A fixed-step twin of a bouncing ball: how wrong are its restitution and time to rest, and does calibrating on the first bounce fix it?

research/impact-twin

A fixed-step twin of a bouncing ball: the true rest time (v1/g)(1+e)/(1−e) is hypersensitive to e (2/(1−e²)), the twin's apparent restitution and settling time are off by O(dt) with scheme-dependent sign (−20% to +4% at dt=10⁻², e=0.9), and calibrating e on the first bounce is not a reliable fix.

10 tests

What it found

Truth is closed form: infinitely many impacts in the finite Zeno time T=(v₁/g)(1+e)/(1−e), with d ln T/de=2/(1−e²) (10.5 at e=0.9, 100 at 0.99). A semi-implicit-Euler twin with a penetration test has an apparent restitution biased by the step with scheme-dependent sign (true 0.9: 0.886, 0.895, 0.879 at dt=10⁻²) and its time to come to rest is off by −19.6%, +4.0%, −30.8% (e=0.9; up to 47% at e=0.95), first-order in dt and 0.1–0.3% at 10⁻⁴. The restitution bias explains most of the clamp error (predicted −14.9%, measured −19.6%), but fitting e on the first rebound is not a reliable repair: it helps the clamp twin at dt=10⁻² (−19.6% to −4.8%), does nothing at finer steps and makes the mirror twin worse (+4.0% to +53%). Twin impact counts (238–688) are mostly floor chatter against 43–87 resolvable bounces. Stylised: one dimension, constant restitution, rigid contact, closed-form real system; the crude interpolated-contact variant is the worst here.

A figure from its own output

Figure produced by research/impact-twin: zeno_vs_e — impact-twin/experiments/zeno_vs_e.svg, written by the project's own experiment code.

Paper, code and results

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.