Video Transcript
In 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 distances moved by the objects? Is it (A) object A travels the
greater distance, (B) object B travels the greater distance, or (C) the distances
moved are equal?
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 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 blue solid line, and we might think that this means the red
dashed line represents an object that travels a greater distance. This is not the correct way to
answer this question, however. This is because 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 distance traveled.
Let us consider what a line on a
distance–time graph does mean. Recalling that time is represented
by the horizontal axis, we should consider that a line has two ends. One end of the line has a smaller
value of time than the other end. 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. And 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. The greater the change in the value
of the horizontal axis between the ends of the line, the greater the time that the
object represented by the line moved for.
Next, let’s recall that distance is
what is represented by the vertical axis. This means that the greater the
change in 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 the greater distance. The question does not ask anything
about the time for which either object travels. As we are only considering the
distance that objects travel, to find the distance traveled by objects represented
by the lines, we only need to look at the changes in the value on the vertical axis
of the ends of the lines.
We can see that the value of
distance is the same for the ends of both the red dashed line and for the solid blue
line 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 the motion. We can also see that the value of
distance is the same for the ends of both lines that show the end of the motion. Both the red dashed line and solid
blue line have ends that touch but do not cross the dashed horizontal line. This dashed horizontal line
represents a value of distance, which we have called 𝑑.
As the ends of both lines touch but
do not cross this horizontal line, the value of distance for the end of motion for
both lines is the same. We can see then that the correct
answer is (C). The objects represented by the red
dashed line and solid blue line moved the same distance.
