Using equations of motion (2 steps numerical)

- [Instructor] Let's solve two problems on accelerated motion, a little challenging one this time. Here's the first one. A tortoise accelerates uniformly from rest to 40 meters per second, covering a distance of 100 meters. Calculate the time taken to cover this distance. So let's think of what is to given to us and let's draw a diagram. So let's say here is the road on which the tortoise is moving and let's say here's our tortoise, okay? It starts from rest, it accelerates from rest. So, here it's at rest to 40 meters per second. So, as it accelerates, let's say a little later the tortoise is somewhere over here and it accelerates all the way to 40 meters per second, which means a little later it has some velocity. Covering a distance of 100 meters. Ooh, this means in doing so, it has traveled 100 meters of distance, okay/ Now we're asked to calculate the time taken to cover this distance. So, we need to calculate what t is, how long it took for that tortoise to go from here to here. So what do we do? Well, whenever I should look at questions like these, the first thing that would come to my mind is speed equals distance over time. It's so tempting to use this because I'm given the speed, I'm even the distance, and I need to calculate time. So can I just plug in and figure out what time is from this formula? No, we can't. And the reason we can't do this is because, this equation only works provided the speed is a constant number, it shouldn't change. In our example, notice the tortoise starts from rest. And so that means its initial speed is zero. And then it accelerates, that means speed keeps increasing and increasing and increasing eventually hitting 40 minutes per second. And so since its speed is continuously changing over here, what number would you put? If you put 40, then you know what we're doing? We're assuming that his speed was 40 throughout the entire motion, and that would be a mistake. And that's why you just can't put 40 over here. For that matter, you can put any number because it's continuously changing. And hopefully that this helps us understand why you cannot use this equation because we are in accelerated motion. Okay, so what do we do then? Well, if you look carefully it's given that the tortoise accelerates uniformly. That's the key. Accelerating uniformly means the acceleration is a constant. And we've seen before whenever we have constant acceleration, we have three equations of motion in which V represents the final velocity, U is the initial velocity, a is the acceleration, t is the time and s is the displacement. Now, if you have not seen these equations before, then we have derived them and we've talked about them in lot detail in previous videos. So you can always refer back. So anyways what we can do now is think about what's given to us, which of these variables are given to us and then think about which equation we can choose to find out what t is. So, let's see what's given to us. We are given that the tortoise starts from rest. Therefore its initial velocity is given and that is zero. Rest means not moving, zero velocity. Then we are given it travels 100 meters which means we know it's displacement at the end of this stretch, the displacement is 100 meters. We're also given the velocity at the end of the stretch, which means we know it's final velocity. Final velocity is 40 meters per second. And we are asked to calculate what t is. Now, all we have to do is look at our three equations and see which equation would be helpful given these three things to find t. So can you first try to do this by yourself? Can you see which equation you would pick to figure out what t is given these three things? Go ahead, give it a shot. Now, if you are a little confused because you couldn't pick any of these three equations, then there is a good reason for that. And that's because all the three questions have a in it and a is not given to us. Without finding the value of a, none of these equations would be useful to us. This means that even though it's not often the problem, our first job is to calculate what a is, all right? So that's the first thing to do and then we'll calculate what t is. So again, we'll now look at the three equations see, which of these three equations would help us calculate a given these three things. Again, pause the video and see which equation you would pick. All right let's see. If you look at the first equation, there is a in it, but there's also t and t is also not given to us. So, this equation is useless because we have two unknowns, we can't solve. If you look at the second question again, there's a problem. T is not given to us, we can't solve. But the third question has no t. We know the v in it, yeah we know the v, we know u and we know s yay. So, that's our winner. The third equation is going to help us calculate what a is. And so if we substitute, v is 40 meters per second. And I always like to substitute with units so that I will end up with the right units for the answer as well. So v squared equals u, u squared which is zero. So this per term becomes zero plus two times a, which we don't know, we need to calculate, times s which is 100 meters. So times 100 meters. And now from here onwards, all we have to do is just algebra to figure out what a is. So let's do that. So, let me try to get rid of this two from the right-hand. Now, usually when I used to do this before, I used to shift this to to the left hand side, that's how I thought algebra worked but that's not how it works. In algebra, there is no shifting, okay? So let me tell you what really happens. To get rid of this two, because it's in the numerator, what we'll do is on the right hand side, we'll divide by two and also on the left-hand side we'll divide by two. And now the two divides out and that's how we eliminate two on the right and it ends up being on the left. It's not that the two got shifted, okay? Don't think of it that way. That's that could be a little bit confusing. So, this is how algebra works anyways. Let's simplify 40 squared is 1,600, four squared is 16, and then two more zeros. And the units also gets squared. So meter squared divided by second squared divided by two equals a times 100. 