Using equations of motion (1 step numerical)

- [Instructor] Let's solve two problems on accelerated motion. Here's the first one. A cheetah starts from rest and accelerates uniformly at five meters per second squared for 250 meters. Calculate the time it takes to cover this distance. So, let's think about what's given to us. And you know what? It always help to draw a diagram, so let's see. There's cheetah involved. So let's say here's our cheetah. This thing is our cheetah. And it's given that the cheetah starts from rest. So let's say this is our cheetah initially at rest. And accelerates uniformly at five meters per second squared. So it starts running, maybe, and it speeds up and so its acceleration is given to us as five meters per second squared. And it does that for 250 meters. So let's say it's running towards the right. A little later it would have come somewhere over here and now it's a little fast. And let's say this distance is that 250 meters. We need to calculate the time it takes to cover this distance. So, how long does our cheetah take to go from here to here? That is the question. How much time? Now, my initial thoughts would be can I just go ahead and use the formula speed equals distance over time? This is the first thing that comes to my mind when you're talking about speed and distance and time. So, can I use this to calculate time is the question. And the answer is no, I can't. The reason I can't use this is because this is only useful if speed was a constant. If speed was a constant number. But notice in our example the speed is continuously changing. In this case the speed is increasing because the cheetah is accelerating. It doesn't have a single speed. And since the speed, this number is changing what number would you put over here to calculate distance and time? You can't, right? So this formula is only useful if this speed is a constant, then you can put some number here. But if the speed is changing, well, we can't use this formula. So, what to do? How do I calculate the time? Well, we're in luck because we're talking about acceleration uniformly. And you've seen before, whenever the acceleration is a constant uniform, whenever the acceleration does not change, it's the same over the entire stretch. Then we have three equations of motion where V is the final velocity . U is the initial velocity. A is acceleration. T is time and S is the displacement. Now, if you're not familiar with these equations then we have talked a lot about them. Even derived them in previous videos. So, you know what? It will be a good idea to go back and watch that video. You can refer to that. But these three equations will work whenever our objects are going at constant acceleration. So, you know what we'll do? We'll just think about which of these variables are given to us and then we will see which equation we can use to calculate time. So let's think about this. We know that in this entire stretch the cheetah starts from rest. So, we know it's initial velocity: zero. Because it's at rest. U represents the initial velocity, so we know U is zero. We also know the acceleration. The acceleration is five meters per second squared. The only thing to be careful about the acceleration is it can be both positive and negative. Whenever the speed is increasing, we say it's positive and when the speed is decreasing we will say the acceleration is negative. And so here, since the speed is increasing, it's a positive five. Okay, what else do we know? We know that displacement S. We know that S is 250 meters. And I think that's about it, right? That's all that's given. U, S and A. That's it. And we are asked to calculate what T is. So now, given these three things, we can go to our equation list and think about which equation to choose to calculate what T is. So, can you try this one yourself first? Go ahead, give it a shot! Pause the video and see which of these equations you choose to figure out what T is. Give it a shot. All right, let's see. If you look at the first equation, there's a T in it. So we can use that equation to calculate T. But the problem is there's also V. That's the final velocity. The velocity over here and that's not mentioned to us. We're not given that number, and so because V's also not given, we can't use that equation. Let's look at the second one. We have T in it, so that's good because we want to calculate T. We have S, which is given to us. We have U, which we know. We also have A, which we know. So, you know what? We can go ahead and use this equation. Let's check out the third one. The third equation doesn't even have T in it. So this is useless for us, right? Because there is no T. What's the point? And so we have a winner! We can use S equal to U T plus half A T square and see what T is. So, if we substitute, S is 250 meters. That equals U, which is zero. So this part becomes zero. Plus half. Times A which is 5 meters per second squared. I usually like to put in the units so that we will get the answer with the units. So five meters per second squared into T squared. And now, to calculate T square, we just have to do some Algebra. So again, if you've not done this before, good idea to pause the video and see if you can do this Algebra and figure out what T is. All right, let's do