Video Transcript
Which of the lines on the following
distance–time graph represents the object that travels the greater distance? Is it (A) line 𝐴, (B) line 𝐵, or
(C) both objects traveled the same distance?
We begin by recalling that for a
distance–time graph as shown here, 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,
for which in this case there are two, shows how the distance traveled by an object
changes with the time that the object travels for. This question asked us to compare
the motion of two objects that are represented by two different lines. These lines are called line 𝐴 and
line 𝐵.
We can see that line 𝐵 is longer
than line 𝐴. And we might think that this means
line 𝐵 represents an object that travels a greater distance. However, this is not the correct
way to answer this question. A line on a graph does not show the
shape of the path that an object travels along, and a longer line does not
necessarily mean a greater distance traveled. Let us instead consider what a line
on a distance–time graph does mean. As already mentioned, time is
represented by the horizontal axis. And taking this into consideration,
we should consider that a line has two ends. One end of a line has a smaller
value of time than the other end. And the end with the lesser value
of time shows where and when the motion of the object represented by the line
starts.
The end of the line with the
greater value of time shows where and when the motion of the object represented by
the line ends. Since time is represented by the
horizontal 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. And as distance is represented by
the vertical axis, 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 that
line moved.
In this question, we are asked
which line represents the object that travels the greater distance. The question does not ask anything
about the time for which either object travels. As such, we only need to look at
the changes in the value on the vertical axis at the ends of the lines. We can see that the value of
distance is the same for the ends of both line 𝐴 and of line 𝐵 that show the start
of the motion. Both lines have ends that touch but
do not cross the origin of the graph. These must represent the same
distance for the start of motion.
We can also see that the value of
distance is the same for the ends of both line 𝐴 and line 𝐵 that show the end of
motion. This time, both lines have ends
that touch but do not cross the dashed horizontal line. This dashed horizontal line
represents a value of distance. And as the ends of both lines touch
but do not cross the horizontal line, the value of distance for the end of motion
for both lines is also the same. We can therefore conclude that the
correct answer is option (C). The objects represented by line 𝐴
and by line 𝐵 moved the same distance.