00mi
= (4.00 / 50.0) h
50.0mi / h
Copyright © 2010 Pearson Education, Inc.
t1 + t 2 = 0.213h
Average speed =
8mi
= 37.6mi / h < 40mi / h
0.213h
2-2 Average Speed and Velocity
Average velocity = displacement / elapsed time
If you return to your starting point, your
average velocity is zero.
average velocity =
v av
displacement
elapsed time
Δx x f − x i
=
=
Δt
tf − ti
Copyright © 2010 Pearson Education, Inc.
2-2 Average Speed and Velocity
Graphical Interpretation of Average Velocity
The same motion, plotted one-dimensionally
and as an x-t graph:
Copyright © 2010 Pearson Education, Inc.
2-3 Instantaneous Velocity
Definition:
(2-4)
This means that we evaluate the average
velocity over a shorter and shorter period of
time; as that time becomes infinitesimally
small, we have the instantaneous velocity.
Copyright © 2010 Pearson Education, Inc.
2-3 Instantaneous Velocity
This plot shows the average velocity being
measured over shorter and shorter intervals.
The instantaneous velocity is tangent to the
curve.
Δx
v = lim
Δt → 0
Δt
average velocity =
v av =
Copyright © 2010 Pearson Education, Inc.
Δx x f − x i
=
Δt
tf − ti
displacement
elapsed time
2-3 Instantaneous Velocity
Graphical Interpretation of Average and
Instantaneous Velocity
Copyright © 2010 Pearson Education, Inc.
2-4 Acceleration
Average acceleration:
(2-5)
Copyright © 2010 Pearson Education, Inc.
2-4 Acceleration
Graphical Interpretation of Average and
Instantaneous Acceleration:
Δv
Δt →0 Δt
a = lim
a av
Copyright © 2010 Pearson Education, Inc.
Δv v f − v i
=
=
Δt
tf − ti
2-4 Acceleration
Acceleration (increasing speed) and
deceleration (decreasing speed) should not be
confused with the directions of velocity and
acceleration:
Copyright © 2010 Pearson Education, Inc.
Units of Chapter 2
• Position, Distance, and Displacement
• Average Speed and Velocity
• Instantaneous Velocity
• Acceleration
• Motion with Constant Acceleration
• Applications of the Equations of Motion
• Freely Falling Objects
Copyright © 2010 Pearson Education, Inc.
Review: 2-3 Instantaneous Velocity
This plot shows the average velocity being
measured over shorter and shorter intervals.
The instantaneous velocity is tangent to the
curve.
Δx
v = lim
Δt → 0
Δt
average velocity =
v av =
Copyright © 2010 Pearson Education, Inc.
Δx x f − x i
=
Δt
tf − ti
displacement
elapsed time
Review: 2-4 Acceleration
Graphical Interpretation of Average and
Instantaneous Acceleration:
Δv
Δt →0 Δt
a = lim
a av
Copyright © 2010 Pearson Education, Inc.
Δv v f − v i
=
=
Δt
tf − ti
Question 2.13a Graphing Velocity I
a) it speeds up all the time
The graph of position versus
time for a car is given below.
What can you say about the
velocity of the car over time?
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
x
t
Question 2.13a Graphing Velocity I
a) it speeds up all the time
The graph of position versus
time for a car is given below.
What can you say about the
velocity of the car over time?
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
The car moves at a constant velocity
x
because the x vs. t plot shows a straight
line. The slope of a straight line is
constant. Remember that the slope of x
vs. t is the velocity!
t
Question 2.13b Graphing Velocity II
a) it speeds up all the time
The graph of position vs.
b) it slows down all the time
time for a car is given below.
c) it moves at constant velocity
What can you say about the
d) sometimes it speeds up and
sometimes it slows down
velocity of the car over time?
e) not really sure
x
t
Question 2.13b Graphing Velocity II
a) it speeds up all the time
The graph of position vs.
b) it slows down all the time
time for a car is given below.
c) it moves at constant velocity
What can you say about the
d) sometimes it speeds up and
sometimes it slows down
velocity of the car over time?
e) not really sure
The car slows down all the time
because the slope of the x vs. t graph is
diminishing as time goes on.
Remember that the slope of x vs. t is
the velocity! At large t, the value of the
position x does not change, indicating
that the car must be at rest.
x
t
Question 2.14a v versus t graphs I
a) decreases
Consider the line labeled A in
b) increases
the v vs. t plot. How does the
c) stays constant
speed change with time for
d) increases, then decreases
line A?
e) decreases, then increases
A
v
t
B
Question 2.14a v versus t graphs I
a) decreases
Consider the line labeled A in
b) increases
the v vs. t plot. How does the
c) stays constant
speed change with time for
d) increases, then decreases
line A?
e) decreases, then increases
A
v
t
B
In case A, the initial velocity is
positive and the magnitude of the
velocity continues to increase
with time.
Question 2.14b v versus t graphs II
a) decreases
Consider the line labeled B in
b) increases
the v vs. t plot. How does the
c) stays constant
speed change with time for
d) increases, then decreases
line B?
e) decreases, then increases
A
v
t
B
Question 2.14b v versus t graphs II
a) decreases
Consider the line labeled B in
b) increases
the v vs. t plot. How does the
c) stays constant
speed change with time for
d) increases, then decreases
line B?
e) decreases, then increases
A
v
t
B
In case B, the initial velocity is positive
but the magnitude of the velocity
decreases toward zero. After this, the
magnitude increases again, but
becomes negative, indicating that the
object has changed direction.
2-5 Motion with Constant Acceleration
If the acceleration is constant, the velocity
changes linearly:
(2-7)
Average velocity:
Note: valid only for constant acceleration
Copyright © 2010 Pearson Education, Inc.
2-5 Motion with Constant Acceleration
Average velocity:
(2-9)
x = x0 + vav t
Position as a function of time:
(2-10)
(2-11)
Velocity as a function of position:
v = v 0 + at
(2-12)
Copyright © 2010 Pearson Education, Inc.
2-5 Motion with Constant Acceleration
The relationship between position and time
follows a characteristic curve.
Copyright © 2010 Pearson Education, Inc.