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A certain ball has the property that each time it falls from a height $ h $ onto a hard, level surface, it rebounds to a height $ rh, $ where $ 0 < r < 1. $ Suppose that the ball is dropped from an initial height of $ H $ meters.

(a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels.

(b) Calculate the total time that the ball travels. (Use the fact that the ball falls $ \frac {1}{2} gt^2 $ meters in $ t $ seconds.)

(c) Suppose that each time the ball strikes the surface with velocity $ v $ it rebounds with velocity $ -kv, $ where $ 0 < k < 1. $ How long will it take for the ball to come to rest?

(A). $D=\frac{H(1+r)}{1-r}$

(B). $\sqrt{\frac{2 \mathrm{H}}{\mathrm{g}}} \cdot \frac{1+\sqrt{\mathrm{r}}}{1-\sqrt{\mathrm{r}}}$

(C). $\mathbf{T}=\frac{\mathbf{v}}{\mathbf{g}} \cdot \frac{1+\mathbf{k}}{1-\mathbf{k}}$

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Missouri State University

Harvey Mudd College

University of Nottingham

So for part A, let's go ahead and find that total distance that the ball travels. So here the real initially ball falls eight meters because that's the height that is dropped from, however, after that in rebounds. So let's say climbs two a height R H meters. However, after reaches this max height, it also has to fall that same blank. That same eye then falls. Yeah, also our age meters. Answer that we multiplied by our again for the second rebound. So our square age and then immediately after reaches that I and it falls by that same height right after and so on. And we would keep repeating this so that whole distance would just be the sum of all these heights that I felt the first was a tch. And then after that, we have to r H. Because the height and in the fall and then to our age r squared age and and so on. And all these terms will all get two's except the first one because the height and in the fall. So this is the geometric son. Now we can go ahead and let's go. Actually, to the I'LL go to the next page. So first let's rewrite that sum and now is use the fact that this is a geometric Siri's. So we have a formula for the geometric series. You take the first term in the Siri's and then divided by the common ratio in this cases are and that could be simplified just by adding the fractions. So here will get one plus are over one minus R. So just come and multiply this age of here by one minus R one minus R and then just add those fractions together. And this will be the Toro height that the total distance that the ball travels. So let's go on toe part be on the next page. So first, let's let's calculate the length of each fall. Then we'LL take the sum some of all the of all the length of the balls for a given end when knees took compu the time tn time needed our time required. Let me let me see the time it takes So fall let's say H r and meters. So here, using the hint, then we can go ahead and solve this for t n. So do some algebra there to solve this for TN and then we have squares of our but that will go to the M power. However, the reason we just need the length of the fall is due to the following. It takes the same amount of time to climb than it does to fall. All right that Oh, since it takes the same amount of time to climb HRN meters as it does to fall the same the same height, we have using the same reasoning that we didn't part a the total time. So let's do like t for time, total time. So we have the first time and then after that, we start multiplying by two and again just for the same thing. And partly the reason we're multiplying by two year is due to the fact that we have a rise and fall. So that's why we're doubling the time. However, for the first one, it was just an initial drop. It didn't There was no rebound before the initial drop. So there's on ly a one Teo now, using the fact that I really wrote the formula for t n on the previous page, sold me right out again to refresh your memory but he could just for wind there the tape. But here already already writing this. Let me let me write that. That's Tien Now we take this infinite sum and here let me take a step back Sorry, I have to race is formula to go backwards. Let me keep the t zero here And now let me rewrite right the remaining part and save my notation. Excuse me now, using the fact that we had the formula for the T I This is to age over G and then we had the end squared of arts the end power But this is geometric series. So this Siri's will just be of t zero plus geometric is all It's always the same formula used the first term So plugging that I equals one that should be in either So you have too radical too h over g and then squared of our and then one minus. The common ratio in this case is square of our So this can be simplified and then even using the formula for t zero itself I should use it Here is well so this is just radical to age over G all inside the radical. So go ahead and add these two terms together here it's Simplify that, and that should just become square room to H over. G times one plus radical are one minus radical har, and that's our final answer for part B. Let's go on to the next page for parties all that we take a look at that part of the problem. So this will be a kind of similar party, so it just hasn't bee. We'LL find the time. So let's say time required to travel. So this is each trip, I guess, however, you wanna work it. So each trip up and down so and then that down and then once again will take a sum. So we we would need, first of all when these time requires to decelerate, because that's what's going on here. Two okay, and the meters per second two. No motion to a hole so equivalently we can Also this's the same is accelerating from zero two k and B. We want to know the time required. So we're going from. We have cheats he and equals K and the but that could be solved. That's equivalent to CNN being knd over. Geez, over in Parsi this is our formula. Now we find t that total total time So this is just as before in part B, we'LL start bringing in those two's after we get the one Now this I can also rewrite this in a different way instead of writing is that you must have some right now So here we can just write This is zero Let me be consistent t zero Let's go find that right now t zero is just the over ji Then here we have this infinite sum. So the first term First I should go ahead and just write that out K V over G two k square V over G and so on. Now this Let me come up here. Theo Virgie, this is a geometric theory. So the first term, all divided by the common ratio, which in this case is just k. So let me go on to the next page. One could add those fractions together and they should all simplify too. V over g times one plus que over one minus k. And that's our answer for Parsi. And this is this. Come solves the whole problem