Calculating displacement from v-t graphs

- [Teacher] My cousin told me that her baby girl started walking yesterday. And I was so excited I asked her did you record it? How far did she walk? And she said yes I recorded it. And I was like show me, show me! And she showed me this. A velocity time graph of her baby walking. So, let's see if we can use this graph and figure out how far she traveled. We've seen velocity time graphs before. It's a graph that tells us the velocity of an object at any time. It's like looking at a speedometer. So, imagine we had speedometer attached to the baby. Let's see what the graph is saying. The graph says at time equal to 0 at time 0 the baby has a speed of 20 meters per second. So, it's already starting at 20 meters per second. And then as the time passes notice the speed remains a constant. It does not change. It remains the same until five seconds. But then notice what happens. Then the graph goes down. Which means the speed decreases. Can you see that? I mean if you were to go to somewhere like six seconds and you'll go up and you go to the left, can you see that the speed is decreasing? So, the speed keeps decreasing decreasing until at 10 second the speed goes to zero. So, if we could look at that speedometer at times zero it will be 20 meters per second and it will stay like that for the first five seconds. And then it will start decreasing and decreasing and at the 10 end of 10 seconds it goes to zero. So, now we can kind of even visualize what that baby is doing. Imagine here is our baby girl. She just learned to walk. So, let's attach this speedometer to her. Let's see what she's doing. At times zero she already has 20 meters per second speed. I don't know how that happened. That information is not there what happened before that. But that's she already has a speed of 20 meters per second. And so she continues with that for five seconds. That's for the first five seconds and then she slows down slows down and comes to rest at the end of 10 seconds. And our goal now is to figure out how far she has reached from her initial location. So, how do we calculate it? Well, we can start with what we know about velocity. We know velocity is displacement over time. And by the way if you're wondering why I kept using the word speed earlier, well velocity is just speed with direction. Which means if I said the kid has moved at 20 meters per second, that's speed. If I say it's moving at 20 meters per second upwards, that's velocity. So, don't worry too much about that. But anyways, we know velocity is displacement over time and we need to calculate how far that kid has traveled. That means we need to calculate the displacement. So, we can multiply by time on both sides of this equation and we will get displacement as velocity into time and we can calculate, right? Just one problem. You see, this is easy to calculate if velocity is a constant. Like for the first five seconds. For the first five seconds I can plug in velocity as 20, time as five, and calculate it. No problem. The problem comes for the next five seconds. Because the velocity is changing, it's decreasing. If it's decreasing, what number will I plug in? And how will I calculate displacement? That's the big question we wanna try and answer over here. So, you know what we will do? We will worry about that a little bit later. Let's first calculate form zero to five seconds and see what we get. So, let's calculate the velocity or displacement sorry from zero to five seconds. From zero to five seconds the velocity is a constant which is nice. So, if I plug in velocity is just 20 meters over per second, multiplied by time which is five seconds, that will give me... Second cancels, 20 times five is 100. I get 100 meters. So, that's the displacement in the first five seconds. But now, what to do for the next five seconds? From five to 10 seconds, how to calculate? That's the big question, right? Well, here's what we'll do, we'll try to figure out what this means in the graph. So, think about this, 20 meters per second, what is that in the graph? What have we calculated? What does that 20 represent in our graph? Well, in our graph if you just focus on this rectangle, because you're only calculating from zero to five, that 20 represents the length of this rectangle, right? This is that 20. And what does this five represent? Well, that represents the breadth of this rectangle. And so when you multiply and we get 100, what does that 100 represent over here? In other words, what do we get in a rectangle when you multiply length by breadth? You know this one. It's the area. When you multiply these two yo get the area and that area is 100, right? That's our 100 meters. This means to calculate displacement we just have to calculate the area under the graph. That's how you do it. Why does it work? It works because displacement is the product of velocity and time. And in our graph when you multiply velocity and time you're basically multiplying two lengths in our graph and that gives us the area. And so that's the secret to calculating displacements and from a velocity time graph. We just calculate the area under it. So, from five to 10 seconds, I know we were going to use the formula, instead I just go to the graph and I'll say, the area from five to 10 seconds, that represents this area. Okay, let me use the white again. This area. That represents the displacement from five to 10 seconds. So, we can calculate that area. In fact, I'm pretty sure you can calculate that area as well. So, you know what, pause the video go ahead see if you can calculate the video this area. All right, let's do this. So, that area which represents the displacement is going to be half it's area of rectangle area of a triangle is half into base into height. So it's going to be half into base, and the base is this length which is five, into height, and the height is this length, again 20. So, this is five seconds into 20 meters per second. And what is that equal to? That is equal to let's see second cancels, two goes 10 times, you get 50 meters. Tada! Done. And so this means the total displacement is 100 plus 50 which is 150 meter. So, somehow this kids ended up walking 150 meters in 10 seconds. All right, let's try another. We have another velocity time graph totally different and we are asked to calculate the displacement from two to four seconds. Again, if you look over here notice that the velocity is changing. It starts with zero and then it's speeding up. It's speeding up. So, we're not gonna use this formula directly. Instead we will calculate the area under this graph because that represent displacement. And since we are asked from two to four seconds we should only calculate the area from two to four under this graph. How do you do that? Well, two is here, four is over here. So, we calculate area only in that section. That represents the displacement from two to four seconds. So, this area gives us the answer. So, you know what? Can you give it a try? See if you can somehow figure out the area of this shape. All right, let's do this. So, what shape is this? This is a trapezium. So, we need to use the formula for the area of trapezium. Now, if you don't like trapeziums another thing which we can do is we can draw a line over here and calculate this as the sum of two areas. One triangle and a rectangle. So again, if now you feel that okay now you can do this again pause the video and try. All right, let's do this. Let's do this. So, the displacement is going to be that total area which is the area of the triangle. What's the area of the triangle? It's half into base... What's the base of the triangle? Well, that's two. Two seconds. Two seconds into height. What's the height of that triangle? If you look carefully this represents the height, right? This one. What is that equal to? Well, if you look to the left this is 40, this is 20, so this height is 20 meters. 20 meters per second. 20 meters per second plus... That's the area of the triangle. Now we have to do the area of the rectangle. Again, it's going to be length into breadth. Let's start with the breadth. Breadth is again two, so that's two seconds, into length. Into length. Again, this length. What is that equal to? That's this length, there is this length which is zero 20, that's also 20. 20 meters per second. And so we can go ahead and calculate now. This two cancels and the second cancels and we get 20 meters over here plus you get two times 20 that's 40 meters over here. And so total displacement is 60 meters. And so in this case displacement from two to four second is 60 meters. So, I guess the moral of the story is if your cousins ever provide you a velocity time graph representing their kids in motion, then you can figure out how far they walked just by calculating the area under that graph.