100 meters, oops, let's use the same color times 100 meters. Let's move this thing down a little bit. We don't need the diagram anymore, it's just algebra. Okay, now again we need to get rid of 100. Again there is no shifting, what we do is we divide by 100 on both sides. So we'll divide by 100 over here the right hand side, and we'll also divide by 100 over here. So, in the denominator we get multiplied. So, this 100 divides out and now we can find out what a is. So, from this a equals, let's see. The 100 cancel, zeros, zeros cancel out, 16 divide by two is eight and a meter cancels out, be careful with the unit. The meter cancels out, we end up with meters per second squared, and that's the unit of acceleration. And so we have found what acceleration is. So acceleration is eight meters per second squared. And now that we know what a is, we can figure out what t is by choosing one of these equations. Again, great idea to pause the video and see which equation you would choose now to calculate t and see if you can again actually go ahead and calculate what the value of t is. All right, let's see. If you look at the first equation, we have v we know that u is something we know, a is also not something we know and we have to calculate t so we can go ahead and use this equation. In fact, second equations also can be used. You can use this equation because s is given, u we know, a we know we can calculate t. Third equation is useless because there's no t in it. There's no point in using the third equation. So we can use any of these two equations, but since the first equation is a little simpler, let's use that. And again, if we substitute v is 40 meters per second, that equals u is zero. So that thing becomes zero, plus a, we just found out is eight meters per second squared times t. And now I'm pretty confident you can do the algebra yourself. To calculate what t is, we need to get rid of this. So we have to divide by eight on both sides and when you do that, we get t as five seconds. I'm pretty sure you can check that yourself, five seconds. So this means if we try to fit everything in one board now, let's see. All right, if you fit everything, this means that our tortoise took five seconds to cover the distance of 100 meters. That's a pretty fast tortoise if you ask me. All right, let's do one more. Here's the second question. In fact, this is pretty similar to the first one, so go ahead and see if you can try this yourself first. All right, let's see and let's make a drawing as we read this. When the driver of a moving truck slams on the brakes. So let's draw this. Here is our truck and it's already moving. It's already moving. It decelerates uniformly at four meters per second square and comes to a stop in five seconds. Ooh, so that means it breaks and eventually it comes to a stop somewhere let's say over here, it comes to rest. And we are given what the deceleration is, that's four meters per second squared. And so we can say that the acceleration is negative four meters per second squared. Whenever the velocity is decreasing, its change is negative. And as a result, our acceleration becomes negative. So deceleration is negative acceleration and this happens in five seconds. So we're given that it takes five seconds for this to happen from here to here. We have to calculate the distance traveled while breaking. So we need to figure it out how far the bus or whatever that the truck moved. So we need to calculate the displacement over here. The displacement and the distance is the same over here, it only changes when things turn back, if this truck was to turn back then this displacement would be a little smaller than the distance traveled. Anyways, again we can think about what is given to us and what we need to calculate. So let's see. We don't know it's initial velocity, we know it's moving but we don't know it's initial velocity. We are given its acceleration, so we know what a is, that's minus four meters per second squared. We are given the time, it takes five seconds to come to a stop. And since we know it comes to a stop which means we also know the final velocity, zero. And that's all that's given, and we need to calculate what s is. And again you know the drill, (mumbles) to pause the video and see which equation you would go for to calculate what s is, all right? And again you'll see that we cannot solve this directly because this time even though acceleration is given, u is not given to us, initial velocity is not given. Why is that important? Because u is present in all the three equations as well. So without finding u, we cannot find what s is. And so this means, even though it's not asking the question our first step is to calculate what u is and the rest of the steps are very similar. So let's think about which question you would go for to calculate u. Let's look at the first equation we have v, which we know, u we need to calculate, a we know, time also we know. We can use the first equation. Just to check, we can't do the second or the third equation because in both these equations we have s which is something we don't know. So we have to go for the first equation to calculate what u is. And if you substitute which I'm pretty sure you can do this now all by yourself, you will see, u turns out to be 20 meters per second. And now that we know what u is, we can go ahead and figure out what s is. Again the same drill. You pick one equation which helps us calculate s now and let's do this quickly. First equation is useless because there is no s in it. You can use the second equation because we know everything else. We know u, t and a, all of that are given. We can also use the third equation if you want, because again v is given, u is given. So pick any of these equations and substitute. And if you do that, you find s equal to 50 meters. I'm pretty sure you can try that on yourself. You can use any of these two equations. Both of them will give you the same answer. And so the important thing is, whenever we're dealing with accelerated or decelerated motions, we cannot use speed equals distance over time because the speed is continuously changing. So, you can't put any number over here but if they are accelerating or decelerating uniformly which is the case that we will see in most cases, then you can use one of these three equations. So, all we have to do is write down what is given to us see what is asked and see which equation to choose.