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Take rightward velocities as positive just before the right-wall collision. The incoming particle velocity is $v$ and the wall velocity is $-V$. Their total momentum divided by total mass gives the centre of mass velocityIn an elastic collision, relative velocity reverses, so the particle's signed outgoing velocity is $v_{\mathrm{out}}=2v_{\mathrm{cm}}-v$. For $M>m$ it is negative, and its speed isConsequently the exact change in speed isFor $m/M\to 0$ with $v,V$ fixed, this proves the requested leading increment $v^{\prime }-v\simeq 2V$. Reflection gives the same increment at the left wall.

This approximation is not uniform when $v/V$ is allowed to become arbitrarily large. To neglect recoil relative to the increment $2V$ requiresThe inequalities $m\ll M$ and $v\gg V$ alone do not imply it: for example, $m/M=10^{-3}$ and $v/V=10^3$ give an exact zero speed increment. The subsequent slowly moving-wall approximation therefore uses the consistent joint range $1\ll v/V\ll M/m$, or treats the walls as effectively prescribed massive reflectors. The assumed slow change of wall speed over many impacts embodies the same negligible-recoil requirement.

Within this approximation, the separation derivative is the right-wall velocity minus the left-wall velocity:For a time window containing many crossings but little separation change, $\ell /v\ll \Delta t\ll \ell /V$, the flight time between successive wall collisions is approximately $\ell /v$. Thus their number and the cumulative speed change arePassing to the averaged differential equation givesHence the particle-speed invariant between slowly approaching walls isIt is an adiabatic invariant for the rapid elastic bouncing motion while the separation and wall velocities vary slowly on the flight-time scale.