Chapter 2 – Motion in One Dimension
Term
Frame of
Reference
Position
Distance
Displacement
Definition
Example
A point of
origin for
describing
motion; serves
as a designated
starting point
for defining
motion.
The railroad
tracks
The separation
between an
object and the
frame of
reference;
precisely
describes the
location of an
object.
The car is
currently
located 200 m
east of the
railroad tracks.
A measure of
how much
ground an
object has
covered; tells us
how far we are
actually
moving.
The truck is
125 m from the
tracks.
The change in
position of an
object; tells us
where we are
located relative
to the starting
point.
Since crossing
the tracks, the
car has moved
200 m to the
east.
Symbol
Unit
Mathematical
Relationship
2
Term
Average
Speed
Average
Velocity
Instantaneous
Velocity
Definition
Example
The average
rate at which an
object covers
ground; a
measure of how
fast an object is
moving along
its actual path;
defined as
distance per
unit time.
The average
rate at which an
object changes
its position; a
measure of how
fast and in what
direction an
object is
moving away
from its starting
point; defined
as displacement
per unit time.
The velocity of
an object at
some particular
instant.
The average
speed of the car
during its trip
was 50 mph
(22.2 m/s).
The average
velocity of the
truck during its
trip was 35
mph (15.6 m/s)
to the east.
At one
particular
instant, the car
was moving at
40 mph (17.8
m/s) to the
east.
Symbol
Unit
Mathematical
Relationship
3
Term
Average
Acceleration
Instantaneous
Acceleration
Definition
Example
The average
rate at which an
object changes
its velocity; a
measure of how
fast and in what
direction the
velocity is
changing.
The average
acceleration of
the car is 4.5
mph/s (2 m/s2)
to the east.
The
acceleration of
an object at
some particular
instant.
At one
particular
instant, the
acceleration of
the car was
2.25 mph/s (1
m/s2) to the
east.
Symbol
Unit
Mathematical
Relationship
4
One-Dimensional Motion Example Problems
1. A car drives 40 miles east, turns around, and then drives 30 miles west.
(a) Find the distance traveled by the car.
(b) Find the car’s displacement.
2. Assuming that the trip described in Problem 1 takes 2 hours, determine the following:
(a) The average speed of the car.
(b) The average velocity of the car.
3. A bus travels 280 km south along a straight path with an average velocity of 80 km/h to the
south. The bus stops for a 30 minute break, then travels 140 km south with an average
velocity of 70 km/h to the south.
(a) How long does the total trip last?
(b) What is the average velocity for the total trip?
4. A car traveling at 7 m/s accelerates uniformly at 2.5 m/s2 to reach a speed of 12 m/s. How
long does it take for the acceleration to occur?
5
Acceleration, Velocity, Displacement, & Time Relationships
Chapter 2 Worksheet
1. A battery-operated vehicle travels with a constant velocity of 30 m/s over a 10 s time
interval. Complete the table below, and then use this data to answer the questions that
follow:
Time
Elapsed
(s)
Total
Displacement
(m)
Velocity
(m/s)
0
0
30
1
30
2
30
3
30
4
30
5
30
6
30
7
30
8
30
9
30
10
30
Acceleration
(m/s2)
(a) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your
acceleration data, how could you have predicted the shape of this graph? What is the
slope of the graph? How does the slope of this graph relate to the acceleration of the
vehicle? What is the area under the graph? How does the area under the graph relate to
the displacement of the vehicle over this time interval?
Velocity (m/s)
Velocity vs. Time
40
30
20
10
0
0
1
2
3
4
5
Time (s)
6
7
8
9
10
6
(b) Plot a graph of total displacement vs. time. What is the shape of the graph? By
examining your graph of velocity vs. time, how could you have predicted the shape of
this graph? What is the slope of the graph? How does the slope of the graph compare
with the velocity of the vehicle over the time interval?
Displacement (m)
Total Displacement vs. Time
300
240
180
120
60
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
(c) Using only the velocity and time values, how could one determine the total displacement
of the cart after any length of time?
(d) Assuming this motion was to continue indefinitely, how far would the cart have traveled
after 100 s?
7
2. A motorized car accelerates at a constant rate of 10 m/s2 over a time interval of 10 s.
Complete the table below, and then use this data to answer the questions that follow:
Time
Elapsed
(s)
Total
Displacement
(m)
Velocity
(m/s)
Acceleration
(m/s2)
0
0
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
(a) Plot a graph of acceleration vs. time. What is the shape of the graph? What is the slope
of the graph? What does the slope of the graph tell you about the motion of the car?
What is the area under the graph? What is the significance of the area under the graph?
Acceleration (m/s2)
Acceleration vs. Time
15
10
5
0
0
1
2
3
4
5
Time (s)
6
7
8
9
10
8
(b) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your
graph of acceleration vs. time, how could you have predicted the shape of this graph?
