= Solution
The laboratory flight time is $R/v_B$. By <time dilation>, the accumulated <proper time> is $\tau=R/(\gamma_Bv_B)$. The definition of the rest-frame <half-life> gives at least one-half survival when $\tau\leq T$, or $\gamma_Bv_B\geq R/T$. Equivalently,
$$
\frac{v_B^2}{1-v_B^2/c^2}\geq\frac{R^2}{T^2}.
$$
Solving for the minimum speed gives
$$
\boxed{v_{B,\min}=\frac{cR}{\sqrt{R^2+c^2T^2}}.}
$$
Under the usual exponential decay law, the survival probability is exactly $2^{-R/(\gamma_Bv_BT)}$. This is <relativistic decay survival to a detector>.
Back to article page
