 This is the position time graph of an object in motion with a changing velocity and we know that the slope of the tangent at any point, the slope of the tangent at any point gives us the instantaneous velocity of the object at that time instant. So the slope of this line gives us the instantaneous velocity at this time instant. In this video, we will see why does the slope give us the instantaneous velocity and also look at an example where we try to study the position time graph and try to figure out the instantaneous velocity of the object in motion at different time instants. We know that we can figure out the average velocity. Let's say we want to figure out the average velocity between these two time instants. We know that this is given by delta x divided by delta t. And that really is, that really is the slope of this line. Slope of this line gives us the average velocity with some time interval delta t. This right here is the average velocity. And if you want to figure out the average velocity between this point and somewhere over here, let's say this point. Then again, we have some delta t, we have some delta x and the average velocity is delta x by delta t, which is again the slope of this line. This right here would be delta t and this right here would be delta x. This is your delta x and this is your delta t. So we see that when we decrease the time interval, the points, so this point came closer to this point. This point is now over here. When we decrease the time interval delta t, maybe it was four seconds to begin with and now it is two seconds. So the points are coming closer when we decrease the time interval. And if we keep on decreasing the time interval, if we make it something super small. So let's say we can find average velocity between this point and this point. Maybe delta t over here is 0.5 seconds. So the points are coming closer to each other as the time interval decreases. And we can ask ourselves, what is the velocity at one particular instant? Maybe at this time and maybe let's say this time is, let's call it two seconds. What is the instantaneous velocity at two seconds? So we can try and approximate it a bit and we can say that we always found velocity from a position time graph by figuring out the slope. So if you want to figure out the velocity at two seconds, we are again looking at some sort of slope, delta x by delta t. But now we do not really have a delta t, right? We are only interested at one particular instant. We can approximate and say that we need to choose, we need to choose a delta t, which is extremely, extremely small, something super small, almost, almost tending to 0. That delta t is almost 0. So there are two points are almost on top of each other at the two second mark. And we are able to figure out the slope and the instantaneous velocity at that particular time instant. And turns out when you do make delta t extremely small, you get a line which is touching the curve only at one point, only at one point like this. This is called a tangent. This is a tangent to this curve at time two seconds. And the slope of this line, the slope of the tangent, that gives us the instantaneous velocity. For this case, at time two seconds, because we have a tangent to the curve at time two. And there's a way of writing this in calculus. So when delta t, when delta t becomes almost 0, you also have extremely super small delta x, which is almost tending to 0 also. So the way of writing this in calculus is we write dx divided by dt. This is the instantaneous velocity. Now here we see that the tangent is making an acute angle with the x-axis. It has a positive slope. We know that such lines have a positive slope, the ones that make an acute angle with the x-axis. A positive slope really tells us that the instantaneous velocity is in the positive x direction. And now we will take an example. We will look at a position time graph and try to figure out the instantaneous velocity at different time instants. We will try to figure out the magnitude and the direction of the object that is moving just by looking at the position time graph. So here we have one. We have a position time graph and we have these different time instants. Two, four, six seconds, eight seconds. x-axis is time in seconds, y-axis is position in meters. And we have one value of position, which is 10 meters. And we have a rabbit who is moving towards a carrot. This graph is describing this rabbit's motion. Now let's try to figure out the instantaneous velocity of this rabbit at all of these different time instants. We already have one snapshot of the rabbit's motion at the time instant zero when time is zero. And we see rabbit at this position and the carrot at this position. If you want to figure out the instantaneous velocity at time zero, we draw a tangent to the curve at that time instant. So the tangent at time zero looks like this. And what can we say about the instantaneous velocity just by looking at this tangent? Well, we can see that the line is making an acute angle with the x-axis, which means that the slope is positive. And it has some value of the slope. We don't know what the slope is at different points. But it has some finite value. It's not zero. It's not the horizontal line. So that means that the