# Calculating Max Height of a Bouncing Basketball

• nobodyuknow
In summary, the basketball loses a consistent amount of bounce height with every bounce. To calculate the height it will bounce to, you need to know the speed the ball hit the ground and the speed it bounced.
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.

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.

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.

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

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

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

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?

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.

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.

## 1. How do you calculate the maximum height of a bouncing basketball?

To calculate the maximum height of a bouncing basketball, you will need to know the initial height at which the ball was dropped, the coefficient of elasticity of the ball, and the height of the rebound after each bounce. You can then use the formula Hmax = (e^2 * H0), where Hmax is the maximum height, e is the coefficient of elasticity, and H0 is the initial height.

## 2. What is the coefficient of elasticity and how does it affect the maximum height of a bouncing basketball?

The coefficient of elasticity is a measure of how well an object can store and release energy when it experiences a force. In the case of a bouncing basketball, a higher coefficient of elasticity means that the ball can store more energy upon impact and release it more efficiently, resulting in a higher maximum height after each bounce.

## 3. Can the surface on which the basketball bounces affect the maximum height?

Yes, the surface on which the basketball bounces can affect the maximum height. A softer surface, such as a rubber court, will absorb more of the ball's energy upon impact and result in a lower maximum height. A harder surface, such as a concrete court, will reflect more of the ball's energy and result in a higher maximum height.

## 4. How many bounces should be considered when calculating the maximum height of a bouncing basketball?

To get an accurate calculation, it is best to consider multiple bounces when calculating the maximum height of a bouncing basketball. The more bounces that are taken into account, the more precise the calculation will be. However, in most cases, considering 3-5 bounces should be sufficient.

## 5. Is the maximum height of a bouncing basketball affected by the initial height at which it was dropped?

Yes, the initial height at which the basketball was dropped affects the maximum height after each bounce. The higher the initial height, the higher the maximum height will be after each bounce. This is because the ball has more potential energy when it is dropped from a higher height, resulting in a higher bounce.

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