Lesson 4: Describing Motion with
Velocity vs. Time Graphs
The Meaning of Shape for a v-t
Graph
The study of 1-dimensional kinematics has been concerned
with the multiple means by which the motion of objects can
be represented. Such means include the use of words, the use
of diagrams, the use of numbers, the use of equations, and
the use of graphs. Lesson 4 focuses on the use of
velocity vs. time graphs
to describe motion. The specific features
of the motion of objects are demonstrated by the shape and
the slope of the lines on a velocity vs. time graph. The
first part of this lesson involves a study of the
relationship between the motion of an object and the shape of
its v-t graph.
Consider a car moving with a
constant, rightward (+)
velocity of +10 m/s. As you
learned in Lesson 1, a car moving with a
constant velocity is a car moving with zero acceleration.
If
the velocity-time data for such a car were graphed, the
resulting graph would look like the graph at the right. Note
that a motion with constant, positive velocity
results in a line of zero slope (a horizontal line has zero
slope) when plotted as a velocity-time graph. Furthermore,
only positive velocity values are plotted, corresponding to
a motion with positive velocity.
Now consider a car moving with a
rightward (+), changing
velocity – that is, a car that is moving
rightward and speeding up or accelerating.
Since the car is moving in the positive direction and
speeding up, it is said to have a positive
acceleration.
If
the velocity-time data for such a car were graphed, the
resulting graph would look like the graph at the right. Note
that a motion with changing, positive velocity
results in a diagonal line when plotted as a velocity-time
graph. The slope of this line is positive, corresponding to
the positive acceleration. In addition, only positive
velocity values are plotted, corresponding to a motion with
positive velocity.
The velocity vs. time graphs for the two
types of motion – constant velocity and changing velocity
(acceleration) – can be summarized as follows:
Positive
Velocity
Zero
Acceleration
Positive
Velocity
Positive
Acceleration
The Principle of Slope for a v-t Graph
The shapes of the velocity vs. time graphs
for these two basic types of motion – constant velocity
motion and changing velocity motion
(i.e., accelerated motion) – reveal an important principle.
The principle is that the slope of
the line on a velocity-time graph reveals useful information
about the acceleration of the object. Whatever characteristics
the acceleration has, the slope will exhibit the same (and vice versa).
If the acceleration is zero, then the slope is zero (i.e., a
horizontal line). If the acceleration is positive, then the
slope is positive (i.e., an upward sloping line). If the
acceleration is negative, then the slope is negative (i.e.,
a downward sloping line). This principle can be
extended to any motion conceivable.
The slope of a velocity-time graph reveals information
about the object's acceleration. But how can you tell
if the object is moving in the positive direction
(i.e., positive velocity) or in the negative direction
(i.e., negative velocity)? And how can you tell if the
object is speeding up or slowing down? The answers to these questions
hinge on your ability to read a graph.
Positive Velocity vs. Negative Velocity
Since the graph is a velocity-time graph, the
velocity is positive whenever the line lies in the
positive region (positive y-values, i.e. above the x-axis)
of the graph. Similarly,
the velocity is negative whenever the line lies in the
negative region (negative y-values, i.e. below the x-axis) of the graph.
As you learned
in Lesson 1, a positive
velocity means the object is moving in the positive
direction; and a negative velocity means the object is
moving in the negative direction. So if an object is
moving in the positive direction, the line is located in
the positive region of the velocity-time graph (regardless if it is
sloping up or sloping down). Likewise, an object is
moving in the negative direction if the line is located in
the negative region of the velocity-time graph (regardless if it is
sloping up or sloping down). Finally, if a line crosses
the x-axis from the positive region to the negative
region of the graph (or vice versa), then the object has
changed directions.
Acceleration vs. Deceleration
How can you tell if the object is speeding up (acceleration) or
slowing down (deceleration)? Speeding up means that the magnitude (the
value) of the velocity is increasing. For instance, an
object with a velocity changing from +3 m/s to + 9 m/s is
speeding up. Similarly, an object with a velocity changing
from -3 m/s to -9 m/s is also speeding up. In each case, the
magnitude of the velocity (the number itself, not the sign
or direction) is increasing; the speed is getting larger.
Given this fact, an object is
speeding up if the line on a velocity-time graph is changing
from a location near the 0-velocity point to a location further away
from the 0-velocity point. That is, if the line is moving
away from the x-axis (the 0-velocity point), then the object
is speeding up. Conversely, if the line is moving
towards the x-axis, the object is slowing down.
