bouncing-ball-for-time-suppose-a-rubber-ball-when-dropped-from-a-given-height-returns-to-a-fraction-

Question

Bouncing ball for time Suppose a rubber ball, when dropped from a given height, returns to a fraction $p$ of that height. In the absence of air resistance, a ball dropped from a height $h$ requires $\sqrt{2 h / g}$ seconds to fall to the ground, where $g \approx 9.8 \mathrm{m} / \mathrm{s}^{2}$ is the acceleration due to gravity. The time taken to bounce up to a given height equals the time to fall from that height to the ground. How long does it take for a ball dropped from 10 m to come to rest?

Geena Pullo · Accepted Answer

Hi. So in this question were pretty much looking at how long it takes for a ball to come. Teoh rest after being bounced from a height of 10 meters. So our initial height is gonna be 10 meters p is the fraction of that initial height. Um, the time, both ways. So when the ball is bouncing up, Andi falling down are both equal to the square root of two times H over G. And our acceleration is gravity. Pretty much so, Um, those are the values that we have right now. So first thing we&#x27;re gonna dio as we can, um, find out, um, ignoring for right now the initial drop. Um, we can easily figure out what the total time for the end bounce was. So the first thing we have to dio is figure out what the bounce would be, what that would look like. So we know that the initial is 10. So our initial list 10 and any bounce after that would be times P because peas a fraction of that, so we would have tend to the P. And then it would continue. Of course, if we tend to the P um squared on, then it would be 10 p cube. And that would just keep on continuing until we get 10 to the P to the end. So this would be for our end, um, bounce from 10 meters. That would be the height. So we would know Then, Um what are height is for end. So I&#x27;m going to just say h on is equal to 10. PM on. We know that the amount of time it takes is for the total going, um, down and then up or up and down in this case was gonna be up and then down to that height, um, it would be two times the number we have right here. So what we can say is our time for end. I&#x27;m going to say t n is going to be eat Teoh too high. We&#x27;re root of two. And this case, we&#x27;re gonna put our h and here divided by G. So this is really just equal Teoh two times the square root of two times 10 times p to the end over G. So now what we can do is we can calculate the total time because we have an end formula which will make it much easier to figure out how long it takes before the ball comes to arrest from 10 meters. So we know what our initial amount of time would be. Um, because it&#x27;s just given by this formula. So our initial would Onley be the amount of time it takes for the ball to drop the ball will never again reached that height to stock going up to that height. So our initial would just be the square root of two times H zero over G, which is equal to the square root of two times 10. Cause that is our initial over G, which is just gonna be equal to the square root of 20 over J. And now we can use both of these pieces of information in order to form an equation. So we have our initial and we&#x27;re going to say that it&#x27;s 20 over g. It&#x27;s gonna be plus a Siri&#x27;s. So we&#x27;re gonna have Infiniti. We&#x27;re gonna have a new equal, uh, 21 We&#x27;re going to say, um, that that is going to be of to times to times 10 to the p on over G. So just using our formula up here that we have because we&#x27;re trying to find total monotypes. We have our initial time, and then this is the amount for up to and our end bats. Now, from here, we can simplify our equation of it. Weaken Be right. This so we can write this as 20 over G. Um, what? And now what we can do is we can, um, again we&#x27;ll have our signal here for now. But we&#x27;re going to just take our two out, and we Can we write this as being 20 over g? Um, on That is gonna be times p on in there. So from here, what we can do is we&#x27;ll have our 20 over G. We&#x27;re gonna add that again to our Siri&#x27;s. But we&#x27;re gonna do is we&#x27;re gonna separate these two. So we&#x27;re gonna have R P. And and now from here we&#x27;re gonna do is recording too. Take out, um, are 20 g from inside Siri&#x27;s. We&#x27;re just going to write it on the outside. It does not have any, um, variables that are needed in this case. So doesn&#x27;t have to be in the Siri&#x27;s. It&#x27;s just a constant, um, on now from here we&#x27;re going to Dio is we&#x27;re going to put this equation into formula. That I&#x27;m sure you&#x27;ve seen a lot in this chapter, which is a sorry. A over one minus are. So as we wrote up here, um, we know that we could take out our 10 um, and knees equations. It&#x27;s not necessary. But r r P is really are Siri&#x27;s here. So we have p and then we have plus piece squared. Plus, he cubed, continuing on to pee to the at, um so we can see that r p is really where are Siri&#x27;s is gonna come from, and right now we have it as the square root of Peter the end. So we do need to write out our new Siri&#x27;s. So we&#x27;re gonna have, uh, just pee, and then it&#x27;s going to be Plus, he squared plus cubes. So what we can see is that are a value is going to be the 1st 1 severe a value. And now we need to calculate our our value, which is just going to be our 2nd 1 divided by first. So it&#x27;s just going to be equal to the square root of P. So now we can rewrite. Oh, this&#x27;ll down here. Actually, Onley going to be rewriting this part and then we&#x27;re going to put the other part into our handy formula over here. So we&#x27;re going to rewrite this as 20 over G plus two times 20 over G, and then we have our A value is a square root of P, and that is over one minus the square root of P. And from here, we&#x27;re just going to start simplifying our equations. Um, so right now we have, um, to parts that have the square root of 20 over g so we can just take out our 20 over she on. That&#x27;s just be a one here and then plus to times are Formula Square with P over one, minus p on from here. Um, we can simplify even more because we can say that this is gonna be equal 21 minus the square root of p over one minus the square root of P class to times the square root of P over one minus the square root of P on. This is then equal to square root of 20 over g times, one plus the square root p over one minus the square root of P. Andi, that is gonna be our answer. This is Thea. Amount of seconds. So it takes this money seconds before the ball will come to arrest.