What is the slope of the graph? What is the significance of the slope of the graph? What
is the area under the graph? What is the significance of the area under the graph?
Velocity (m/s)
Velocity vs. Time
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
(c) Plot a graph of total displacement vs. time. What is the shape of the graph? By
examining your graph of velocity vs. time, how could you have predicted the shape of
this graph? Find the slope of the graph over the time interval from 0 to 10 s. What is the
significance of the slope of the graph in terms of the car’s velocity over that interval?
Displacement (m)
Total Displacement vs. Time
500
400
300
200
100
0
0
1
2
3
4
5
6
7
8
9
10
Time (s)
(d) Using only the given acceleration, initial and final velocity values, and amount of elapsed
time, how could one determine the total displacement of the cart after any length of time?
(e) Assuming this motion was to continue indefinitely, how far would the cart have traveled
after 100 s?
9
Displacement, Velocity, & Acceleration from Graphs
Chapter 2 Worksheet
1. The slope of a position-time or total displacement-time graph over any interval of time gives
the _______________ _______________ during that interval.
2. The tangent to a position-time or total displacement-time graph at any point gives the
_______________ _______________ at that point.
3. The graph below shows the total displacement of an object from a given starting point over a
time interval of 20 seconds. Use the graph to answer the questions that follow:
Position (m)
Position vs. Time
20
15
10
5
0
0
5
10
15
20
Time (s)
(a) What is the object’s position at t = 10 s?
(b) How far did the object travel between t = 5 s and t = 15 s.
(c) What is the object’s displacement between t = 15 s and t = 20 s?
(d) What is the total displacement of the object over the entire 20 s time interval?
(e) Find the average velocity of the object between t = 5 s and t = 10 s.
(f) What is the instantaneous velocity of the object at t = 7 s.
(g) Describe the motion of the object between t = 10 and t = 15 s.
10
4. The slope of a velocity-time graph over any interval of time gives the _______________
_______________ during that interval.
5. The tangent to a velocity-time graph at any point gives the _______________
_______________ at that point.
6. The area under a velocity-time graph between any two times gives the _______________
during that interval.
7. The graph below shows the velocity of an object over a time interval of 40 seconds. Use the
graph to answer the questions that follow:
Velocity (m/s)
Velocity vs. Time
12
8
4
0
0
5
10
15
20
25
30
35
40
Time (s)
(a) What is the velocity of the object at t = 15 s?
(b) Describe what is happening to the velocity of the object at t = 25 s?
(c) Find the average acceleration of the object between t = 0 and t = 5 s.
(d) Find the average acceleration of the object between t = 5 s and t = 30 s.
(e) Find the average acceleration of the object over the entire 40 s time interval.
(f) Find the instantaneous acceleration of the object at t = 35 s.
(g) How far did the object travel between t = 5 s and t = 15 s?
(h) What is the object’s displacement between t = 15 s and t = 20 s?
(i) Find the displacement of the object over the entire 40 s time interval.
11
8. The area under an acceleration-time graph between any two times gives the
______________________________ during that interval.
9. The graph below shows the acceleration of an object over a time interval of 8 seconds. Use
the graph to answer the questions that follow:
Acceleration
2
(m/s )
Acceleration vs. Time
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
Time (s)
(a) What is the object’s instantaneous acceleration at t = 5 s?
(b) What is the change in acceleration of the object between t = 3 s and t = 7 s?

(c) Describe what is happening to the velocity of the object between t = 4 s and t = 6 s.
(d) What is the object’s change in velocity between t = 0 s and t = 4 s?
12
Equations of Motion for Uniform Acceleration
Uniform or Constant Acceleration is acceleration that does not change with time. The velocity-time graph of a body experiencing such
motion is a straight line. The following equations are valid only for such cases:
(1)
(2)
(3)
(4)
Equation:
vf = vi + at
d = ½ (vf + vi)t
d = vit + ½ at2
vf2 = vi2 + 2ad
Relates:
(must know
3 of the 4)
Initial Velocity
Final Velocity
Acceleration
Time
Displacement
Initial Velocity
Final Velocity
Time
Displacement
Initial Velocity
Acceleration
Time
Initial Velocity
Final Velocity
Acceleration
Displacement
Not Needed:
Displacement
Acceleration
Final Velocity
Time
Since aavg =
v
and
t
v = vf – vi,
v  vi
a= f
.
t
Derivation:
Solving for vf, we obtain
the result
vf = vi + at.
Previously, we found that
d = vavgt.
Earlier we found that
vf = vi + at.
And, for an object
experiencing constant
acceleration,
vavg = ½ (vf + vi).
Substituting this for vf in
eqn (2),
d = ½((vi+at)+vi)t
= ½ (2vi+at)t.
