How many bounces does a bouncy ball bounce

if a bouncy ball could count bounces

I love the sound of a nice bouncy ball as it is nearing the end of its bouncing: that speed up to infinity as the individual bounces merge into a musical note of rising pitch, diminishing volume, and the whole thing races off past perception.

1. How many bounces before it stops?

2. Why does it seem to stop so soon? Should a really nice ball could just keep bouncing with ever decreasing height until a Boltzmann energy limit is reached where the bounce height gravity potential equals the ambient thermal energy?

Figure 1, the bouncy ball and microphone

Use a Microphone to record the bounces

Bounce the ball on the table, close to the microphone record with Audacity:

Figure 2, the audio track of the bouncing ball.

How to plot the data?

Try plotting (number of bounces) vs ( Time ). Manually locate the bounces and their times.

Figure 3, Number of Bounces as a function of Time in seconds

The gist of the plot in Figure 3 agrees with intuition - the number of bounces looks like some geometric increase. Jump over to www.desmos.com for a quick manual plot-fit. Use an exponential function, with a few adjustable parameters, Figure 4.

Figure 4, a manual curve fit, using simple exponential function. Is it coincidence that the time-scale-factor, “b”, is unity? Would that be true for all bouncy balls?

Answer to Question 1?

The curve fitting is good enough that, assuming any time range that I hear the bounces, I can say how many bounces. Looking at the sound file, it seems many fewer bounces than I would have guess just from the sound.

And there is the strange fact that the curve fit gives a time-scale-factor of unity - - the count increase is a simple exponential by time in seconds ~exp(time/1.0). Is that a universal property of counting, or just a coincidence for my ball? Both options seem unlikely.

What about Question 2 - when exactly does the bouncing stop, and why?

I think the stopping of bounce is directly related to how much the ball deforms on the bounce.

Figure 5, the bouncing ball deforms slightly. When the amount of deformation equals the height it would have bounced, then there is no more bouncing.

Maybe when the height of the bounce equals (R0-Rb) then there is no more bounce.

My ball bounces to ~0.85 of whatever height it is dropped from. I do not have any photographic equipment that can capture the deformation at bounce. Hence I will use my assumption, work the other way, using the time it takes to stop bouncing to estimate the amount of deformation.

Dropped from 20cm, I can see 35 bounces on the microphone-sound plot. The height after 35 bounces is 20cm * (0.85^35) = 0.06cm. NOPE. That does not make sense to me. The ball is not deforming by 0.6mm when it is in tiny-bouncing mode near the end.

Will continue thinking on the mechanism for the end of bouncing.