Kinematic quantities: Accumulations of change

0:00- [Instructor] 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.

0:20And 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 zero to two seconds, the acceleration is minus one.

0:38So the average acceleration is minus one meters per second square and the time interval is two seconds and therefore the change in velocity of this object from zero to two seconds is minus two meters per second. Now remember this is not the velocity, it's the change in velocity.

0:57It tells me that the velocity is reduced by two meters per second. Okay, alright now the question is what does this represent geometrically? Well, let's see the minus one meters per second square over here is basically the length of this rectangle and the two second is basically the width.

1:13So 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 science.

1:35Now 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?

1:52Well, we can divide this into tiny, 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 are adding them up, okay.

2:13But 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, with this approaching zero delta T is approaching zero, this submission ends up becoming perfect.

2:37And 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(t) graph. And by the way, these are the bounds of the integral.

3:00Over here for example, T one would be zero and T two would be two. It tells me from where to where we are evaluating the area. 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.

3:16So let's start with zero to two seconds. We already computed that the change in velocity is minus two meters per second. It's basically the area of this particular rectangle. So from that, how can we plot the velocity time graph? Well, I know from zero to two seconds the velocity has reduced by two meters per second.

3:35What 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 two seconds that velocity reduces by two, which means from zero it becomes minus two meters per second.

3:56So I know my graph starts from here and ends over here, but what would that graph look like? Is it gonna be a curve? Is it gonna 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.

4:19In other words, it needs to be a straight line. So now I have figured out that the velocity time graph for zero two 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 five meters per second.

4:41Well then the graph would've started here, and at the end of two seconds it would've reduced by two from five to three. So this is what it would've 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.

4:55Depending on that the graph can shift. But anyways, for our purposes, let's assume that the initial velocity was zero. So this is the graph from zero two seconds. Okay, what would the graph look like? Let's proceed now, what does it look like from two to five seconds?

5:11What 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. Alright, so from two to five seconds the acceleration is zero. So I immediately can say that the velocity is not changing.

5:26You can also see that there is no area, so the change in velocity is zero. What does that mean? That means from two to five seconds the velocity stays, whatever it was, it's constant. It was minus two at the beginning of this, it means from two to five it stays minus two meters per second.

5:45Okay, let's proceed. What would the change in velocity look like from five to seven seconds? Well, we have to calculate the area under this graph. And what's the area over here? Well, it's two times one that is plus two. So the change in velocity now is two meters per second.

6:02So velocity increases by two meters per second from five to seven seconds. So at the end of five seconds we are at minus two. If it increases by plus two, it goes to zero. So at the end of seven seconds it needs to go to zero, which means my graph goes from here to here.

6:17And 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.

6:33But anyways, let's proceed. So seven to nine seconds. Again, there is no acceleration. Change in velocity is zero. So the velocity stays the same, it was zero at seven seconds. So until nine seconds the velocity stays zero. And then from nine to 14 seconds, what is the change in velocity?

6:51We'll calculate the area under this graph. Well how much would that be? Well, that's one, two, three, four, five. Five times one. That's five. So chain velocity is five meters per second, so it increases by five. So we are at zero by the end of 14, we need to increase by five.

7:07So we go to five meters per second as a straight line and that is how we build the velocity time graph. Alright, 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.

7:28So 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 genera what we can do to find this area is we can take tiny, tiny rectangles and we can then sum it up, and that will give you an approximate value.

7:51And then if we 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.

8:08So 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 a position time graph. Okay, alright, so let's start with zero to two seconds because I can see a nice triangle over here.

8:27What is the area over here? Well, the area of a triangle is half times the base, which is two seconds times the height, which is minus two, take care of the signs. Okay, and that ends up being just two minus two meters. So the change in the position is minus two meters.

8:48So 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. 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.

9:01That information has to be given, but let's say in this particular case it's zero. Then I know at the end of two seconds the position reduces by two meters, which means from zero at the end of two seconds it'll come down to minus two. Okay, so my graph starts from here and ends over here.

9:17But 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.

9:37It's increasing from, you know, it's reducing from zero to minus two, 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.

9:53Do you think this is the right curve? Pause and think about it. Alright, here's how I'd 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 equals zero, which is clearly not the case.

10:12The velocity at time equal to zero 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 the opposite. We are going from zero to negative value. So this is not the right curve.

10:26The 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.

10:41And 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 zero to two 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 three seconds from two to five seconds.

10:58Okay, pause and think about it. All right, so I'm gonna now calculate what the change in position from two to five seconds is. For that we need to calculate the area and that's a nice rectangle for us. So the area would be one, two, three, 3 times minus two that is minus six.

11:14So the change in position is minus six, meter position reduces by six meters further okay. So that means at the end of five seconds the position further reduces by six meters, it was already minus two. If you further reduce it by six it goes to minus eight. So by the end of five seconds we have to come here.

11:31But 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.

11:49Beautiful, right? Okay, let's do the next couple of seconds from five to seven seconds, how much is the change in position? Well that's the area under this triangle and which is the same as this triangle area. So it's half into base. So base is two, height is also minus two and half into base, into height will give you again minus two meters.

12:11So that means the position by the end of seven seconds further reduces by two meters from minus eight it'll go to minus 10. How does the graph look like? Well this time notice we should start with a very high. Yeah, we should start with a negative velocity and end with zero.

12:28So 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, alright. From seven to nine seconds, what is the change in position? Well, this time there is no area, so the area under the curve is zero.

12:49So changing position is also zero. So the position is gonna stay wherever it was. So it was at minus 10. At the end of seven seconds, until nine 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.

13:06Okay, finally, from nine 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, one, two, three, four, five. So it's five times five times half. So it's 25 divided by two, so that's 12.5. It's positive.

13:22So the changing position is 12.5 meters. That means now the position increases by 12.5. It is already at minus 10. If you add to 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.

13:43So our curve looks like this, and you can confirm the loop 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 times equals zero.

14:04The 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, and 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?

14:23Well, the graph says look for the first couple of seconds, the car has a constant negative acceleration. So the car starts accelerating then in the 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'll then have a positive acceleration.

14:43Then again, zero acceleration and then again positive acceleration. So let's see what that animation looks like. Are are you ready? The car starts accelerating in the negative 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.

15:11And 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.

15:28Okay, 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 equal zero, the velocity is zero, okay? As time passes by it's magnitude increases in the negative direction, now it's magnitude stays a constant.

15:46Now it's magnitude reduces, it goes to zero, the car comes to a stop, and now the velocity becomes positive. It's 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 it's magnitude is increasing, increasing, increasing, increasing.

16:05Once 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.

16:23Are you ready? Here it 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.

16:54And 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.