= Relativistic decay survival to a detector
{title2=$P_{\mathrm{survive}}=2^{-R/(\gamma vT)}$}
A freely moving unstable particle of speed $v$ travels a laboratory distance $R$ in time $R/v$, accumulating <proper time> $R/(\gamma v)$. For rest-frame <half-life> $T$ and <exponential decay>, its survival probability is the displayed expression. At least half survive exactly when $\gamma vT\geq R$. The minimum speed is therefore
$$
v_{\min}=\frac{cR}{\sqrt{R^2+c^2T^2}}.
$$
The relevant flight length is $\gamma vT$, rather than $\gamma cT$ for a particle whose speed is appreciably less than $c$.
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