The Last Bounce
A dropped ball never quite stops bouncing, and stops after 4.81 seconds. The rattle at the end is the tail of a series you can hear converging.
What it is
A ball, dropped. Every arc it will take, drawn all at once, with a tick on the floor for every contact.
Listen rather than look. The bounces come closer and closer together until they run into each other and become a rattle, and then it’s over. Everybody has heard that sound. Nobody counts it.
The dashed line is where it stops. The ticks pile up against it and never cross it.
How it works
A ball comes back to a fixed fraction of the height it fell from. That fraction never changes, so the height never reaches zero, so there is always another bounce. Infinitely many of them.
The gaps between them shrink by that same fraction each time, and a geometric series with a ratio under one adds up to something finite. So the bounces don’t run out. The time does.
The total is exact: the first drop times (1 + e) / (1 − e). No approximation and no limit on
how many bounces are counted.
| bounciness | stops at | half the time | 90% | 99% | 99.9% | ratio measured |
|---|---|---|---|---|---|---|
| 0.45 | 1.41 s | 2 bounces | 4 | 7 | 10 | 0.4500 |
| 0.80 | 4.81 s | 4 bounces | 11 | 22 | 32 | 0.8000 |
| 0.92 | 12.82 s | 9 bounces | 29 | 56 | 84 | 0.9200 |
Four bounces get an ordinary ball halfway through its life. Getting to the last thousandth takes thirty two, and getting to the end takes all of them.
What surprised me
The rattle is most of the bounces and almost none of the time. For an ordinary ball, the first four contacts use up half of the total, and everything after them is crammed into the other half, with the last twenty of the thirty two audible ones inside the final tenth of a second.
So what you hear as one sound at the end - the brrrp - isn’t the ball giving up. It’s the majority of the bounces, arriving too fast to count. The part that sounds like an event and the part that sounds like a texture are the same process at two rates.
Where you stop listing them is a decision, and it shows. The list has to end somewhere, so it ends when the gaps fall under two milliseconds, which gives 29 bounces for an ordinary ball and 76 for a lively one. Both of those numbers are properties of my threshold rather than of the ball, and the ball’s own answer is the same in every case: all of them.
I nearly put the listed count on the poster as though it meant something. It’s on the page with what it actually is written next to it.
Two milliseconds is also roughly where hearing gives up. Contacts closer together than that stop being separate events and start being timbre, which is the same threshold day 61 ran into from the other side with echoes. The place the maths needs a cutoff and the place the ear imposes one are, by coincidence, about the same place.
What I would do next
Drop it on something that isn’t rigid. Real restitution isn’t a constant: it falls at higher impact speeds, so the first bounces lose more than the model says and the last ones lose less, and a real ball settles on a slightly different schedule from a geometric one.
The measurable version of that is comparing the recorded gaps from a real drop against the straight geometric fit, and the interesting part is which end they disagree at. That needs a microphone and a table, and it would turn a piece of arithmetic into an experiment.