Question Video: Comparing the Speeds and Velocities of Objects Using a Displacement-Time Graph Physics

The change in the displacement of two objects with time is shown in the graph. The gray arrows in the diagram are the same length. Do the two objects have the same velocity? Do the two objects have the same speed?

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Video Transcript

The change in the displacement of two objects with time is shown in the graph. The gray arrows in the diagram are the same length. Do the two objects have the same velocity? And do the two objects have the same speed?

If we look at our graph, we can see displacement on the vertical axis and time on the horizontal axis. Both objects start with a displacement of zero. The displacement of the object represented by the blue line increases with time. And the displacement of the object represented by the red line decreases with time. So let’s first consider the velocity of the two objects.

The key thing to remember is that velocity is the slope of a displacement–time graph. If we look at our two lines, the blue line has a positive slope and the red line has a negative slope. These two slopes cannot possibly be the same, and therefore the two velocities can’t be the same. So the answer to the question “Do the two objects have the same velocity?” is no.

Now let’s consider “Do the two objects have the same speed?” Recall that speed is the magnitude, that is, the positive value, of velocity, or equivalently the magnitude or size of the slope of a displacement–time graph. So in this case, it doesn’t matter that one slope is positive and the other is negative; we just need to know whether the sizes are the same. We know that both objects start with a displacement of zero. And since these two gray lines, we’re told, are the same length, we know that the object represented by the blue line ends up with a displacement the same distance above the horizontal axis that the object represented by the red line is below.

In other words, the displacements have changed by the same amount, one in a positive direction and one in a negative. So if the slope of the blue line is 𝑎, then the slope of the red line must be negative 𝑎. But since speed is the magnitude or size of the slope, we don’t care about the negative sign in this case. So the two objects do have the same speed. Therefore, the answer to the question “Do the two objects have the same speed?” is yes.

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