18:16
If we are given an acceleration time graph, how can we use that to plot the velocity time and the position time graph? That's what we want to explore in this video. So let's begin. Okay, so where do we start? Well, we know that average acceleration is basically rate of change of velocity. And if we rearrange this equation, we get change in velocity as average acceleration multiplied by the time interval. So look I can use that to calculate how much the velocity of an object has changed. So for example if I consider from 0 to 2 seconds the acceleration is minus1. So the average acceleration is -1 m/s squared and the time interval is 2 seconds and therefore
the change in velocity of this object from 0 to 2 seconds is -2 m/s. Now remember this is not the velocity it's the change in velocity. It tells me that the velocity is reduced by 2 m/s. Okay. All right. Now the question is what does this represent geometrically? Well, let's see. The minus 1 m/s squared over here is basically the length of this rectangle and the 2C is basically the width. So look when we multiply these two we are multiplying the length with the width. We are getting the area. So this represents the area of this rectangle which means the area under the acceleration time graph gives us change in velocity but important it is a
signed area. We have to take care of the signs. Now this is awesome because look at what this means. Okay here the shape was a nice rectangle. Okay so we could directly use algebra to figure this out. But what if it wasn't a nice shape? What if we had a shape that looked like this? Now, how would we calculate the area over here? Well, we can divide this into tiny rectangles. Calculate the area of each of these tiny rectangles and then add them up. In other words, we now have to do a summation. This is the acceleration at each point and you're multiplying the acceleration with the tiny width of each rectangle and then we're adding them up. Okay? But of course, this is not an exact value because look, there are gaps over here.
So this is an approximation. Now that approximation will get better if we use tinier and tinier width and use more rectangles and you do more summations. And then eventually in the limiting case where the width is approaching zero, delta t is approaching zero, this summation ends up becoming perfect. And now we call this an integral to be specific a definite integral. So this means in general integrating the acceleration over time gives you change in velocity and what it represents geometrically is basically the area under the a graph. And by the way these are the bounds of the integral over here. For example t1 would be zero and t2 would be two. It tells me from where to where we're evaluating the area.
Okay. Now that we know how to figure out the changes in velocity, let's see if we can use that in our original problem to plot the velocity time graph. So let's start with 0 to 2 seconds. We already computed that the change in velocity is minus 2 m/s. It's basically the area of this particular rectangle. So from that, how can we plot the velocity time graph? Well, I know from 0 to 2 seconds, the velocity has reduced by 2 m/s. What I don't know is what the initial velocity is. That information is not given here. So that is something that we need to have. And let's assume in this particular example that the initial
velocity was zero. Then I know at the end of 2 seconds that velocity reduces by two which means from zero it becomes -2 m/s. So I know my graph starts from here and ends over here. But what would that what would that graph look like? Is it going to be a curve? Is it going to be a straight line? How do I figure this out? Well, remember that the slope of this graph gives me the acceleration, right? And I know that the acceleration is a constant, which means the slope needs to be a constant. In other words, it needs to be a straight line. So now I have figured out that the velocity time graph for 02 seconds should look like this. Constant negative slope as you can see.
