How many bounces does it take for a ball to travel 1854.94320091 feet?

[darshanpatel]

Homework Statement

A ball is dropped from a 100 feet and has a 90% bounce recovery. How many bounces does it take for the ball to travel 1854.94320091 feet?

Homework Equations

-None-

The Attempt at a Solution

I know the ratio is .9 and the 'a one' value is 180, so I plugged those values in this equation that is the equation for a Finite Geometric Series:

S sub n= a1(1-r^n/1-r)

I subtracted 100 feet because that doesn't follow the ratio.

then 1754.94320091=180((1-.9^n)/(1-.9))

I got 35 for 'n' but I was wondering if it would be 36 because of the initial bounce?

 

[tiny-tim]

hi darshanpatel!

(try using the X² button just above the Reply box )

I know the ratio is .9 and the 'a one' value is 180, so I plugged those values in this equation that is the equation for a Finite Geometric Series:

S sub n= a1(1-r^n/1-r)

I subtracted 100 feet because that doesn't follow the ratio.

then 1754.94320091=180((1-.9^n)/(1-.9))

I got 35 for 'n' but I was wondering if it would be 36 because of the initial bounce?

let's see …

(1 - 0.9³⁵)/(1 - 0.9)

= 1 + … + 0.9^what? ​

 

[darshanpatel]

I did this using the finite geometric series formula:

S_n=a₁((1-rⁿ)/(1-r))

Subsituted Values: a₁=180 r=.9 and S_n=1754.94320091

1754.94320091= 180((1-.9ⁿ)/(1-.9))

and solved for 'n' which is the number of bounces.

n=35.000002 <---- I think, I am not sure how many zero's I am off, but that is what it was.

Now, because we solved for 1754 feet and not 1854, like it asks. Does that mean we add one more bounce to get 36, or is it 36 becuase you round up from the 35.000002 becuase you can't have a fraction of a bounce.

Oh, Btw tiny-tim, thanks for the tip, I always wanted to know how other people were doing it.


My Logic for the problem:

First bounce is 100 feet, and then the 90% bounce recovery starts from there, so that means the series starts from there.

meaning:
a₁=180
a₂=162
a₃=145.8
.
.
.
a_n

+100 from the intial start.

From that, I am wondering would it be 35 bounces or 36 bounces?

 

[tiny-tim]

A ball is dropped from a 100 feet and has a 90% bounce recovery. How many bounces does it take for the ball to travel 1854.94320091 feet?

First bounce is 100 feet, and then the 90% bounce recovery starts from there, so that means the series starts from there.

meaning:
a₁=180
a₂=162
a₃=145.8
.
.
.
a_n

+100 from the intial start.

From that, I am wondering would it be 35 bounces or 36 bounces?

i think the question is badly worded

i would have said it starts from the beginning, 100 feet up, but the answer is not compatible with that

so i suppose it starts from the first contact with the floor, and therefore that contact is not counted

 

[Nessdude14]

Let's write out the sequence over the first few values of n, and maybe you'll be able to see the pattern.

Each time the ball bounces, the height it travels to is 90% of the last height, with the initial height of 100. The distance the ball travels is equal to twice the bounce height, so the sequence for the distance the ball travels on the nth bounce is:
a₁=.9*100*2
a₂=.9*.9*100*2 which equals .9²*100*2
a₃=.9³*100*2
a₄=.9⁴*100*2

So when you sum that sequence, that gives you the distance traveled minus 100 feet. If you think about what this means, you'll find that the "n" you solve for is the number of bounces you're looking for. If your answer is 9.99 bounces, then that means the ball will reach the distance before it makes the 10th bounce, this requiring only 9 bounces.

 

[Nessdude14]

Sorry about my last message, I made a mistake in the original post, so if you read that, please read the updated version.

 

[darshanpatel]

But wouldn't the 100 feet be counted as a bounce too? Because the question is asking for 1854.94320091 feet and not just 1754.94320091 feet? I solved for 1754.94320091 and got 35.000002, but what about the other hundred feet because 35 bounces only equals 1754.94320091 feet and not 1854.94320091 feet?

I tried to fix the question:

A ball is dropped from a height of 100 feet. The ball bounces to 90% of its previous height with each bounce. How many bounces does it take for the ball to travel 1854.94320091 feet?

The thing you did above is wrong because 90% of 100 is 90, but that is only going up, then ball comes down too, so the total distance is 180 feet for both, going up and coming down.

But the sequence would be like this: 100, 180, 162, 145.8, ect...

So you don't start the series from the initial bounce. But instead from the second bounce, being 180. So 180 then becomes the first bounce. And then you use the formula for finite geometric series to find the number of bounces.

But, because you start the series from the "real-life" second bounce, I think you add one more bounce because of the initial 100 foot bounce.

Does anyone know what I am talking about or no?

lol, this question has been bugging me for a couple of days know.

It can be either 35 bounces or 36.

If someone can shed some light on this problem, they will take a lot of stress off my back.

Thanks.

 

[Nessdude14]

The ball travels 100 feet before the first bounce, and then it travels another 180 feet after the first but before the second bounce. It only requires one bounce for the ball to travel 280 feet, because only after it has traveled 280 feet does it make the second bounce. Thus, my above explanation is not wrong.

If you re-read my previous post, you will find that for each step in the sequence, I multiplied by 2, making the travel distance not 90, but 180 feet between the first and second bounces. Apparently you overlooked this. When writing out the terms of your sum, it's best not to simplify it, so you can clearly see what's changing between each term.

Then, at the end of my post, I explain that if you sum the sequence given, it will give you the travel distance minus 100 feet. This means that you will need to add 100 to your final sum.

If you need any more clarification, please explain what has confused you.

 

[darshanpatel]

Does that mean it is actually 35 bounces or 36?

Here's a diagram like in my book:

http://imgur.com/gQWbc

 

[Nessdude14]

Does that mean it is actually 35 bounces or 36?
Here's a diagram like in my book:
http://imgur.com/gQWbc

The distance in question is reached after the 35th bounce, just as it hits the ground, so the answer is 35.

 

[darshanpatel]

But because the bounce is 35.000002, wouldn't that require rounding up? IN a real-life situation or no?

 

[Nessdude14]

But because the bounce is 35.000002, wouldn't that require rounding up? IN a real-life situation or no?

If it were true that the answer was n=35.000002 bounces, then yes, you would need to round up. However, I calculated the sum to be exactly equal to 1854.94320091 as it hits the ground after the 35th bounce (at exactly n=35). That means that it does not require a 36th bounce to reach the given distance.