Kara Merfeld · Answer

So we take the ball and throw it up at 20 meters per second. You won&#x27;t know when it&#x27;s at its apex when it stops moving. Um, so basically, we know that, um Let&#x27;s see. I guess the best equation of years would be the V two is to be one plus acceleration times Delta t um and this is set up so that we want this this second velocity in this form, we really don&#x27;t We&#x27;re after right now is Delta T you&#x27;re gonna switch this up to, say, Delta V, which is just a change of velocity and that is equal to a Delta T. So really, all we did was move the view one over the other side there and then delta T we have so Delta V over a Miss Delta T and that&#x27;s just dividing both sides by it. So now if the change in velocity has to be 20 meters per second, right, Um, we have to be careful about positives and negatives here, so the change in velocity is gonna be pointing downward. But the acceleration is always is gonna be pointing downward to so we can have those negatives cancel each other or we can just pretend that they never were negative in the first place. You just kind of redefine, um down as being positive. So now change velocity and our do it for it to go from 20 miles, 20 meters per second, down zero has to be 20 meters per second. And then the acceleration is the acceleration of gravity. So that&#x27;s 9.8 meters per second. Squared traders of Castle in here that meter&#x27;s cancels with that meters. That second cancels with the second squared in this second transitions up to the top because it&#x27;s in the denominator of the denominator. So it&#x27;s the same. It&#x27;s the numerator. So now essentially, we&#x27;re gonna cut you over here kind of this 9.8 is 10. Then we have 20. Divide by 10 and that&#x27;s too. So we&#x27;re going to say this is roughly equals two, two. Um, and that&#x27;s themes on. That is seconds

Ankit Gupta · Answer

in this question were given that an object is dropped from initial height on Earth and Jupiter surface, and the height of an object after time T is given by H is equal to minus one. Divide by two g d square. Bless each note where each note is the initial height on dhe. He&#x27;s the time. Jeez, the acceleration due to gravity. And here we are asked to calculate the objects time that it takes to reach the ground on each planet on dhe. Acceleration due to gravity in case of Jupiter has given us 23.1 meters per second. Squared on for IRT. It has given us 9.8 meters per second squared and the value off each. Note that initial height as 100 meters, first of all will calculate the time in case it&#x27;s Jupiter for Jupiter. When the ball reaches the ground, the value of each will be zero. So Sposito is equal. Its toe minus one bite Do G hair is 23.1 meters per second squared. So 23.1 be square Bless Reginald Reginald, this 100. So our aim is to kill grit, be so after a simplification, we can write one by two. 23.1. The square is equal to 200. From here, the value off be will be two times 100 divided by 23.1 square group and the very off key. From here we will get it as 2.9 seconds. And if you calculate in case off art so we will get same Etch will be zero when the ball it is to the our surface. So zero is equal to minus one by two The value of G in case off artists 9.8 meters per second squared d squared plus h Nordeste 100. So from here we can write one bite to 9.8. The square is equal to 200 and the value of tea from here will would be to times 100 Divide by 9.8 square root on After a simplification, we will get the value s 4.5 seconds

J Hardin · Answer

So for part A, let&#x27;s go ahead and find that total distance that the ball travels. So here the real initially ball falls eight meters because that&#x27;s the height that is dropped from, however, after that in rebounds. So let&#x27;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&#x27;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&#x27;s go. Actually, to the I&#x27;LL go to the next page. So first let&#x27;s rewrite that sum and now is use the fact that this is a geometric Siri&#x27;s. So we have a formula for the geometric series. You take the first term in the Siri&#x27;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&#x27;s go on toe part be on the next page. So first, let&#x27;s let&#x27;s calculate the length of each fall. Then we&#x27;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&#x27;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&#x27;t part a the total time. So let&#x27;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&#x27;re multiplying by two year is due to the fact that we have a rise and fall. So that&#x27;s why we&#x27;re doubling the time. However, for the first one, it was just an initial drop. It didn&#x27;t There was no rebound before the initial drop. So there&#x27;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&#x27;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&#x27;s will just be of t zero plus geometric is all It&#x27;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&#x27;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&#x27;s our final answer for part B. Let&#x27;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&#x27;t bee. We&#x27;LL find the time. So let&#x27;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&#x27;s what&#x27;s going on here. Two okay, and the meters per second two. No motion to a hole so equivalently we can Also this&#x27;s the same is accelerating from zero two k and B. We want to know the time required. So we&#x27;re going from. We have cheats he and equals K and the but that could be solved. That&#x27;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&#x27;LL start bringing in those two&#x27;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&#x27;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&#x27;s our answer for Parsi. And this is this. Come solves the whole problem

