 Moving things have energy. We call this the kinetic energy. But the question is how much kinetic energy does a moving thing have? For example, if we knew the mass of this ball, let's say it was 2 kilogram and let's say it was coming in coming in at 10 meters per second, how much kinetic energy does it have? And what would that number mean? That is what we want to try and find out in this video. Now there are more than one ways to derive things, right? So over here we'll take a little different approach compared to the standard textbook derivation. We will use a definition of energy to derive things. So what is energy? Energy is the capacity to do work, right? So what is kinetic energy? Kinetic energy would be the capacity of moving things to do work. So for example, if this cricket ball can do a maximum of let's say 500 joules of work when it hits something. Okay, let's say maximum. Then that's its capacity. Capacity means maximum and therefore we would say the kinetic energy of this ball is 500 joules. On the other hand, if you find out that when it hits something it can only do let's say 3 joules of work as an example. Then that's its maximum capacity. Then that would be its kinetic energy, 3 joules. So we have to figure out how much work this ball can do when it hits something, maximum work. That would be its kinetic energy. Okay, so how do we calculate work? Well, we calculate work as force acting on an object multiplied by the displacement of that object. And we've talked a lot about this energy and work in previous videos. So if you need more clarity, great idea to go back and watch them. So let's say this ball goes and hits something. Let's say it goes and hits my hand. Let's say I'm trying to catch the ball. You might know whenever you try to catch a ball, when you catch it, eventually your hand moves back a little bit. So let's say after catching my hand has come over here. Now can you see the ball has done work on the hand? It has pushed on the hand with some force. It hits your hand with some force. Let's call it as force on the hand. And notice it has made your hand move. It has displaced your hand from here to here. And therefore it did work on your hand. And by the way, since the ball has come to a stop after catching it, that means it can't do any more work. So it has done the maximum work it could. So whatever work it does on your hand, by definition that would be the kinetic energy of this ball. Okay? So let me just write that down. The kinetic energy of this ball will equal the work done by the ball on your hand. So work done on the hand by pushing it and by moving it. And now before we substitute and see how to calculate, you know what we'll do? We'll first write these in general. Instead of two kilograms, let's say the mass of that object is m kilograms m. And similarly, instead of say 10 meters per second, let's write it in general. Let's say it's coming in with some velocity u. So that will get a nice equation. All right. So we just have to calculate how much this work is. And we know how to calculate work. It is the force in your displacement. So the work done on the hand will be the force on the hand. So that is this force on the hand multiplied by the displacement of the hand multiplied by s. So what do I do next? Well, I want to get rid of the forces and bring out bring in masses and velocities into the picture, right? Which means I need to connect force and motion. How do we do that? Well, there's one equation that connects force and motion, which you have studied earlier, Newton's second law. Newton's second law basically says that the force acting on an object or net force acting on an object equals mass times the acceleration. Hey, I can use this, right? But there's a problem. If I directly substitute f equals m a here, I will end up with the mass of the hand because you think force of this is the force on the hand. It will be the mass of the hand and the acceleration of the hand. Hey, that's not something that I want. I don't know what mass of the hand. I don't know the acceleration. I'm not interested in it. I don't want that. That's not what I want. So what do I do? Well, think about this. When the ball pushes on the hand with some force, we know from Newton's third law that the hand pushes back on the ball with an equal and opposite force. So the ball also experiences a force. I'll call it as force of the ball. And this is exactly equal to this. And it's in the opposite direction. So we can now substitute the force of the hand to be equal to the force of the ball because of Newton's third law. It's equal, but it's also opposite, right? It's also opposite. And how do we write opposite in mathematics? We write a negative sign, all right? Into the displacement. Whose displacement is this, by the way? This is the displacement of the hand, but guess what? This is also the displacement of the ball, right? Because ball also gets displaced by the same amount. So what we have done now is we have brought in the properties of the ball. We now know the force of the ball and the displacement of the ball. And that will be easier to bring in mass and the velocity of the ball. So what do I do next? Well, next, I can apply Newton's second law to this one. This is the net force acting on the ball. There's no other force acting. So I can say the force acting on the ball