this. Since I want what T is, I need to get rid of these things. Now, you know earlier what I should do? I should think that, you know, to get rid of this say two in the denominator, I have to shift this up to the left. But that's not how Algebra works. Here's how the Algebra works. To get rid of this two, because it's in the denominator, I will multiply by two on the right side and also multiply by two on the left side. Now that I've multiplied this two and two, it's canceled. And that's how there's a two that ends up on the left side. It's not that the two got shifted up, this is how it works. There's no shifting business, okay? All right. So anyways, once we multiple we get now two times 250 is 500. So this becomes 500 meters. Equals... this two is gone. Five meters per second squared times T squared. And now again, I need to get rid of five meters per second squared. Again, there is no shifting. We don't shift. Now what we do is, because this five's in the numerator, and I want to get rid of this, I divide by 5 meters per second square on the right hand side. And, of course, if I do that on the right-hand side, I should also do it on the left-hand side. And again notice this cancels out. And now I'm left only with T squared. That's what I wanted. So, let me write that T squared to the right side. So, let me write it over here. So, T squared equals... let's see what we end up with over here. So 500 divided by 5. The five cancels and we get 100. Five goes one time and you get 100. So, T squared becomes hundred. And let's see what happens with units. Be careful with the units. There's a meter here and there's a meter in the denominator that cancels. And we have one over second square in the denominator that ends up being second squared in the numerator. You can just pause and just check that. And so anyways, we end up with this and now since you want to calculate what T is, well there's a square, so we can take square root on both sides. So if we take square root on both sides, we will get T equals square root of 10. 100 is 10, square root of S squared is S. So, there we have it. That's our answer. So, this means the cheetah took 10 seconds to cover that distance of 250 meters. Let's do one more. Can you first read this question and try to solve this all by yourself? Again, pause the video and give it a shot. All right, let's see. A train moving at 20 meters per second. So, let's quickly draw this as we read. So you imagine here's our train. And this train is already moving, let's add this way, at 20 meters per second. Brakes and decelerates uniformly at eight meters per second squared. So, it's braking. It's slowing down. And so there's a deceleration and that number is 8 meters per second squared. So, it's decelerating at this rate. And whenever things are decelerating, we say that it has a negative acceleration. Okay? So it's slowing down. How much distance does it travel before coming to a stop? So, as it slows down, let's say, imagine a station has arrived. A slows down, slows down, slows down. Eventually, it'll stop right? Somewhere over here let's say it stops. So, it comes to rest. The question now is how much distance does it travel before coming to a stop. So we need to calculate what this is. How far does it travel? And that's S for us. Okay. So, how do we do this? Same thing. It's a uniform acceleration. It's deceleration, but the point is it's uniform, so we can use one of these equations. So, let's see what's given. We know the initial velocity because we know initially it's at 20 meters per second. So, that's twenty. We know the accelration:A. That's minus eight meters per second squared. What else is given? We know finally it comes to rest. So that means we know it's final velocity this time. That's zero. And we have to calculate what S is. Again, you know the drill. Pause the video. See which equation you would choose to figure S out. All right, let's look at this. The first equation is useless because it has no S in it. The second equation, although it has S, it also has a T. Which is not given to us, we don't know the time. And so this is useless. The third equation has a V which is good. U which is good. A which is good and S, we want to find out. Yay, we have our winner! The third equation! So again, we have to just substitute. V is zero, so V squared is also zero. Equals U squared. Let's use the same color. Twenty meters per second the whole squared. Plus two times A. We have to be careful! It's negative eight meters per second squared times S. And I'm pretty sure you can do the Algebra yourself, right? So, you're gonna again pause the video and do the Algebra and just to save time, I'll just tell you what you get. If you do the Algebra, you end up with S equals 25 meters. So, go ahead. Try this on your own. Convince yourself that you actually get 25 meters. And this means the train traveled 25 meters before coming to a stop. And so the important thing, is that whenever we are given the objects are accelerating or decelerating uniformly, in such cases we cannot use this formula: speed equals distance by time because the speed is continuously changing. But we can use one of these three equations. So, we'll figure out what is asked, what is given to us and then choose what equation to use.