Sherrie Fenner · Answer

So we have the equation Each he wolves negative 16 e squared. What&#x27;s H so? And they&#x27;re getting us. H Savo. He wants 96 Ask in part. A after distance hit the ground. So half of 96. 48 people&#x27;s negative. 16. He&#x27;s weird plus 96. Subtract 96 for both signs. We can&#x27;t negative for eight equals negative. 16. He&#x27;s where? E draft negative science. Of those. I survived by 16 and I get He&#x27;s squared. He was three. So t would eat What? This rare root buster minus of three. Again, we&#x27;re talking time, so we&#x27;re only looking at the positive value. But I want to make sure that whenever you take a square root you&#x27;re taking plus reminding So the square root of three. And that&#x27;s about one point that they three seconds. So for Part B, we&#x27;re not using half the distance where you think the whole distance evolved to the ground, so I have to be zero negative. 16. He square plus 96. Subtract 96. I have naked of 96 equals negative 16. He&#x27;s layered. Cancel the negatives on both sides, which gives me 96 equal 16. Please where divide both sides by 16 and I get he&#x27;s squared equal. Six. They&#x27;re four e equals, plus or minus is where Room of six. Hey, equals plus or minus This Where of six. But since we&#x27;re talking about time, our that is the positive 30 equals this We&#x27;re route of six, which is about three white No. Two point for five second.

Karuna Ramroop · Answer

All right. So for this question, we have been given the acceleration function. They told us that the acceleration off the ball we&#x27;re gonna call dysfunction A of T is a constant negative 9.8. So the acceleration doesn&#x27;t change. That is a constant function. Then we&#x27;ve been asked to find the velocity, functions and the position bump shins. So we know that acceleration is the derivative of velocity. So to find the velocity from the acceleration, we&#x27;re going to take the integral of the acceleration function. So we&#x27;re gonna do the integral of our acceleration function. Negative 9.8 tt. And that gives us negatives. 9.8 t plus some constant. We don&#x27;t know what that constant is, but we can figure it out. We were given some initial conditions for velocity. They said that the initial speed of the goal was 40 meters per second. Sue, when the time was zero. The initial speed Waas 40. So we&#x27;re gonna use this to help us find our velocity function. We know that the of zero equals 40. So that tells us that negative 9.8 time zero waas. Whatever. See, Iwas is equal to 40 because we know that this is our velocity functions the integral of the, um, acceleration function. So we just have to solve this equation for C so negative 9.8 times zero is just zero. So this just becomes C equals 40. So now we have our velocity function. We&#x27;ll see. Function is negative. 9.8 t plus 40. Now, we just need to find the position, function or height option. We&#x27;re gonna call that each of teeth, so we know that the position function is just the integral off the velocity function. Looking at these functions going in this direction, we take derivatives to keep going. But going in this direction we&#x27;re taking the intervals are doing the opposite of the derivatives. So to find our position function, we&#x27;re gonna take our velocity function. We know that this position function is equal to the integral of the velocity function, so that is equal to the integral of negative 9.8 t plus 40. So when we do, the integral we get, um, we&#x27;re gonna use power. So we&#x27;re gonna take the experiment, we&#x27;re gonna add one, and then we&#x27;re gonna divide by that some. So when we do that? We get negative. 4.9 t squared was 40 t and then plus seat. Whatever she is so to find. See, we&#x27;ve been given the another initial condition for our heights. We&#x27;ve been told that when the time is zero, the initial height of the bowl is 1.5 meters. So we&#x27;re gonna use this to figure out what&#x27;s he is. So if we know that each of zero equals 1.5 and we know that this is R. H, our height function, we&#x27;re just gonna plug that in. So we get negative 4.9 times zero swear plus 40 times zero, Let&#x27;s see, is equal to one boy. So I&#x27;m looking at that first term. We&#x27;re gonna start simplifying things. Zero squared to zero and zero times negative. 00.9 is still zero. So this full term disappears and then 40 times zero is also zero. So this germ disappears and we&#x27;re just left with C equals 1.5. So then our hft function is just negative. 4.9 tease where US 40 t plus 1.5. So this is how we find our velocity and height functions

Problem

Multiplier effect Imagine that the government of …

Problem 85 Hard Difficulty

Answer

$$\sqrt{\frac{20}{g}}\left(\frac{1+\sqrt{p}}{1-\sqrt{p}}\right)$$

Related Courses

Calculus for Scientists and Engineers: Early Transcendental

Related Topics

Discussion

Top Calculus 2 / BC Educators

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 9

Video Transcript

William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$$\sqrt{\frac{20}{g}}\left(\frac{1+\sqrt{p}}{1-\sqrt{p}}\right)$$

Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

Integration Review - Intro

Definite vs Indefinite

William Briggs, Lyle Cochran, Bernard Gillett Calculus for Scientists and Engineers: Early Transcendental

Anna Marie V.

Kayleah T.

Caleb E.

Kristen K.

William Briggs, Lyle Cochran, Bernard Gillett

Calculus for Scientists and Engineers: Early Transcendental