Video Transcript
The change in displacement of two
objects with time is shown in the graph. The lines plotted on the graph are
parallel. Which of these 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.
This question has given us a
displacement–time graph; that’s a graph that plots displacement on the vertical axis
against time on the horizontal axis. We are being asked about the speeds
and velocities of the two objects whose motion is shown on this graph.
Let’s recall that displacement and
velocity are vector quantities, which means that they have both a magnitude and a
direction. Meanwhile, speed is a scalar
quantity, so it has just a magnitude and no associated direction. The speed of an object is equal to
the magnitude of that object’s velocity. And the velocity of an object is
equal to the rate of change of displacement with time.
Looking at the graph, we can see
that the two objects have different initial and final displacements. The object represented by the blue
line begins at some positive initial displacement and ends up at a displacement of
zero. The object represented by the red
line begins at a smaller positive displacement and ends up at a negative
displacement value.
Since the displacement of both
objects is decreasing as time goes on, then both objects must be traveling in the
negative direction; that is, their velocities must both be negative. So, the direction of the velocity
of each object is the same. But what about the magnitudes? We’ve said that the velocity of an
object is equal to the rate at which its displacement changes with time.
Since a displacement–time graph has
displacement plotted against time, then this means that the velocity of an object is
given by the slope of the corresponding line on a displacement–time graph. We’re told that both these two
lines on the graph are parallel, which means that they have the same slope. Therefore, not only do the two
objects move in the same direction, but the magnitude of their velocity in this
direction is also equal. We have found then that both
objects have the same value of velocity.
Since we know that an object’s
speed is equal to the magnitude of its velocity, and we’ve seen that both objects
here have the same magnitude velocity, then we know that these two objects must also
have equal speeds. Since the objects here have the
same velocity and the same speed as each other, then we can identify the correct
answer as option (D): both their speeds and velocities are the same.
