= Proper-time acceleration by reflected laser light
{title2=$\beta(\tau)=\tau/(\tau+\tau_0)$}
An ideal receding mirror reflecting a source beam of constant <power> $P$ receives <force> $(2P/c)(1-\beta)/(1+\beta)$ after including both Doppler-shifted reflected <momentum> and reduced impact rate. For fixed <rest mass> $M$, use $dt/d\tau=\gamma$ in the <momentum> equation. It reduces to $d\beta/d\tau=(1-\beta)^2/\tau_0$, where $\tau_0=Mc^2/(2P)$. Starting from rest gives $\beta=\tau/(\tau+\tau_0)$. This continuum mirror model neglects individual-photon recoil and finite beam interception.
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