should equal mass of the ball times the acceleration of the ball. And we have displacement. Oops, same color. Let me use displacement. Okay, fine. Is this my final answer? No, this is not my answer because I want to bring in velocities. See, I know the initial velocity of this ball is u. I also know the final velocity is 0, right? So final velocity v, let's call it. It's also 0. I want to bring in velocities over here. I don't want acceleration. I don't want the displacement. So can I somehow connect acceleration, displacement, initial velocity, and final velocity? How do we do that connection? Hey, this brings us back to equations of motion. We've seen that in equations of motion, we can do these connections. So let's quickly recall what were the three equations of motion. Let's get rid of this over here and let's write down the equations of motion. These were the three equations of motion we had. Now we want to choose an equation which connects the velocities, the acceleration, and the displacement. Which of these three equations do that? Can you try and pause the video and check this for a while? All right, let's see. The first one has velocities, but it has time. I don't want that. There is no displacement also over here. I don't want that. Second one has displacement and initial velocity, but it doesn't have final velocity. Again, there's also time which I don't want. That's my main culprit. I don't want time. The third equation, that has final velocity, initial velocity. It also has acceleration, displacement. It has no time. This is our winner. So let's use this equation to somehow get rid of a and s and bring in the velocities. All right, let's see. Let's substitute and see what we get. Now before I do it, it would be a great idea for you to pause the video and see if you can continue this derivation yourself. Because we have what we want. We can use this to get rid of acceleration and displacement. Look carefully and somehow bring in the u into the picture. We can do that. All right, so great idea to pause and see if you can continue derivation from here. All right, let's do this. I know that v is zero, so this becomes zero. And if you look carefully, the a s is over here, which I want to get rid of. So let's find out what a s is from this equation. Okay, so let's keep a s on this side and let's put everything else on the other side. So let's subtract u square from both sides. So I'll get a minus u square over here and that equals two a s and we can divide by, we want to get rid of this two, right? So let's divide by two on both sides. And so that gets rid of the two and now I know what a s is. So a s turns out to be minus u square by two. Let's plug that in over here. So this will be minus m times a s which is minus u square by two and I have gotten rid of everything I wanted and now I have mass and velocity. Also the negative sign cancels out, that's also good. And so what we see now is that the kinetic energy of the ball is m u squared divided by two. And so in general we say the kinetic energy of any object will be half times mass times its velocity squared. Okay let's see, quickly see if it makes sense to us. So the equation is saying that if the velocity increases the kinetic energy will also increase. Does that make sense? Well yeah, right? I mean if this ball were to come and hit you with more speed you would expect it to push you more and move your hand back even more so it would do more work. It has more capacity to do work, more kinetic energy. It also says that if the mass of the object increases then also kinetic energy will increase. Does that make sense? Well let's see. If instead of a cricket ball if a bowling ball were to come and hit your hand with the same speed let's say, don't you think it'll now push with much more force and push your hand much further back? So therefore it has a much higher capacity to do work and therefore it will have more kinetic energy. So hopefully that makes sense right? All right so now that we have the exploration let's go ahead and see how much is the kinetic energy of our cricket ball. What numbers did we take? We said the mass of the cricket ball let's call it as 2 kilograms and we said the speed of the cricket ball or the velocity was 10 meter per second. All right so if we substitute over here we will get mass which is 2 kilograms and I will not substitute the units because I already know the units of kinetic energy. It's the same as that of the work joules right? But of course you can try substituting the units and checking that you'll get the same unit as well but I will not do that everything is in standard units. So two times velocity is square velocity is 10 so 10 squared divided by 2 and so the 2 and 2 cancels and we end up with 10 squared which is 100. So in our case the cricket ball has a kinetic energy of 100 joules and what does that mean? It means that this cricket ball can do a maximum of 100 joules of work when it goes and hits something not more than that. All right so to quickly summarize how did we derive the expression for kinetic energy? We said it's the capacity it has to do work and so we calculate how much work it did, maximum work it did when it hits something and comes to a stop. Then we use the formula for work and then we use Newton's laws and some equations of motion to finally arrive at the kinetic energy.