Video Transcript
Consider the following graph. The motion of object A is
represented by the solid line. The motion of object B is
represented by the dashed line. Which of the following statements
correctly compares the time traveled by the objects? Is it (A) object A travels the
greater time, (B) object A and object B travel an equal amount of time, or (C)
object B travels the greater time?
We recall that for a distance–time
graph, the horizontal axis represents the time for which an object travels and the
vertical axis represents the distance that the object travels. A line on a distance–time graph
shows how the distance traveled by an object changes with the time that the object
travels for. This question asks us to compare
the motion of two objects that are represented by two different lines, a solid blue
line and a red dashed line.
We can see that the red dashed line
is longer than the solid blue line. We might think that this means the
red dashed line represents an object that travels a greater time. This is not the correct way to
answer this question, however. A line on the graph does not show
the shape of the path that an object travels along. A longer line does not necessarily
mean a greater time traveled.
Let us consider what a line on a
distant–time graph does mean. Recall that time is represented by
the horizontal axis. Taking this fact into
consideration, we should consider that a line has two ends. One end of the line has a smaller
value of time than the other end of the line. The end of the line with a lesser
value of time shows where and when the motion of the object represented by the line
starts. The end of the line with a greater
value of time shows where and when the motion of the object represented by the line
ends.
Recalling once again that time is
represented by the horizontal axis and consider that distance is what is represented
by the vertical axis, this means that the greater the change in the value of the
horizontal axis between the ends of a line, the greater the time that the object
represented by the line moved for. This also means that the greater
the change in the value of the vertical axis between the ends of a line, the greater
the distance that the object represented by the line moved.
In this question, we are asked
which line represents the object that travels for the greater time. The question does not ask anything
about the distance which either object travels. Because we’re only considering the
time, to find the times that the objects represented by lines travel for, we only
need to look at the changes in the value on the horizontal axis of the ends of the
lines. We can see that the value of time
is the same for the ends of both the red dashed line and for the solid blue line
that show the start of motion. Both lines have ends that touch but
do not cross the origin of the graph. These must represent the same time
for the start of motion.
We can also see that the value of
time is the same for the ends of both the red dashed line and for the solid blue
line that show the end of motion. Both lines have ends that touch but
do not cross the dashed vertical line. The dashed vertical line represents
a value of time we will call 𝑡. As the ends of the red dashed line
and the solid blue line both touch but do not cross the vertical line, the value of
time for the end of motion for both lines is the same.
We see then that the answer is
(B). The objects represented by the red
dashed line and solid blue line moved for the same time.
