Is there an easy way to calculate this problem?

[goonking]

Homework Statement

A Bouncy ball is dropped from a window, 28.7 meters above the ground. It bounces inelastically off the ground and bounces up again, but not as high as it stated out. If it looses 9 percent of its kinetic energy every time it hits the ground, how many times can it bounce, and still have a maximum height greater than half the original window height.

Homework Equations

1/2 mv^2 = mgh

The Attempt at a Solution

Since I know it loses 9% of its kinetic energy, the potential will be lowered by 9% each time the ball bounces also.

So I did 28.7 - (0.09 x 28.7) = 26.1

then 26.1 - (0.09 x 2.61) = 23.76

then 23.76 - (0.09 x 23.76) = 21.627

and so on, I keep repeating this until I get an answer below 28.7 / 2 = 14.35 meters.

And after all that, I get the answer to be 7 bounces.Is there a faster way to do this problem?

 

[ehild]

The question is, how many times can it bounce, and still have a maximum height greater than half the original window height.
The seventh bounce produces less height than half the initial. Count the bounces till the height is still greater than half the initial height, but the next bounce produces less than the half.
What is the relation between the k-th height and the k+1 height? What sequence do the heights make?

 

[goonking]

The question is, how many times can it bounce, and still have a maximum height greater than half the original window height.
Count the bounces till the height is still greater than half the initial height, but the next bounce produce less than the half.
What is the relation between the k-th height and the k+1 height? What sequence do the heights make?

what is k-th and k+1?

 

[ehild]

k represents the k-th bounce. It is just a positive integer. For example, the height after the third bounce is 0.91 times the height after the second bounce.

 

[goonking]

k represents the k-th bounce. It is just a positive integer. For example, the height after the third bounce is 0.9 times the height after the second bounce.

the previous bounce is always 1.01 times higher.

28.7/26.11 = 1.01

26.11/23.76= 1.01

23.76/21.62 = 1.01
etc etc

so I can just keep dividing by 1.01 to get a new height.

 

[ehild]

Why do you divide by 1.01? If the energy loss is 9 percent it means that the energy after the next bounce is 0.91 times the previous one. Dividing by 1.01 produces 0.99 times the previous one, which is 1 percent loss.

The next bounce is 0.91 times the previous one. Considering the heights as a sequence h₀, h₁, h₂... h_k what is the relation between one term and the next one? What kind of sequence is it?

 

[goonking]

Why do you divide by 1.01? If the energy loss is 9 percent it means that the energy after the next bounce is 0.91 times the previous one. Dividing by 1.01 produces 0.99 times the previous one, which is 1 percent loss.

The next bounce is 0.91 times the previous one. Considering the heights as a sequence h₀, h₁, h₂... h_k what is the relation between one term and the next one? What kind of sequence is it?

oh, I am sorry, I meant 1.1, not 1.01.

 

[ehild]

Instead of dividing, use multiplication. Loosing 9 % means that 91% remains. Dividing by 1.1 means that 90.90909...% remains.They are not the same.
So how do you get the height after the first bounce from the initial height? How do you get the height after the second bounce?

 

[goonking]

Instead of dividing, use multiplication. Loosing 9 % means that 91% remains. Dividing by 1.1 means that 90.90909...% remains.They are not the same.
So how do you get the height after the first bounce from the initial height? How do you get the height after the second bounce?

first height times 0.91 gives new height , then we just keep repeating?

 

[ehild]

Keeps repeating. How many times is the second height of the initial height? How many times is the third height of the initial height? How many times is the k-th height of the initial height?

 

[goonking]

ah

Keeps repeating. How many times is the second height of the initial height? How many times is the third height of the initial height? How many times is the k-th height of the initial height?

the first bounce is 0.91 times the initial height, and the 2nd bounce is 0.82 times the initial , and the 3rd bounce is .73 the initial.

I'm assuming k-th is the height we want it at, which is 14.35 which is .5 times the initial.

 

[ehild]

ah

the first bounce is 0.91 times the initial height, and the 2nd bounce is 0.82 times the initial , and the 3rd bounce is .73 the initial.

How is it symbolically? If the initial height is H the first is
h₁= H*0.91,
the second is
h₂=H*0.91*0.91=H*0.91²,
the third is
h₃=H*0.91³,
the k-th is
h_k=H*0.91^k.

So you have the relation

h^k ≥ 0.5 .

What is the maximum value of k so as the inequality is valid? How do you express k with logarithm?

 

[goonking]

logarithm

log(14.35)/log(28.7) = .7935

is that it?

 

[ehild]

No. Take the logarithm of both sides what do you get?

 

[goonking]

No. Take the logarithm of both sides what do you get?

you mean log h^k ≥ log 0.5

?

 

[ehild]

Yes. Do you know how to take the logarithm of something on the k-th power?

 

[goonking]

Yes. Do you know how to take the logarithm of something on the k-th power?

Sadly, no. I will just look it up.

 

[ehild]

Haven't you studied logarithm yet?

 

[goonking]

Haven't you studied logarithm yet?

yes, but it was a while ago and somethings have been forgotten

 

[ehild]

log(a b) = log (a) +log(b)
log(a/b) = log (a)-log (b)

log (a^k) = k log(a) ...

 

[goonking]

log(a b) = log (a) +log(b)
log(a/b) = log (a)-log (b)

log (a^k) = k log(a) ...

what do the eclipses mean?

 

[ehild]

you mean the dt

what do the eclipses mean?

you mean the dots? They mean only that there are some more identities.