Particle-speed invariant between slowly approaching walls ($v\ell \approx \mathrm{constant}$)

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 $(m/M)(v/V)\ll 1$.

// Target: classical-mechanics.bigb