Video Transcript
The change in velocity of two
objects with time is shown in the graph. Which one of the following
statements about the speeds and velocities of the two objects is correct? (A) Their speeds are the same, but
their velocities are different. (B) Their velocities are the same,
but their speeds are different. (C) Both their speeds and
velocities are different. (D) Both their speeds and
velocities are the same.
In this question, we are given a
velocity–time graph and asked to identify the correct statement about the speeds and
velocities of the two objects whose motion is shown on this graph. We can recall that velocity is a
vector quantity, which means that it has both a magnitude and a direction. In order for two objects to have
the same velocity as each other, they must have velocities with equal magnitudes and
equal directions.
On a velocity–time graph, the
height of a line on the velocity axis at a particular point tells us the value of
that object’s velocity at that particular time value. Velocity values above the
horizontal axis are positive, meaning the object is moving in the positive
direction, while velocity values below the horizontal axis are negative.
Looking at the graph, we can see
that the object represented by the blue line starts out with some positive value of
velocity. At this same initial instant, the
object represented by the red line starts out with some negative velocity value. Since the red line begins further
below the horizontal axis and the blue line begins above it, that is, since this
distance here is larger than this one here, then the magnitude of the initial
velocity of the red object is larger than that of the blue object.
We have found then that initially
the red and blue objects move in opposite directions with different magnitudes of
their respective velocities. Therefore, they do not have equal
velocities. The same must also be true at any
later point in time. Since the two lines on the graph
appear to be parallel, this means that they have the same slope as each other. That is, the two velocities change
at the same rate. Since they start out as different
velocity values, then the two velocities will remain different at all later values
of time.
Before this instant, the two
objects have different directions. After this point, the red object
has begun moving in the positive direction, that is, the same direction as the blue
object. However, the magnitude of the red
object’s velocity is still not the same as that of the blue object’s velocity since
the lines still have different heights on the velocity axis at any given instant in
time.
We have found then that the two
objects have different velocities. That means we can eliminate the
answer choices (B) and (D), which claim that the velocities are the same.
Now we need to consider the speed
of the two objects. Let’s clear some space on screen to
do this. Let’s recall that, unlike velocity,
speed is a scalar quantity. That means it has a magnitude but
no associated direction. An object’s speed is equal to the
magnitude of its velocity. It is possible for two objects with
different velocities to have equal speeds. For example, if two objects move in
opposite directions, but the magnitudes of their velocities are equal, then they
will have the same speed despite having different velocity values.
However, that is not the case
here. We have seen that, in general, the
magnitudes of the velocities of the red and blue objects are not equal. Therefore, the speeds are not
equal. So we can say that the speeds of
these two objects are not equal for the duration of time shown on this graph.
We could, in fact, use the
information from the velocity–time graph to draw a speed–time graph for the two
objects. Since speed is the magnitude of the
velocity, then positive velocity values will look the same on a speed–time graph,
while negative velocity values will be replaced by the equivalent positive
values.
The blue object has a positive
velocity at all values of time shown on the graph. So the speed–time graph for the
blue object will look just the same as its velocity–time graph. The red object has a positive
velocity after this point. So the speed–time graph for this
object will look just the same as its velocity–time graph after this point. Before this point, the red object’s
velocity is negative, and the speed is equal to the magnitude of this negative
velocity, which is a positive value. So the speed of the red object in
this interval looks like this.
Now, at this instant here, where
the red and blue lines cross, the two objects instantaneously have the same
speed. But at all other time values, their
speeds are different. So in general then, we have found
that the two objects have different speeds.
We can therefore eliminate answer
option (A), which claims that their speeds are the same.
Since we’ve seen that the
velocities and speeds are both different, we can identify the correct answer as
option (C). Both their speeds and velocities
are different.