Okay. And by the way, what if the velocity was not zero? What if the velocity was let's say initially, let's say it was 5 m/s. Well, then the graph would have started here and at the end of 2 seconds, it would have reduced by two from five to three. So, this is what it would have looked like. Okay? So, depending upon the initial velocity, which is not given over here, just to be clear, it is something that we need to know. Um, depending on that, the graph can shift. But anyways, for our purposes, let's assume that the initial velocity was zero. Okay, so this is the graph from 0 to 2 seconds. Okay, what would the graph look like? Let's let's let's proceed now. What does it look
like from 2 to 5 seconds? What is the change in velocity over here? Why don't you pause the video and think about this and then see if you can you know continue this graph. All right. So from 2 to 5 seconds the acceleration is zero. So I immediately can say that the velocity is not changing. You can also see that there is no area. So the change in velocity is zero. What does that mean? That means from 2 to 5 seconds the velocity stays whatever it was it's constant. It was minus2 at the beginning of this. It means from 2 to 5 it stays -2 m/s. Okay, let's proceed. What would the change in velocity look like from 5 to 7 seconds? Well, we have to calculate the area under this graph. And what's the
area over here? Well, it's 2 * 1 that is + 2. So the change in velocity now is 2 m/s. So velocity increases by 2 m/s from 5 to 7 seconds. So at end of 5 seconds we at minus2. If it increases by plus2 it goes to zero. So at the end of 7 seconds it needs to go to zero. Which means my graph goes from here to here. And again since the acceleration is a constant the slope of this graph needs to be a constant. Does that make sense? Therefore it needs to be a straight line. Okay let's proceed. Again feel free in between to pause and you know continue and finish this yourself. But anyways let's proceed. So 7 to 9 seconds again there is no acceleration. Change
in velocity is zero. So the velocity stays the same. It was zero at 7 seconds. So until 9 seconds the velocity stays zero. And then from 9 to 14 seconds what is the change in velocity? Well we'll calculate the area under this graph. Well how much would that be? Well that's 1 2 3 4 5. 5 * 1 that's five. So change velocity is 5 m/s. So it increases by five. So we are at zero. By the end of 14 we need to increase by five. So we go to 5 m/s is a straight line. And that is how we build the velocity time graph. All right. Now that we have the velocity time graph, how can we use this to predict what the position time graph looks like? Well, we know that average velocity is the rate of change of position. So if we rearrange this
equation, we will get the change in position as the product of average velocity and time interval. This is just like before this represents the area under the velocity time graph. And just like before in general what we can do to find this area is we can take tiny rectangles and we can then sum it up and that will give you an approximate value and then if you take the limiting case where the delta t is approaching zero you get the perfect value and that represents the integral. So look integral of the velocity over time gives you change in position and geometrically it's the area under the velocity time graph. So I can use that to calculate how much the position has changed and
from that I can figure out what the position time graph looks like. So let's go ahead and use this to draw our position time graph. Okay. All right. So let's start with 0 to 2 seconds because I can see a nice triangle over here. What is the area over here? Well, the area of a triangle is half time the base which is 2 seconds time the height which is minus2. Take care of the signs. Okay. And that ends up being just 2 - 2 m. So the change in the position is -2 m. So what does that mean? How do I draw the graph? Well, again, I need to know what the initial position is. That's not given to me. All right. So let's assume that the initial position is zero. There's no reason for it to be zero. We're just assuming it or it
needs to be given to you. That information has to be given. But let's say in this particular case, it's zero. Okay. Then I know at the end of 2 seconds the position reduces by 2 meters which means from zero at the end of two seconds it'll come down to minus2. Okay. So my graph starts from here and ends over here. But the question is what does that graph look like? Is it a straight line? Is it a curve? Well can it be a straight line? Well remember this time the slope of the position time graph gives us the velocity. If it was a straight line the slope would be a constant. That would indicate the velocity is a constant. But clearly the velocity is not a constant. It's increasing from you know it's it's reducing from 0 to minus2.
Therefore it cannot be a straight line. The slope needs to be changing and therefore it has to be a curve. So what kind of a curve would it look like? Well let's just try. Maybe the curve looks like this. Do you think this is the right curve? Pause and think about it. All right. Here's how I like to check it. Think about what the slope over here looks like. Well, we'll draw a tangent line and think about its slope. Look, it has a very high negative slope, which means this represents a very high negative velocity at time t=0, which is clearly not the case. The velocity at time t equal to 0 is zero. So, clearly immediately I know this is wrong. And look, the slope goes from some negative
value to zero, which is not what's happening over here. It's exactly opposite. We're going from 0 to negative value. So, this is not the right curve. The right curve should be in such a way that it should start with zero slope and then it should have some negative slope at the end over here. And that curve looks like this. Again, we can double check over here. See, initial slope is zero. So, the velocity is zero. And then look what happens to the slope as we go forward. It becomes steeper and steeper in the negative direction. And that's exactly what we want over here. So, from 0 to 2 seconds, that's what the
