 Here, we have a car which starts from rest at time 0. We have taken snapshots, pictures of this car at different time instants. At time 2 seconds, 4 seconds, 6 seconds, 8 seconds. Let's combine all of them and see what they look like. So here it is at T, 2 seconds, here it is at 4 seconds, this is 6 and this is 8 seconds. We see that the distance that the car is covering in 2 seconds, in an interval of 2 seconds, it keeps on increasing. So that means that the velocity with which the car is moving, even that must be increasing. Let's try and show this type of motion where the velocity is changing and in this case it is increasing. Let's try and show this motion on a position time graph. So on the x-axis here we have time, we can add 2, 4, 6 and 8 seconds. And let's try and see where the car would be at different time instants. So at T equals to 0 when the motion is just beginning, it is at the origin. At T equals to 2 seconds, if we try and trace where the car is, the car could be at this point, this position. At T equals to 4 seconds, this is where it would be. This is where it would be at T equals to 6 and this is where it is at T equals to 8 seconds. Let's also write some positions on the y-axis. This does not mean that the car at 2 seconds is at a position of 40 meters or then at 4 seconds it is at a position of 80 meters. These are just random numbers which are put to introduce a scale. So now if we try to join the different positions of this car at different time instants, let's see how the shape would look like. First let's make these dotted lines disappear. Okay, so if we join this, there it is, if we join this. Now we don't really see a constant straight line, we see different straight lines but the slopes are increasing. So this is a straight line with some slope, this is again a straight line with a slope but the slope of this line is more than more than the slope of this line and we see lines getting steeper and steeper. Now let's say if we try to took snapshots, if we try to take pictures after every 1 second. So now we have a smaller time interval after which we are taking a picture, after which we are taking a snapshot and that interval is just 1 second now. This is where the cars will be at these different time instants. Now here again if we try to join all of these positions, we keep on getting straight lines but the slope of these straight lines is again increasing after every 1 second. Or we can say that the line has gotten slightly smoother, not smooth enough because the line is still taking a sharp turn after every second. But if we kept on reducing the time interval after which we took a snapshot, if we had some super powerful camera which could take a picture, a snapshot after a very, very small time interval, then we will get the position of the car after extremely small time intervals, could be 0.0002 seconds, something extremely super small. And then if we joined all of those points, then we can get a smooth curve, we can get a smooth curve like this. This position time graph shows that the motion of the car or the motion of any object is changing. It's moving with the velocity which is changing and in this case it is increasing. Now what if we had a case which looked like this, here the case is exactly the opposite of the previous one. Why don't you pause the video at this point and try to imagine what could the graph, what could the shape of the position time graph look like for this kind of motion. Alright, hopefully we have given this a shot. Now here we see that the distance that the car is travelling after every 2 seconds that is decreasing. It's covering lesser and lesser distance in the same amount of time, which means that the velocity with which it is moving, which means that the velocity with which it is moving, even that is decreasing. The velocity is still not constant, but now it's not increasing, it's decreasing. And because the velocity starts from some value, so it's not really at rest at time equals to 0, it's going towards the state of rest, it's going towards the point where its velocity is 0 as time progresses. So if we try and show this type of motion on a graph, we can follow the same approach, we can try and trace the positions of these cars at all of these time instants at 0, 2, 4, 6 and 8. So this is how they could look like, at t equals to 2, this is where it is, this is where it is at 4, this is at 6 and this is at 8. Now if we try to join these positions, and again if we took snapshots of pictures at smaller time intervals, we can get more data, we can get more data on the position of the car at all of those time instants. So if we try to join all of those points, we don't really see all of them on the screen right now, but again if you try to join them, we will get a smooth curve and the smooth curve would look like this. The blue curve over here shows that the velocity is not constant and it is decreasing. But how do we talk about velocity in this case? Velocity is constantly changing, how do we give some numbers to the velocity? At least when we had a straight line, we knew that the velocity was not changing, it was one value, it was a constant and we could calculate that velocity by figuring out the slope, the slope of the straight line. So if this point, if this point was let's say, let's say this is x2, y2 and this point is let's say x1, y1, then the slope of this line, the slope of this line was y2 minus y1 divided by x2 minus x1. Or we can also write this as delta y by delta x. This really told us the rate of change of y with respect to x. In the context of a position time graph, this told us the average velocity because this would just become delta x by delta t. And average velocity is the same throughout when the graph is a straight line, the slope is not changing. It's just one velocity with which the car is moving, so that just becomes the average velocity. So how do we talk about average velocity when we have a curved line? Let's try and remove this part first and even this. Now let's say that we are interested in figuring out the average velocity in a time interval of delta t of let's say 5 seconds from 3 to 8. So going by the same approach, delta t would be 5 and delta x would be the position at time 8 minus the position at time 3. So let's see how that would look like. Here, here, this right here is, this is delta x and this right here is delta t, which is which is 5. And delta x, we can say that this is 160 minus 40, so 120. And this might not really be 160, we're just assuming it to be as 160 at 8. Let's just assume that maybe it's 140, 150, we don't know. And again, at time, because of 3 seconds, this could be less than 40, more than 40. Again, we don't know. These are just random numbers. And we're just trying to figure out the average velocity in this time interval. So delta x by delta t, this is nothing but the slope of this line. So the average velocity between time t equals to 3 to 8 seconds, the average velocity, this comes out to be equal to, this is delta x by delta t. So this is 120, 160 minus 40 divided by delta t, that is 5. So comes out to be equal to 24, 24 meters per second. The average velocity between time 3 to 8 seconds is 24. We can try and take a different time interval. Let's take a time interval of 4 seconds from 3 to 7. In that case, we really need to find the slope of this line. So this would be, this is, we need to figure out this delta x. So that would be 120 minus 40 divided by, divided by delta t, which is 7 minus 3, that is 4. So delta x by delta t in this case, that is 120 minus 40, that is 80 divided by 4, around 20 meters per second. Notice how we are getting different average velocities for different time intervals. 24 meters per second and 20 meters per second. And turns out if we took a time interval of 2 seconds, we will get some different average velocity and it also depends which time interval are we taking. For example, for this 4 second time interval, if we took something from like, I don't know, from 1 to 5 seconds, maybe we would have gotten a different average velocity. There is one more way of talking about velocity when it comes to curves. There is one special quality about these curves. The velocity is changing at every, at any given time instant. It's continuously changing. It's never the same as it was in the past. So talking about average velocity does give us some idea, but it does not really tell us the velocity of the car at any given single time instant. Let's say I'm interested in figuring out the velocity at 3 seconds. Average velocity will not tell me what the velocity is at 3 seconds. But we saw over here, when we decreased the time interval, when we went from 5 seconds to 4 seconds, this point, this point came closer, this point right here, it came closer to this point. We can see how much it came closer. So if we keep on reducing the time interval, if we push delta T towards 0, if we make it extremely small, if the time interval is almost 0, then the two points will keep on coming closer to each other, closer and closer to each other. And when delta T is almost 0, they are almost on top of each other. And your line does not really cut the curve in two parts. In fact, it starts touching the curve at just one point. So if we are interested in figuring out the velocity at 3 seconds, what we really need to do is figure out the slope, figure out the slope of this red line, which is touching the curve just precisely at one point, which is at time 3 seconds. This velocity at any given time instant is called instantaneous velocity. And we will talk more on this in other videos.