Video: Analyzing the Distance-Time Graph for an Object That Changes Speed Repeatedly

The distance–time graph shows the change in the distance moved by a person walking in the time interval from 𝑡 = 0 seconds to 𝑡 = 6 seconds. What speed does the person move at between 𝑡 = 0 s and 𝑡 = 2 s? What speed does the person move at between 𝑡 = 4 s and 𝑡 = 6 s? How far has the person moved at 𝑡 = 5 s?

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Video Transcript

The distance–time graph shows the change in the distance moved by a person walking in the time interval from 𝑡 equals zero seconds to 𝑡 equals six seconds. What speed does the person move at between 𝑡 equals zero seconds and 𝑡 equals two seconds?

The graph shows distance on the vertical axis and time on the horizontal axis. So recall that speed is given by the slope of a distance–time graph and also that the slope is given by the vertical difference divided by the horizontal difference. So we need to find the slope of the graph between 𝑡 equals zero seconds and 𝑡 equals two seconds. So let’s first find 𝑡 equals two seconds on the horizontal axis. And we find that the distance at this point is two meters and at 𝑡 equals zero seconds, the distance is zero meters.

Therefore, the slope is given by the distance two minus zero divided by the time two minus zero, which is just two divided by two or one. Therefore, the speed is equal to one. And for the units, we take the units of the vertical axis divided by the units of the horizontal axis, so that’s meters per second. So between 𝑡 equals zero seconds and 𝑡 equals two seconds, the person moved at one meter per second. Next, we need to find what speed does the person move at between 𝑡 equals two seconds and 𝑡 equals four seconds. So here, we need to find the slope of the graph between 𝑡 equals two seconds and 𝑡 equals four seconds.

So we can first find the point of the graph where 𝑡 equals four seconds. And we find that the distance is equal to six meters. And at 𝑡 equals two seconds, we find that the distance is two meters. So speed is equal to the slope, which is the vertical difference of six minus two, divided by the horizontal difference of four minus two. That gives four divided by two, which is two. And again, the units are meters per second. So the answer to “What speed does the person move at between 𝑡 equals two seconds and 𝑡 equals four seconds?” is two meters per second.

The next part of the question asks, “What speed does the person move at between 𝑡 equals four seconds and 𝑡 equals six seconds?”

On this section of the graph, we can see that the line is flat. In other words, it has a slope of zero. So the key point to remember here is that a flat line with a slope of zero on a distance–time graph indicates a speed of zero. So between 𝑡 equals four seconds and 𝑡 equals six seconds, the person is moving with a speed of zero meters per second.

Next, we need to find what average speed does the person move at between 𝑡 equals zero seconds and 𝑡 equals six seconds.

Here, we need to recall that average speed is the total distance divided by the total time. So we first need to find the total distance traveled, which is the distance at 𝑡 equals six seconds, and for that, we find a total distance traveled of six meters. And the distance of the starting point at 𝑡 equals zero seconds is zero meters. Therefore, the average speed is the total distance of six meters divided by the total time of six seconds. So the average speed the person moves at between 𝑡 equals zero seconds and 𝑡 equals six seconds is one meter per second.

Next, we need to find what average speed does the person move at between 𝑡 equals zero seconds and 𝑡 equals four seconds.

If we look at the endpoint here of 𝑡 equals four seconds, we find that the total distance traveled is the same as it was at 𝑡 equals six seconds since the person is stationary for the last two seconds. So the total distance traveled here is again six meters but covered in a smaller amount of time, which means the speed will be higher. So we can write down our total distance traveled, which is six meters, divided by our total time of four seconds. So the average speed is six divided by four, which is 1.5 meters per second. So 1.5 meters per second is the average speed between 𝑡 equals zero seconds and 𝑡 equals four seconds.

Next, we need to find the average speed that the person moves between 𝑡 equals two seconds and 𝑡 equals six seconds.

So once again on the graph, we can find that at 𝑡 equals six seconds, the person had moved a total distance of six meters. However, this time our starting point is not zero. We’re starting from 𝑡 equals two seconds, by which time the person had already moved a distance of two meters. So this time, our average speed is our total distance moved, which is six minus two meters, divided by the total time taken, which is six minus two seconds. So that’s four divided by four, or one meter per second. So the average speed the person moves at between 𝑡 equals two seconds and 𝑡 equals six seconds is one meter per second.

Next, we need to find how far has the person moved at 𝑡 equals three seconds.

To answer this, we find 𝑡 equals three seconds on the horizontal axis, move up to our blue line, and then across to the vertical axis where we can read off the total distance of four meters. So at 𝑡 equals three seconds, the person has moved a distance of four meters.

The final part of this question asks, “How far has the person moved at 𝑡 equals five seconds?”

Again, we can read this directly off the graph by finding five on the horizontal axis, moving up towards our blue line and then across towards the vertical axis where we can read off the distance of six. And so the answer to how far has the person moved at 𝑡 equals five seconds is six meters.

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