graph looks like. Okay? Feel free to pause the video and think about what the graph looks like for the next 3 seconds from two to 5 seconds. Okay, pause and think about it. All right, so I'm going to now calculate what the change in position from 2 to 5 seconds is. For that, we need to calculate the area and that's a nice rectangle for us. So the area would be 1 2 3 * -2 that is - 6. So the change in position is - 6 m. Position reduces by 6 m further. Okay. So that means at the end of 5 seconds the position further reduces by 6 mters. It was already minus2. If you further reduce it by six it goes to minus 8. So by the end of 5 seconds we have to come
here. But again how does the graph look like? How does it go from here to here? Well this time notice the velocity is a constant meaning the slope of the position time graph should be a constant. Constant slope is a straight line. So that means the graph is going to be a straight line. Beautiful, right? Okay, let's do the next couple of seconds. From 5 to 7 seconds, how much is the change in position? Well, that's the area under this um triangle and which is the same as this triangle area. So, it's half into base. So, base is two. Height is also minus2 and um half into base into height will give you again -2 m. So, that means the position by the end of 7 seconds further reduces
by 2 m. from 8 minus 8 will go to minus 10. How does the graph look like? Well, this time notice we should start with a very high we should start with a negative velocity and end with zero. So we start with a negative slope and the slope should go to zero. And so the graph would look like this and we can confirm the slope is going to zero. Okay. All right. From 7 to 9 seconds, what is the change in position? Well, this time there is no area. So the area under the curve is zero. So change in position is also zero. So the position is going to stay wherever it was. So it was at minus 10. At the end of 7 seconds at the until 9 seconds, the position will stay the same. So your graph will
just be a straight line that looks like this. The slope remains zero because velocity remains zero. Okay. Finally, from 9 to 14 seconds, how much is the change in position? We'll evaluate the area under this big triangle. And how much would that be? Well, 1 2 3 4 5. So it's 5 * 5 * half. So it's 25 divid by two. So that's 12.5. It's positive. So the change in position is 12.5 m. That means now the position increases by 12.5. It was already at - 10. If you add 12.5, it would go to 2.5. So by the end of 14 seconds, our graph should end up over here. And how should that graph look like? Well, the velocity is increasing. So the slope should go from zero to some positive value. So our curve looks like this. And you can
confirm the slope increases. Okay. So let's see if we can visualize this. Let's say that these graphs represented the motion of a car. Okay. So what would the motion look like? Let's assume the car's initial position is over here and its initial velocity at time equals zero. The velocity is zero. Let's also assume that the forward direction is positive for positions, velocities and acceleration and the backward direction to be negative. Okay, this is just a convention. We are free to choose our own conventions. But under these circumstances, under these conventions, sorry, what is the graph saying? Well, the graph says look for the first couple of seconds the car has a constant
negative acceleration. So the car starts accelerating in a negative direction. Then for the next three seconds it has zero acceleration meaning it moves with constant velocity. For the next couple of seconds it then have a positive acceleration. Then again zero acceleration and then again positive acceleration. So let's see what that animation looks like. Are you ready? The car starts accelerating in the negative direction. And now its acceleration goes to zero. It's moving with constant velocity. But now look the car is still moving in the negative direction. So its velocity is still negative but the acceleration has become
positive. Since acceleration, velocity are in the opposite directions, the car slows down and eventually it comes to a stop. The acceleration goes to zero. And now look, the acceleration becomes positive again. But this time even the velocity is positive because the car is also moving in the positive direction with both pointing in the same direction. This time the car speeds up. So it keeps speeding up. The animation is a little choppy, but you get the point. Okay, let's repeat the same animation, but this time let's concentrate on the velocity. Again, the forward direction is positive velocity, backward direction is negative velocity. So at time t equals 0, the velocity is zero. Okay, as time passes by, its
magnitude increases in the negative direction. Now its magnitude stays a constant. Now its magnitude reduces. It goes to zero. The car comes to a stop and now the velocity becomes positive. Its magnitude keeps increasing. So the car keeps speeding up. This time let's look at the position. Okay, as time passes by look, its position is negative and its magnitude is increasing, increasing, increasing, increasing. Once it hits minus 10, it stays a constant for a while and then it decreases. The magnitude decreases, decreases, decreases, goes to zero and then it increases again in the positive direction. Now we'll look at the same animation one final time, but we'll look
at all the three vectors together. Are you ready? Here goes. There it is. Acceleration, velocity, and positions. All three. Cool. So, long story short, when you have an acceleration time graph, if you find the area under the curve, you get the change in velocity. This also represents the definite integral of the acceleration over time. Using these changes in velocity, we can plot the velocity time graph. And now if you calculate the area under the velocity time graph, you end up with changes in position. And that represents the definite integral of velocity over time.
And using these changes in position, we can plot the position time graph.
