Position-time graphs

- [Instructor] Check out this race between a snail and a rabbit and see if you can tell which one was faster. Clearly the snail was faster than the rabbit. And we could easily tell that because we're looking at an animation. But in this video, we'll see how to put this entire information that both are moving and that the snail is moving faster than that rabbit, all of that information in a single picture, not an animation, but a single picture. In physics this picture is called a position-time graph and the secret to drawing such a picture is we take photos of this motion at different different moments in time and stitch 'em together. Here, let's take an example and it'll make a lot of sense. But before we do that, let's put some markings on this track. You can think of these as meters, 10 meters, 20 meters. It will be easy to keep track of location. And let's only focus on the rabbit first. We'll think about the snail later. What we'll do is, when the rabbit comes at this position, zero, we will pause the video, so let's do that. So the rabbit comes here, we'll pause the video. And since time is also very important to understand how fast something is moving, we'll have a timer with us. And let's say we start our timer from here. So right now our timer is zero, and when I play this video, our timer will start. So, we'll take our first picture at this moment. And we'll keep it to the right. So let's do that, let's take a picture, there it is, and since this picture is taken at time to equal to zero, we'll put a zero over here somewhere at the bottom. Oh, it's coming on top of the rabbit, let's move this thing a little bit to the left. Excellent, now we'll play the video and we'll pause again when the time hits one second. So we'll pause for every second and we'll take a picture like this. So keep an eye over here, when it hits one second, we'll pause again, here goes, paused. Let's take another picture and keep it to the right. And this picture was taken at one second, so let's mark that at the bottom. And we'll continue, we'll keep on doing this. For every second, we'll take a picture. There it is, now look at this picture. This picture tells you where the rabbit is at every instant of time. For example, I know at two seconds, the rabbit is at 20. At four seconds, the rabbit is at 40. And so, because the picture tells me the position of the rabbit at every instant of time, this graph is usually called the position-time graph. Sometimes it's also called the distance-time graph because it's giving me the distance of the rabbit from the zero mark. Also, before we proceed, notice that all these rabbits are lying on a straight line, a single straight line. That's not a coincidence. That's happening because our rabbit is moving in a uniform motion meaning it's having a constant speed. And that is, it's covering equal distances every second. We'll see if the rabbit is changing its speed, then this wouldn't be a straight line. All right, now notice that we took the photos at every second interval, right? But there's no reason to do that. We could've also taken the photos at every half a second. We could've taken photos at .5 second, 1.5 second, 2.5 seconds, and if we did that, and if you put them over here, then even those rabbits will be on that same straight line. Again, because the rabbit is moving with a uniform speed. And we can put more photos in between. And so, if you put photos of this rabbit at every single instant of time and put them all together, then we have constructed our position-time graph. Now this is great, but how do we draw this on a piece of paper for example? I mean, writing all of this and drawing all the rabbits is going to take a lot of time, right? Well we can reduce the work. First of all, notice we don't have to write 10, 20, all the markings everywhere, we can just write it once to the left, so let's take all of this and put it to the left. That's sufficient, because now, again, if I want to know what is the location of the rabbit at three seconds let's say, all I have to do is go to three seconds, look at where the rabbit is, and the marking is to the left. So I know, ha, it's at 30 meters, so great. Secondly, we don't have to draw rabbits at all. We can just draw a straight line representing all of these rabbits together. So let's do that, let's get rid of the rabbit and draw a straight line. And now that the rabbit is gone, we can bring the zero back to where it was suppose to be, excellent. And we can also get rid of this background. That's also not needed. And there we have it, this is how we draw it on a piece of paper. Usually we like to draw one axis over here. We call that as our position axis because it tells me the positions. And we'll draw another line over here and we call that as our time axis. And this is how we draw it on a piece of paper. Again, just to be clear, for example I want to know, looking at this graph, where the rabbit is at, say three seconds, then all I'll do is come to three seconds, go up and I'll see the rabbit is over here. But where is that? To understand that, I'll go to the left and then I'll understand, aha, it's at 30 meters. So the rabbit is at 30 meters. Also, another important thing I want to point out, is that when I asked you to look at