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Question Video: Comparing the Times of Travel of Two Objects Science

Consider the 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? [A] Object A travels the greater time [B] Object A and object B travel an equal amount of time [C] Object B travels the greater time

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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.

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