Modeling a Bouncing Ball

First, in my defense I was just making a model based on the data I had. Of course, I am talking about my first super bounce ball post.

Just to get you up to speed, I wanted to see how fast you would have to throw a ball at the ground so that it would bounce 75 feet high (like it says on the packaging). To explore the bounciness of this ball, I made this plot of the height vs. the bounce number.

In my normal fashion, I wanted to build a model for bouncing based on my data. Granted, this data only had 4 bounces - but still it was my data. From this I claimed that the data was linear.

Next comes Frank Noschese (from Action-Reaction). He points out that the rebound height as a function of bounce number should be:

Where n is the bounce number. Ok - that makes sense if indeed each bounce is 0.8 times has high as the previous one. The problem is that this does not agree with the standard method of investigating bounces. Usually, people look at the coefficient of restitution. This is defined as the ratio of rebound speed to initial speed:

Does this model apply to this super ball? What about other balls? Notice that this is different from my initial model where I said there was a constant ratio of initial to final bounce height. So, what if I use this coefficient of restitution - what does this say about the bounce height. Suppose I drop a ball and it bounces back up.

Since it is much easier to measure the height than the velocity, I would like to get the velocity as a function of height. If I use the work energy system on the falling ball (starting at h₁) and include just the ball as the system, then the work done is:

Using the same idea, I can get a similar expression for the relationship between h₂ and v₂. So, the coefficient of restitution in terms of height would be:

So, the ratio of initial to final bounce height should still be constant - but not the coefficient of restitution.

More balls, more data

My problem with the original data was that I didn't let it bounce enough. I fixed that with a longer video. So, how about a plot? If this ball does a constant coefficient of restitution, then starting height vs. bounce height should also be a linear function.

The above is actually data for two bounce runs mixed together. I will call the slope of this function the coefficient of heightistution where:

Two important point:

• The slope is constant - so the coefficient of heightistution and the coefficient of restitution are constant.
• The coefficient of restitution is the square root of the slope (R = 0.808).
• One more bonus point: using this value of R, I would have to throw a ball down with a speed of 26 m/s so that it bounces back with a speed of 21 m/s. This is the rebound speed needed to make the magic 75 foot rebound.

What about some other balls?

If it is worth bouncing the superball, it is worth bouncing some other balls. Here is a plot for those other balls.

Interesting that both the racquet ball (old racquet ball at that) and the black bounce ball have a larger coefficient of heightstitution than the super bounce ball. The hard pink plastic ball was pretty much the worst at bouncing (on this type of surface).

Just in case you need this for something (or future Rhett might need it), this is some other data about the balls.