Video Transcript
This is a displacement–time graph
for two objects. Which of these statements is true
about both objects nine seconds after they begin to move? (A) The speed of object one is
greater than the speed of object two. (B) The speed of object two is
greater than the speed of object one. (C) The speed of object two is
equal to the speed of object one.
In this question, we have a
displacement–time graph for two objects. And we want to determine what
happens to the speed of both objects nine seconds after they begin to move.
We begin by noting that this is a
displacement–time graph and not a distance–time graph. We can recall that displacement is
a vector and distance is a scalar. So to get the speed, we will need
to take the absolute value of the slopes on the displacement–time graph.
So let’s begin by considering
object one, the blue line, first. We want to find the slope of the
line nine seconds after object one moves. The formula for the slope of a
straight line is given by the vertical difference over the horizontal
difference. This line is straight from time 𝑡
equals zero seconds to 𝑡 equals 12 seconds, so we can choose any two points along
this line, as long as they are on the same straight line segment.
For convenience, let’s choose the
start of the line at 𝑡 equals zero seconds and our endpoint at 𝑡 equals 12
seconds. At 𝑡 equals zero seconds, the
displacement of object one is equal to zero meters. And at 𝑡 equals 12 seconds, the
displacement of object one is equal to eight meters. So the slope of the line at nine
seconds is given by eight meters minus zero meters over 12 seconds minus zero
seconds, which equals 0.67 meters per second.
And to find the speed, we need to
take the absolute value of this slope. Doing this, we find that the speed
of object one at a time nine seconds after it has moved is equal to 0.67 meters per
second.
Now let’s consider object two, the
red line. Since this line is curved, we must
create a tangent line at 𝑡 equals nine seconds to determine the slope at that
moment in time. We do this by taking a small
portion of the slope at that point and continuing it along the whole graph.
This tangent line is, however,
completely flat. It has no change in the
displacement axis. We see that for the tangent line,
time goes from zero seconds to 20 seconds. And the displacement goes from nine
meters to nine meters. So the slope of the line at nine
seconds is given by nine meters minus nine meters over 20 seconds minus zero
seconds, which equals zero meters per second.
And to find the speed, we need to
take the absolute value of this slope. Doing this, we find that the speed
of object two at a time nine seconds after it has moved is equal to zero meters per
second.
We can now see that the speed of
object one after it has moved nine seconds is greater than the speed of object two
after it has moved nine seconds. This means that options (B) and (C)
are incorrect. So option (A) must be the correct
answer, which is what we have calculated. The speed of object one is greater
than the speed of object two at a time of nine seconds after both objects begin to
move.
