Lesson 4: Describing Motion
with Velocity vs. Time Graphs
The Meaning of Slope for a v-t
Graph
As discussed in the previous section of Lesson 4, the
shape of a velocity vs. time graph reveals pertinent
information about an object's acceleration. For example,
if the acceleration is zero, then the velocity-time graph
is a horizontal line (i.e., the slope is zero). If the
acceleration is positive, then the line is an upward
sloping line (i.e., the slope is positive). If the
acceleration is negative, then the velocity-time graph is
a downward sloping line (i.e., the slope is negative). If
the acceleration is large, then the line slopes up
steeply (i.e., the slope is large). Thus, the shape
of the line on the graph (horizontal, sloped, steeply
sloped, mildly sloped, etc.) is descriptive of the
object's motion. This section of Lesson 4 will
examine how the actual value of the slope of any straight line
on a velocity-time graph corresponds to the acceleration of the
object.
Example 1
Consider a car moving with a constant
velocity of +10 m/s. A car which is moving with a constant
velocity has an acceleration of
0 m/s/s.
The velocity-time data and graph would look like the
table and graph below. Note that the line on the graph is
horizontal. That is, the slope of the line is 0 m/s/s. Here, it
is obvious that the slope of the line (0
m/s/s) is the same as the acceleration (0 m/s/s) of the
car.
Time
(s)
Velocity
(m/s)
0
10
1
10
2
10
3
10
4
10
5
10
So in this case, the slope of the line
is equal to the acceleration of the velocity-time graph.
Example 2
Consider a car moving with a
changing velocity. A car which moves with a changing velocity has
an acceleration.
The velocity-time data for this motion shows that the
car has an acceleration of +10 m/s/s. A graph of
this velocity-time data would look like the graph below.
Note that the line on the graph is diagonal — that is, it
has a slope. The slope of this line, when calculated, is
10 m/s/s. Once again, the slope of the
line (10 m/s/s) is the same as the acceleration (10
m/s/s) of the car.
Time
(s)
Velocity
(m/s)
0
0
1
10
2
20
3
30
4
40
5
50
Example 3
Let's examine
a more complex case. Consider the motion of a car which
travels with a constant velocity (a = 0 m/s/s) of 2
m/s for four seconds and then accelerates at a rate of +2
m/s/s for four seconds. That is, in the first four
seconds, the car does not change its velocity (the
velocity remains at 2 m/s) then the car increases its
velocity by 2 m/s each second over the next four seconds.
The velocity-time data and graph are displayed below.
Observe the relationship between the slope of the line
and the corresponding acceleration value during each four-second interval.
Time
(s)
Velocity
(m/s)
0
2
1
2
2
2
3
2
4
2
5
4
6
6
7
8
8
10
From 0 s to 4 s: slope = 0 m/s/s
From 4 s to 8 s: slope = 2 m/s/s
A motion such as the one above further
illustrates the importance of the principle of slope: the slope of the
line on a velocity-time graph is equal to the
acceleration of the object. This principle can be used
for all velocity-time graphs in order to determine the numerical
value of the acceleration.
Check
Your Understanding
The velocity-time graph for a two-stage rocket is
shown below. Use the graph and your understanding of
slope calculations to determine the acceleration of the
rocket during the listed time intervals. When finished,
depress your mouse on the pop-up menus below to
see the answers. (Help with Slope
Calculations)