Combining these,
d = ½ (vf + vi)t.
Simplifying,
d = vit + ½ at2.
Solving eqn (1) for t,
we obtain
v  vi
t = f
.
a
Now substituting this into
eqn (2),
v  vi v f  v i
d = f
2
a
=
v f 2  vi 2
.
2a
And rearranging,
vf2 = vi2 + 2ad.
13
Acceleration Due to Gravity
Chapter 2 Worksheet
A ball is fired up it the air with an initial velocity of 49 m/s, and its velocity and displacement is
monitored over a 10 s time interval. Complete the table below, and then use this data to answer
the questions that follow:
Time
Elapsed
(s)
Total
Displacement
(m)
Velocity
(m/s)
Acceleration
(m/s2)
0
0
49
–9.8
1
–9.8
2
–9.8
3
–9.8
4
–9.8
5
–9.8
6
–9.8
7
–9.8
8
–9.8
9
–9.8
10
–9.8
(a) Plot a graph of acceleration vs. time. What is the shape of the graph? What is the slope
of the graph? What is the significance of the slope of the graph? What is the area under
the graph? What is the significance of the area under the graph?
Acceleration (m/s2)
Acceleration vs. Time
20
10
0
-10
-20
0
1
2
3
4
5
Time (s)
6
7
8
9
10
14
(b) Plot a graph of velocity vs. time. What is the shape of the graph? By examining your
graph of acceleration vs. time, how could you have predicted the shape of this graph?
What is the slope of the graph? What is the significance of the slope of the graph? What
is the area under the graph? What is the significance of the area under the graph?
Velocity (m/s)
Velocity vs. Time
50
25
0
-25
-50
0
1
2
3
4
5
6
7
8
9
10
Time (s)
(c) Plot a graph of total displacement vs. time. What is the shape of the graph? By
examining your graph of velocity vs. time, how could you have predicted the shape of
this graph? Find the slope of the graph over the time interval from 0 to 5 s. What is the
significance of the slope of the graph in terms of the ball’s velocity over that interval?
Displacement (m)
Total Displacement vs. Time
125
100
75
50
25
0
0
1
2
3
4
5
Time (s)
6
7
8
9
10
15
Acceleration Due to Gravity
Problem:
A ball is thrown up into the air with an initial speed of 20 m/s from the top of a bridge located 30
m above the surface of the river below.
(a)
How long will it take the ball to reach the top of its path?
(b)
Relative to the bridge, how high will the ball rise?
(c)
From the moment of its release, how long will it take the ball to hit the water?
(d)
How fast will the ball be moving when it strikes the water?
16
Chapter 2 Worksheet – Review of Motion in One Dimension
1. The following graph shows the velocity of a moving object as monitored over a time period
of 8 s. Use the graph to answer the questions that follow:
Velocity (m/s)
Velocity vs. Time
4
2
0
-2
0
1
2
3
4
5
6
7
8
Time (s)
(a) Determine the distance traveled by the object between t = 0 s and t = 8 s.
(b) Determine the displacement of the object between t = 0 s and t = 8 s.
(c) What was change in the object’s velocity between t = 2 s and t = 6 s?
(d) Find the average acceleration of the object between t = 3 s and t = 8 s.
(e) Calculate the average acceleration of the object over the entire 8 s interval.
2. The VW Beetle goes from 0 to 60 mph with an acceleration of 2.35 m/s2.
(a) Using the fact that there are 1.6 km/mile, convert the final velocity to m/s.
(b) Starting from rest, how many seconds should it take the VW Beetle to reach this final
velocity?
(c) A dragster can go from 0 to 60 mph in a mere 0.600 s. What is the acceleration rate (in
m/s2) of the dragster?
3. A golf ball is dropped from rest into a river from a bridge 55 m above the water. A short
time later, a second ball is thrown downward with a speed of 11.9 m/s, and happens to strike
the water at the same moment as the first ball.
(a) How long did it take the first ball to reach the water?
(b) How long was the first ball falling before the second was thrown?
(c) With what velocity did the first ball strike the water?
(d) With what velocity did the second ball strike the water?
4. A woman on a bridge 100 m high sees a raft floating at a constant speed on the river below.
She drops a stone from rest and is successful in hitting the raft. The stone is released when
the raft has 6 m more to travel before passing under the bridge.
(a) How long does it take the stone to reach the water?
(b) At what constant speed is the raft traveling?
Answers: (1) 14 m, 6 m, –6 m/s, –0.8 m/s2, 0, (2) 26.7 m/s, 11.4 s, 44.5 m/s2, (3) 3.35 s, 1.00 s,
–32.8 m/s, –34.9 m/s, (4) 4.52 s, 1.33
17
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Chapter 2 – Motion in One Dimension