position-time graphs, I should think that this line literally represents the path of an object. For example over here, I would think that the rabbit is actually traveling this way. But that's not what's happening. You get it, right? I mean, if you bring back the original picture, you see that the line connects the rabbits at different time instants. The rabbit is always traveling upwards. So this is the path in which the rabbit is traveling. This line does not represent the path, it's just a connection of the rabbits at different, different time instants. And now that we understand position-time graphs, let's see if we can draw one for the snail. If you go back to our animation, what is the difference? The difference is the snail is a lot faster than the rabbit. But it's also traveling with uniform speed. It also has a constant speed, which means its graph should also be a straight line. But since it's faster, it's gonna be a little different. So, can you pause the video for a while and think about or visualize how different would it be? What would be the graph for the snail? All right, let's do it. Let's do it the same way we did it for the rabbit. So let's bring back the racetrack and the rabbit so that it will become easier to understand. Just like with the rabbit, when the snail comes to zero, that's when we'll take our first picture. And in this animation, both the snail and the rabbit come to zero together. And that's when we call T equal to zero. So, at time equal to zero, our snail is also at zero. So let's take that first picture and put it over here. And just like with the rabbit, we will take pictures every second, so next picture we'll take at one second. Ooh, notice the snail is much faster than the rabbit. So let's put that picture at one second. We're putting it under one second. Again, let's play it for another second. It's now reached at 40, so at two seconds, it'll be at 40. And after this, it'll go out of our frame, so let's stop over here. Again notice that these snails lie on the same straight line because it's traveling with a constant speed. And again, if we were to take pictures in between, all of those pictures would lie on a straight line. And now we can replace that with a single line. And if you bring back our old graphical picture, this is what the graph looks like for our snail. And what we see, is that the graph of the snail is a little bit more slanted compared to that of the rabbit. In physics terms, or in math terms we say this graph has more slope compared to this one. And we say that because if you think of it as a mountain, it's not, but if you think of it as a mountain which you are climbing, then can you see that climbing this mountain will be more difficult because it's more sloping compared to this mountain? And that's why we say this has more slope than this. But anyways, this means that in a position-time graph, if the graph has more slope, it's faster. If it has less slope, it's slower. And that's why position-time graphs are awesome. Because just with a glance, you can tell which objects are faster and slower. All right, the last thing we'll do is, remember we said that this graph is a straight line only when objects are traveling with a constant speed? Well now let's look at what happens if they're not traveling with a constant speed. If they're not going at a uniform motion. And again, we'll concentrate on one of them. Let's look at the snail. So let's get rid of the rabbit. And let's bring back the snails and the racetracks. Okay, what if our snail doesn't travel with a constant speed, but let's say it's slows down and stops. Let's say it looks somewhat like this. Slows down and stops, non-uniform motion. How would this graph change? Well here's how I like to think about it. Because the snail is traveling slower, the distance that it travels every second keeps decreasing. So let's say in the first second it traveled 20 from here to here, then in the next second, it won't travel another 20. Maybe it'll travel only another 10. So, this will be somewhere back over here. Maybe in the next second, it'll travel only another five. It means it'll be somewhere over here, as an example. And so on, and now notice the graph looks like this. Which means they're not on the same straight line. And again, if we were to fill in, in between, then notice the graph looks like a curve. And so this is what the graph looks like for an object which is slowing down. And again, it kind of makes sense, right? Because notice, if you look at the slopiness of this graph, initially it's pretty high, if you were to climb this mountain, very high slope, very high speed. But then as you go further, notice the slopiness decreases. The graph is becoming more and more flat, indicating that it's speed is reducing. And so, in short, if objects are traveling with a constant speed in a uniform motion, then their position-time graphs are straight lines. And if they're traveling faster, they will end up having more slope. On the other hand, if objects travel with non-uniform speed, like the snail slowing down, then the graph will be a curve. And now I'm pretty sure you can draw position-time graphs for all sorts of things like objects speeding up, or maybe even going backwards. Or even the ones which are not moving at all. Have fun!