Calculating Max Height of a Bouncing Basketball

[nobodyuknow]

Hey guys,

I was wondering, is there a formula I can use to calculate the height of a which a basketball can achieve in a controlled environment situation, therefore, factors such as wind, debris, spin do not influence the results provided.

A brief explanation of how the formula works would be greatly appreciated.

EDIT: I have calculated the Velocity by using:

GPE = KE
mgh = 1/2mv^2
0.45*9.8*3 = 1/2*0.45*v^2
13.23 = 0.225*v^2
13.23/0.225 = v^2
v = sqrt58.8
v = 7.668115805m/s

I'm just unsure how to imply this information into calculating it's maximum height achieved.

 

[Stonebridge]

The height it reaches after this bounce will depend on the amount of energy lost in the collision with the ground.
If the collision is perfectly elastic (it never is in real life) there is no overall loss of energy and the ball bounces up at the same speed as it struck the ground. This means it rises to exactly the same height as it was dropped from.
To know the height it will rise you need to know the coefficient of restitution of the collision.
http://en.wikipedia.org/wiki/Coefficient_of_restitution
The coefficient ranges from 1 (perfectly elastic - same speed after bounce) to zero (ball does not bounce).
It is numerically equal, in this case, to the ratio of the speed after the bounce to the speed before.
So the upward speed after the bounce will be the coefficient times the speed before impact.
This upward speed will enable you to calculate the height of the bounce.
You should be able to find a coefficient value for a typical basketball.

 

[nobodyuknow]

Hmm how do I calculate the Coefficient of Restitution? or am I really confused?

I have worked out that the basketball loses about 19% bounce height every time it bounces freely from a height of 3m.

 

[Stonebridge]

Your original question was asking how to calculate the height to which the ball bounces. To do this you need a value for the coefficient.
I'm confused as to what, exactly you want to know. You say you have measured the height it bounces to. So what is it you need to find? If it's the coefficient you want to calculate, then you need to know the speed the ball hit the ground, and the speed it bounced.
Your calculation above is correct for the speed it hit the ground.
To find the speed it bounced, use the same formula, but with the lower height in mgh.
Say, h₁ is initial height and h₂ is bounce height.
The coefficient is the ratio of the speed after bounce V₂ to the speed before V₁. [=V₂/V₁]

you know mgh₁ = ½mV₁²
and mgh₂ = ½mV₂²
so the coefficient can be found from h₁ and h₂

 

[ACPower]

The coefficient is 0.9, since you measured h2/h1 to be 0.81, and V2/V1 = sqrt(h2/h1) = sqrt(0.81) = 0.9

 

[Stonebridge]

I was hoping the poster would be able to work that out for himself!

 

[nobodyuknow]

Haha thanks a lot Stonebridge and ACpower for providing the answer. I just analysed my information again and realized that the ball didn't lose a consistent amount each time. It was actually losing between 26% - 20% therefore, it isn't losing a consistent amount of energy after each bounce.

But I assume that if I found an average and worked out the co-efficient with that would be viable?

 

[Stonebridge]

It's perfectly normal for results of such an experiment to show a statistical variation like that.
Yes, find the average value of a number of results to get your best estimate.

 

[sophiecentaur]

I'd also expect a systematic departure from the simple model. The height reached by a bouncing ball is affected by lots of things. The air inside compresses and the envelope stretches and flexes. The harder it's pumped up, the more ideally it will behave as the majority of the energy will be stored in the air. The energy loss as the envelope flexes will be pretty non-linear with height / flexing, I should imagine.
I suppose what I'm implying is that the term 'coefficient of restitution' is an over-simplification in a case like this. It's value depends on the other variables so it's not really a 'coefficient', as such. You just have to work it out for each case.
It works well and can be used to make predictions for collisions involving materials with a well behaved modulus.

Some students of mine did a project involving bouncing pingpong balls from up to about 20m. We reckoned that the effect of the air resistance showed itself at heights where the terminal velocity was being approached - a definite and repeatable curve to the restitution with height graph.
You could try a beach ball instead - I would imagine its terminal velocity would be pretty low.