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Hey guys, So for this video, I want to cover the conservation of mechanical energy, which is probably one of the most important principle that we learn in all of physics. So you're going to use this whenever you're asked to calculate the mechanical energy as an object or system, which we'll talk about in just a second moves between two different points like we have in this example right here. So let's check this out. The whole idea is that we have this ball that's at 100 m and it's going to fall down to the ground like this. So the Y final is gonna be zero and we want to calculate the mechanical energy before and after. So let's take a look here for part A We're going to calculate the mechanical energy at the top of the building. Remember the mechanical energy of any object or a system is really just gonna be the sum of kinetic and potential energies. K plus you we're gonna talk more about systems in a later video for now. A system could be as simple as our ball right here. Okay, so really the mechanical energy initial it's just gonna be the kinetic initial plus the potential initial. We can expand out both of these terms because we know what those equations are. So this is gonna be one half mv initial squared plus MG Y initial. Alright, so, we can see here is that if we're dropping this ball from this height, the initial velocity is going to be zero. Which means that we can actually eliminate our kinetic energy initial term. So this V initial is going to be zero. That whole term goes away. So all of our mechanical energy initial is actually gravitational potential. And we can calculate this because we have all the numbers this is 29.8 and this is 100. So you get an initial gravitational potential of 1960 which means that your total mechanical energy initial Is 1960 jewels. All right, let's move on to the second part. For the second part. We want to calculate now the mechanical energy final once it reaches the bottom. So when it reaches the bottom here, right before it hits the ground, it has all of this velocity here, which means has some kinetic energy and our final height is going to be zero. Right? So, we're just gonna expand the terms exactly how we did this before. So this is going to make a final plus you final. Which is going to be one half M the final squared plus MG Y. Final. So, now what happens in the mechanical energy? Final term is because our why final is going to be zero. We no longer have any gravitational potential energy. We're not we don't have any heights. So what happens is we do have some kinetic energy but no gravitational potential. So now all of our mechanical energy is now kinetic energy. To calculate this, we're gonna have to figure out what this v final is and I'm gonna go ahead and just give you the shortcut. This really just comes down to a vertical motion equation. So we're just gonna have to write our five variables. We're gonna have to pick out an equation. Just remember this equation right here and we get a speed of 44.3. So we can do is I can basically say this is one half of two times. This is going to be 44.3 squared. And what you get is the mechanical energy final. Once you plug it into your calculator and your round is gonna be 1000 jewels. Okay so we get the same exact number here. The mechanical energy at the top is the same as the mechanical energy at the bottom. So what happened basically what happened is that we had our mechanical energy initial which was all gravitational potential. And it basically just became all kinetic energy at the bottom. But the numbers were the same, it remained the same. So when a system is mechanical energy gets transferred between potential and kinetic energy. And there's no loss. We say that that energy is conserved and that's exactly what the principle of conservation of mechanical energy is. It says that the M. E. Initial is equal to M. E. Final. Now we usually are not going to know what the total mechanical energy is, our initial and final. So the way that we're always going to write this, if you're always gonna write this as kinetic initial plus potential initial equals kinetic final plus potential final. This is what's called the conservation of mechanical energy. It's gonna be super helpful for us to solve problems that we haven't been able to solve before and we can also solve some problems that we have seen before much quicker. Let me show you another example here. So now we're gonna drop a ball from a 50 m building and we want to use conservation of energy to figure out the speed when it hits the ground. So whenever you have conservation of energy you're going to draw a diagram. So let's go ahead and do this. We have a building like this, we got the ball and it's gonna be falling. So very similar to our example. Before we have an initial height of 50, it's gonna fall down to the ground. All right that again. So it's going to fall down to the ground. Ry final is going to be zero. So we wanna use conservation of energy with the diagram. Now we want to figure out we actually wanna write that equation out. So we're always going to start with K. I. Plus U. I. Equals K. F. Plus U. F. So now we want to do is we want to eliminate and expand out any of the terms. What do I mean by that? So we know from the previous example that if you drop something the initial velocity is going to be zero. And what that means is that your initial kinetic energy is not going to be uh is gonna be zero. Right? So it's all just gravitational potential. You have some gravitational potential because you're at some heights. And then finally what happens is that your kinetic energy final comes from the fact that you have some speed V final, which is what we're actually trying to solve here. So you have some kinetic energy here. But because you're on the ground in your wife final is equal to zero. You have no gravitational potential energy. So these are the only two terms that survive and we can expand them because we know that our Yugi initial is going to be M. G. Y initial and our kinetic energy final is going to be one half M. V. Final square. This is an equation and we can do is we can say that the masses are going to cancel out. In fact that's most most of the time that's gonna happen in these problems. Which is actually really, really important because we were told that this ball had an unknown mess. So we wouldn't be able to solve this if we didn't know that. All right, so now we can just figure out the speed, you can rearrange this equation to solve for the final. What you're gonna get is that the final is equal to the square roots, what you move the one half to the other side, you're gonna get to G. Y. Initial. So, remember I told you that this equation was going to be super important. I even highlighted it up there because these are two equations are actually going to be the same. Notice how we got this equation using vertical motion and motion equations. Now we've got the exact same equation from conservation of energy. So I can do is I just have to plug in the numbers. Now, this is just gonna be two times 9.8 times the initial height of 50. You go ahead and work this out. What you're gonna get Is you're gonna get 31.331.3 m/s. And that is the speed at the bottom. All right, so let me know if you guys have any questions. That's it for the conservation of energy equation. Let's go ahead and take a look at some practice.

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