Question Video: Identifying Which Line on a Speed-Time Graph Corresponds to a Given Line on a Distance-Time Graph Science

Which color line on the speed–time graph shows the motion of the object on the distance–time graph?

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Video Transcript

Which color line on the speed–time graph shows the motion of the object on the distance–time graph?

We are given two graphs to work with in this question. On the left, we have the distance–time graph, which we can tell because it has distance on the vertical axis. On the right is the speed–time graph, and we know this one because speed is the label on the vertical axis. Now the distance–time graph has one line, and this is the motion of the object in the question. What we need to do is figure out which line, the diagonal red line or the horizontal green line, on the speed–time graph corresponds to this motion. We can see, though, that neither of these graphs have scales on their axes, so we won’t be able to approach this problem quantitatively. We will have to do all of our work qualitatively.

Okay, so here’s what we know about the distance–time graph. First of all, it is a straight line. Now we recall that straight lines on distance–time graphs represent objects moving with uniform speed. We can also recall that for an object moving with uniform speed, the slope of the line on the distance–time graph is the speed of the object. Our key observation is that the speed of the object is constant. Now we recall that on speed–time graphs, objects moving with constant speed are represented by horizontal lines. Also on a speed–time graph, the height of the horizontal line is the constant speed of the object. This is the most important distinction between speed–time graphs and distance–time graphs showing uniform speed.

On a distance–time graph for uniform speed, the speed is the slope of the line. But on a speed–time graph for uniform speed, the line is always horizontal and its height is the speed. So we just need to identify the correct horizontal line on the speed–time graph. Well, there’s only one of those. It’s the green line. The red line is not horizontal. So the horizontal green line on the speed–time graph must be the line that shows the motion of the object on the distance–time graph. We can also double-check that the diagonal red line on the speed–time graph is not the correct answer. The motion that we’re interested in has uniform speed; that is, the speed is the same at all times.

So let’s look at two different times on the speed–time graph, say, this time here and this time here. To find the corresponding speeds represented by the red line at these two times, we first draw a vertical line from the horizontal axis to the red line. Next, we draw a horizontal line to the vertical axis. From each of these two intersection points. Where these horizontal lines intersect the vertical axis are the speeds at each of the times we were interested in. In particular, the speed at this first time is represented by this point on the vertical axis, and the speed at this second time is represented by this point on the vertical axis. But the two points on the vertical axis are at different values and therefore represent different speeds.

So what we have found is that the motion represented by the red line has different speeds at different times. But this means that the red line cannot be the correct answer because it has different speeds at different times, but the object we are interested in has uniform speed. That is, it must have the same speed at all times. So we have both found that the green line must be the correct answer and also that the red line cannot be the correct answer.

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