Interpreting motion data | Physics | Khan Academy

Let's learn about position time graphs and position time tables to analyze motion. Let's start by considering a car going at a constant velocity. To create a position timetable, let's take snapshots of it at, say, every five seconds. So here we go, boom!

And let's convert this to our familiar oil drop diagram. It's easier to work with dots to mark positions. All we need is an axis, a vertical axis. Think of it as a number line. We need to start somewhere, so we need to call some point as zero. So let's just call this point as 0, and then we measure all the other points.

Let's say when I measure this, it turns out to be 10 meters. If this distance is 10 meters, we will now say this position is 10 meters. You can imagine we have chosen upwards to be positive, and that's why this is positive 10 meters. Then we'll measure how far this dot is from our origin, and let's say that happens to be 20 meters. In fact, that should be 20 meters because since the car is traveling with a constant velocity, it travels the same distance, right? So if this was 10 meters, this would also be 10 meters.

So the total would be 20 meters, and therefore this position now is 20 meters. Note that we are measuring always from the origin, not from here, from the origin. And same would be the case over here. This would be 30 meters, and this would be at 40 meters mark.

Well, we now have our position and time recordings, so we can go ahead and fill up this position time table. Why don't you give it a shot first?

All right, let's do it. So at time 0 seconds, our position was 0 meters. Then at five seconds, our position is 10 meters. Then at 10 seconds, 20 meters. At 15 seconds, it's 30 meters, and at 20 seconds, we are at 40 meters. Ta-da! This is how we draw our position timetable.

Now, here's a question: if we were not shown the animation, or we were not given this oil drop diagram, could we just look at the position timetable and comment on whether the car is accelerating or not? Sure! All we have to do is look at its displacements.

So, at zero, we have the position 0. In the first five seconds, it travels 10 meters. But look at what happens in the next five seconds: again, it travels from 10 to 20, 10 meters. In the next five seconds, again, it travels 10 meters. So can you see? In every five seconds, it keeps traveling 10 meters. That's constant velocity!

So just by looking at the position time graph, you get the same information as we get from the oil drop diagram. So it's just another way of representing the same data. Guess what? There is another way to represent this data, and that is by drawing what we call a position time graph.

Here, we draw the time on the horizontal axis, and we draw the position on the vertical axis. By the way, we tend to use the symbol X to represent the position. That's just a tradition for us. Now again, before I plot it, can you try to plot the position time graph yourself first and see what kind of graph you end up with?

All right, let's do this. Now ideally, you need a graph paper to have very accurate markings, but since I don't have it, I'm going to draw a few reference lines for myself. Cool! Now at zero time, our position is zero, so that's (0, 0). That's over here. At five seconds, our position is at 10 meters, so we draw here. At 10 seconds, we are at 20. At 15 seconds, we are at 30, and at 20 seconds, we are at 40.

Can you see that all these points lie on a straight line? So this means that when things are going at a constant velocity, their position-time graph would be a straight line. And again, that's happening because at every five seconds, your graph is incrementing by the same amount: 10 meters! Another 5 seconds, 10 meters! Another 5 seconds, 10 meters! And that gives you a straight line.

Finally, before we move on, what do you think is the velocity of this car? Feel free to use any of these because all three are telling you the same story.

All right, so how do we calculate velocity? Well, velocity is displacement over time. And for that, let's see if I can use the position timetable to do that. Well, for this, I just need to pick some time interval, and we are free to choose any time interval we want. So let's pick, say, from 10 to 15 seconds.

So from 10 to 15 seconds, the time interval is 5 seconds. And in that time interval, what is the displacement? Well, you go from 20 to 30. That is 10 meters! And therefore, what is your velocity? Your velocity is 2 meters per second.

And could we get the same thing from our oil drop diagram? Sure! If you go from 10 to 15 in 5 seconds, it's covering 10 meters—the exact same thing! Do we get that? Can we get that from the graph as well? Well, let's see. If we go from 10 to 15, that 5 seconds is actually this length because remember, in the graph, the horizontal represents the time. So this is the 5 seconds, and when we calculate displacement from 20 to 30, we are measuring this length because the vertical represents the position over here.

And so what you notice is velocity over here is going to be the vertical divided by the horizontal. And that quantity, the vertical divided by the horizontal, represents the slope. It literally tells you how steep the graph is!

So this means the velocity, or at least the magnitude of the velocity, is given by the slope of the graph. And this is amazing because that means if I had another graph, let's say that looked like this, I can immediately say this is also an object going at a constant velocity because it's a straight line. But I can now compare it with this one, and I can say, "Hey, this has less slope, and therefore it must be having smaller velocity compared to this."

You can confirm this—for example, in five seconds, our car is over here, but this object is only here, so clearly, it's going slower! But just by looking at the slope, you can identify it. And similarly, if I had another object which had a graph like this, and imagine that's a straight line, then again you can say, "Hey, this is also an object moving at a constant velocity," but it has a much higher slope! Look at it—how steep it is, therefore it must be having a higher velocity compared to this.

So that's amazing! That's why I love position time graphs, because just by looking at the slope, I can compare velocities. That is awesome!

Can we check our understanding by taking a couple of more examples? Okay, here's the next one! This time I'm going to give you a table, and why don't you pause and think about what this object is doing? Is it going with a constant velocity? Is it accelerating or decelerating? Why don't you pause and think about it?

All right, just like before, I will look at the displacement in the first five seconds. It covers one meter. In the next five seconds, oh look! It covers—it goes from one to four! That means it covers three meters. In the next five seconds, look! It covers five meters! It's covering more and more distance. That means this car must be speeding up! Immediately, I know that this object is accelerating.

And if we can go ahead and draw the oil drop diagram for this, it will look something like this: plotted the same story over here, and you see the same thing visually—it's covering more and more distance in every five seconds. And again, I encourage you to pause the video and see if you can plot the graph yourself first.

All right, because I don't have a graph paper, I have drawn some reference lines. So again, for zero, we are at zero; at five, we are at one. Now, at 10, we are at four, so somewhere over here. At 15, we are at nine—that's over here! And at 20, we are at 16.

Is this a straight line? Clearly, no, that's not a straight line! Let's just see that there is no way this can be a straight line. I cannot draw a straight line! Instead, it becomes a curve. And so when objects are accelerating, we get a curve!

But now, let's think about the slope. Clearly, because it's a curve, the slope is changing. But if you see here, the slope is less. The graph is not very steep over here, but as I go forward, look, it becomes steeper and steeper! Imagine it's a mountain that you're climbing—it's easier to climb here, but it gets harder and harder to climb.

So that means the slope is increasing! Ah, that means the velocity is increasing. So you see, again, the graph also gives you the same information that we got from here: velocity is increasing; it is accelerating.

Here's the final one. This time I'm directly giving you the graph. Can you analyze the motion? What is the object doing? Well, because it's a curve, I know that it's not constant velocity; its velocity is changing.

But is it accelerating or decelerating? What is it doing? Well, let's look at the slope. Initially, the graph is very steep—it's very hard to climb this, so very high velocity! But then notice, as I keep climbing, it becomes easier to climb. It becomes less steep. Oh, the slope decreases! This means the velocity becomes smaller.

So that means this object must be decelerating! And looking at the table and the oil drop diagram, they too tell the same story, and I'm sure you can confirm this yourself now!