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Problem 107 Easy Difficulty

The coefficient of restitution $\epsilon$ for a head-on collision is defined as the ratio of the relative speed after the collision to the relative speed before. (a) What is e for a completely inelastic collision? (b) What is $\epsilon$ for an elastic collision? (c) A ball is dropped from a height $h$ onto a stationary surface and rebounds back to a height $H_{1}$ . Show that $e=\sqrt{H_{1}} / h .$ (d) A properly inflated basket-ball should have a coefficient of restitution of $0.85 .$ When dropped from a height of 1.2 $\mathrm{m}$ above a solid wood floor, to what height should a properly inflated basketball bounce? (e) The height of the first bounce is $H_{1}$ . If $\epsilon$ is constant, show that the height of the $n$ th bounce is $H_{n}=\epsilon^{2 n} h .(f)$ If $\epsilon$ is constant, what is the height of the eighth bounce of a properly inflated basketball dropped from 1.2 $\mathrm{m} ?$

Answer

a. $\epsilon=0$
b. $\epsilon=1$
c. $v_{1}=\sqrt{2 g h}$
$v_{2}=\sqrt{2 g H_{1}}$
d. $=0.87 \mathrm{m}$
e.$H_{n}=\varepsilon^{2 n} h$
f. $=0.089 \mathrm{m}$

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Video Transcript

{'transcript': "all right. So for this problem you are dropping a tennis ball on top basketball and this thing hits the ground and in the bus double rebounds and steal hits the tennis ball again on the way up. So essentially so we can have two situations before the collision and then after the collision. So I want to address first. Is the situation just after the corners in which this possible is now moving upwards with a certain velocity? And this tennis ball is now moving again? Downloads off separated them here to make it sadly easier to understand because essentially what happens is first the boys want to hit the ground and the Boston boy is going to hit the ground and then stop moving upwards. But the tennis ball is still going down. So essentially these will be. This will look like this scenario just after the collision with the ground, but before the collision with the ball. So essentially, we can assume that this is much m small. I am. And this is my biggest for the basketball. So this is right before the collision and then after the collision, you have a situation where essentially this Basta boys going to have a certain velocity going upwards. And then there's tennis, boys, We're also going to have a certain velocity going upwards as well. So this is small M. This is again began. So notice that the only thing the only things that you're given here are essentially they tell you that this staying both through high H and military integration of masses and they don't tell you in the house. So whenever we have so we can assume here that this collision happens really fast, which means the imposes very small. So this the velocity of this ball is going to be the same velocity that he had going downwards. So we know exactly what you can calculate exactly what it is in terms of age should we need to. But essentially this. This is an elastic collision where both on anything energy remains of my concern. So if you want to know the initial velocities and the final velocities, we can use those two equations so we can assume in this case I can set up my coordinate system to say that upwards is positive in Donald's negative. And I can assume that this Oh since this is falling down, it has a velocity V by notice that divorce it doesn't depend on the mass. So if you're forced to it, I h it means that this Boston ball is also V because the velocity is just a function off the heights. It's just too g age because of conservation of kinetic energy. Um, so given that we also need equations for the initial Impala velocities for the last six relations. So if you're right out momentum equations, which it's a M one V one initial plus and to the two initial Oh, this is ego to M one V one final plus m to be to final and the Connecticut his equations who also be how m one the one initial squared plus, huh him to V two initial squared will be you go to, uh, M one V one final squared. Plus, so I am to be to final squared. So now you have. So you have these three equations and these you can combine and simplify them into expressions for velocities off off this thing. So we know that if we work on this, we work on this, algebra will find that the two f He's actually going to m one over m one plus m two minus m one minus m two over m one plus I m to much, but this is more divide by one and this is much bye bye. He too. And these air the initial velocities, so anything I have. But here are the initial velocities over here the final velocities and then for V one, if I have also the same seem a similar expression Em wan minus m two over I m one plus and to less to m two over m or less and to also more to black by V two. So these equations come in directly. If you go ahead and simplify these two equations together, you come up with expressions with so notice now that go back to a dagger, Um, we can actually simplify this a little further, so we'll call em. One can be the mosque, the basketball here. So I will this m one and I will call this him to. And so that means that V two is velocity off the ball off the tennis ball after the collision in V one. F is velocity off the basketball after the tradition, but and then these would be the initial velocities. So if so now I can plug in my numbers to see what I get so again in. The other thing that you need to notice here is that we can assume that the mass of the basketball is much larger than the most of the tennis ball in this case. Because this is a symbolic calculation. You don't need to come up with exact numbers. We just want an idea off. Where this where the Senate's boys going to go after this collision? So now with this assumption and you know the velocities in this case, we cannot go back and see what that's going to give us. So for B to Yeah, we can now. So since this is too m one, which is the great um S and M one bless him to Since M one is much bigger than M two, this is essentially just becomes to m one over M Juan times V one. Okay, and then this would be minus. So, since this is very small compared to this, this becomes M one. And then this is just again M one because M two is very small, So this would be minus V two. V two is V two initial. So you are so V two initial is actually pointing in the negative direction in my sign convention. So this velocity is actually negative. So this B minus minus, essentially or minus minus, we want because these things have the same philosophy magnitude. So I can just go that if you want in this case, so you notice that this becomes three. The one and V one again is just this group two d eight. Oh, So who notice that if V if the velocity of the tennis ball is three times the initial velocity off off the basketball, given that the high even that the velocity is directly related to the height here, if you increase this by three, if you If you multiply this by three, then you also have to our march by everything else by three. So essentially, if you know, go back to the equation that gave us this high. We know that m g h is he good to off m the square? So that's what we get this wrong. So this directly leads to that, but notice that if the masses and change, if you are moving, if you're moving at velocity V, you essentially are going to achieve the maximum height off age. But if you're not moving at a velocity three V since this for us is squared here, you actually going to end up with a maximum high off nine h because this would be three square. So if you now. So if I'm now moving at 03 v here square, we should not be equal to MGI that say it's two year. So if I if I now want to find the ratios, I can divide the streak reasons up, I'll find out that my height after moving with this increased velocity is actually going to be nine times the initial that had. So essentially, after this collision, this ball is moving at three times a velocity. This tennis board is moving at three times velocity, and because of that it achieves. It might some height of nine age, so it's going to go much higher than the initial I had. So you can also go back here and try to calculate what the one final is going to be and you will see that again. Since everyone is much greater, this just becomes so. I have a need of you on here. So since this was much greater, this is just going to be M one over m one touch V, which is just V one. And then if you are dividing something very small with something very large here, essentially, your whatever you get is going to be very close to zero. So essentially the velocity of the basketball after this condition is going to be pretty much the initial velocity that he had here, which is so the basketball moves with the same velocity upwards. But the tennis ball afterwards is going to go with a velocity three V one or three V in this case and the Basta boys going to move with velocity three in the mike, some highest age. So the reason this goes even much higher is because again the energies to conserve. Because we year we neglected the action of the ground here. But the ground caused the change in momentum and at the collision with the ball again. That's a change momentum, because this is a much heavier objects of this world impart some momentum onto the tennis ball, which means it's going to go much higher than it did before."}