= Particle-speed invariant between slowly approaching walls
{title2=$v\ell\approx\mathrm{constant}$}
An <elastic collision> with a wall moving toward the particle at speed $V$ changes the particle speed by approximately $2V$ when recoil is negligible. The collision rate is approximately $v/\ell$ for $v\gg V$, and $\dot\ell=-2V$, yielding $\dot v/v=-\dot\ell/\ell$. For a freely moving finite wall of mass $M$, the neglected recoil changes the increment by order $(m/M)v$, so relative accuracy also requires $(m/M)(v/V)\ll1$.
// Target: classical-mechanics.bigb
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