rabbit is moving to the right. Rabbit is moving to the right, moving to the right with some speed. Its velocity is in the positive x direction to the right. So let's show that. Let's show that with an arrow. So rabbit has started moving to the right. Now at time two seconds, again, if you want to figure out the instantaneous velocity, we draw a tangent at that point. And tangent at t equals to two looks somewhat like this. So now what do we see? We see that again the line is still making an acute angle, which means the slope is positive. That means that a rabbit is still moving towards the right in the positive x direction. But now the line is steeper. It's more steep, which means more slope, which means that the rabbit's instantaneous speed is more at this time in strength. So at time two seconds, let's say this is a snapshot and if you were to draw the velocity, we were to show the velocity. This will be a bigger arrow. This will be a bigger arrow because it's moving with a greater velocity. We have a steeper slope and still moving to the right positive x direction. The slope is still positive. So nothing wrong with that. Now at time four, at time four, if you draw tangent at time four, we see the tangent is a horizontal line. So this means it has no slope. It has zero slope, which means the instantaneous velocity at this point is zero. And I forgot to mention it in the beginning. The carrot is at a position of 10 meters from the rabbit. So basically the rabbit has reached the carrot and is eating the carrot and is just standing there and eating it. So it has no velocity, no instantaneous velocity at four seconds. And then we see the graph turning and going like this. So at time six seconds, if we are figuring out the instantaneous velocity at six, we are going to draw a tangent at this point. And when we do that, this is what the tangent could look like. Now here the tangent is making, this line is making an obtuse angle with the x axis, which means it has a negative slope. And negative slope means that the direction of the velocity changed if the rabbit was moving to the right in the first part of the motion. Now it is moving to the left in this part of the motion. So the rabbit has consumed the carrot. It ate the carrot and like with some miracle, it did that in two seconds. It started moving back and we can see that the slope steep, not as steep as this one, still quite steep. So let's show that. Let's show that. This is rabbit is now, rabbit is now moving back and the carrot is gone. And it's now moving with some velocity, not as high as time two velocities, but still quite high. And it's moving in a negative x direction. And now at time eight, we see that the position is not really zero. It's not really zero. And instantaneous velocity, we can figure out by drawing a tangent and studying its slope. Now here the slope is still negative. It's making an obtuse angle and it's less steep than this one. So the speed is decreasing. The magnitude of the velocity is decreasing. And the rabbit can be at this point. This point almost reached its initial position. Still moving with some speed, very tiny amount of speed because it has almost reached its home. And this is how we can use a position time graph to describe an object's motion. Now we can try and get some more information from this graph. We know that at least from zero to two seconds, from zero to two seconds, the slope is constantly increasing. Even if you drew tangents at all of these time instants, no matter how many tangents you draw, you will see that the slope of the tangent increases between zero to two. So that really tells us that between these two time instants, the rabbit is really speeding up. It is increasing its speed. Then between two to four, we see that the slope, the slope, if you try and draw the slope, if you try and draw the slope at these different time instants, you will see that the slope decreases. It's still positive but it's decreasing. That means that the rabbit is now slowing down. It's slowing down until it comes to a state of rest at time four. And now from between four to six, again if you try and draw tangents at all of these different points, you will see that the slope, it's, the tangents look good. I mean we start drawing a tangent here, then here, then here, the slope really it's negative but it's also increasing. So that really means that the rabbit is again speeding up but speeding up in the opposite direction now because the slope is negative. And in the last case, we can say that it's slowing down. It's again, this is, this is slowing down because the tangent slope, it decreases. Still negative, so going in the opposite direction but the slope is decreasing. The line is getting less and less steep which means that the rabbit is slowing down. Now we can have, we can have snapshots of the rabbit's motion. If we can start with all of this information and draw a position time graph also, this can work both ways. But the big idea, the main idea here is that the slope of the tangent in position time graphs, it gives us the instantaneous velocity and also the sign of the slope whether it's positive or negative and the steepness of the slope also gives us with more information about the instantaneous velocity at that time instant.