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Problem 56 Hard Difficulty

$\bullet$ A fun-loving 11.4 $\mathrm{kg}$ otter slides up a hill and then back
down to the same place. If she starts up at 5.75 $\mathrm{m} / \mathrm{s}$ and returns
at 3.75 $\mathrm{m} / \mathrm{s}$ , how much mechanical energy did she lose on the
hill, and what happened to that energy?


108 $\mathrm{J}$
This energy is converted to thermal energy.


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Video Transcript

so in this problem and the order slights up and then slides down back in the hill and returns to this imposition, we should remember one thing. That the tool energy is always conserved at each point off her journey or off her part part. So if we say that total energy is Kaye plus you, which is a kind of the energy plus the potential energy, we know that at each point, the kind of energy plus potential energy will be equal now. And this 11 where the order returns. We see that the if we if we consider this as our origin or let's say we call it Baikal zero and why is the vertical direction then? There's no pretension energy over here because there's the Heidi zero. Essentially, the order comeback comes back to the same place, but we see that the philosophy is now decrease the initial velocity, the velocity at which she starts going uphill is not equal to the velocity of a tte, which she comes back. So the energy here is well, it's cancer, but it's not in the form off kind of energy. So someone off energy has been lost during her sliding down that many for different reasons, like maybe it's for its converted to thermal anti or it's been converted toe frictional energy. But we don't have to worry about that because all we care his that how much energy has been lost. Do you do the kind of energy part since there's no potential energy left here? So initially, let's let's say this is our initial point when the order goes uphill and when she comes back, this is the final point. So at this point, are potential energy is equals, which is we can call it zero. If we said this as our origin, then the change or the energy loss will be the final minus initial, and let's actually call it mechanical energy because we're not considering the heat energy or the fictional energy which is been lost. And that's why we're getting less amount of velocity when she comes back. So let's just see the mechanical part of the energy. So that's going to be the final, which is heart off in. They find and squared minus are actually, Since the final energy is less than the initial energy, we should do the other way so let me rewrite the question. So, yeah, that's going to be the initial energy, which is greater minus the final energy. And that's going to be half off him. The I squared minus half off and the exclaimed, where we can take em comin from both of these expressions. So we'll have the initial squared minus the final squared. And if we use the numbers that's given years, will have half, and then it's 11.4 kg for velocity we have the initial velocity is 5.75 1,000,000 per second, and we have a whole square minus V F, as in 3.75 minutes per second. So using all the numbers, we see that the amount of energy that's lost is 108 Jules. Now that might be lost in the form of heat energy are a friction energy, which we don't care. Thank you