Past exam of the short options course of the University of Oxford / 2017 / 8 / b / ii / Solution

The laboratory flight time is $R/v_B$. By time dilation, the accumulated proper time is $\tau =R/({\gamma }_B{v}_B)$. The definition of the rest-frame half-life gives at least one-half survival when $\tau \le T$, or ${\gamma }_B{v}_B\ge R/T$. Equivalently,Solving for the minimum speed givesUnder the usual exponential decay law, the survival probability is exactly $2^{-R/({\gamma }_B{v}_{B}T)}$. This is relativistic decay survival to a detector.