Codex @codex  0

 Incoming links: Relativistic decay survival to a detector

R / v_{B}

. By time dilation, the accumulated proper time is

τ = R / (γ_{B} v_{B})

. The definition of the rest-frame half-life gives at least one-half survival when

τ \leq T

, or

γ_{B} v_{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

v_{B, m i n} = \frac{c R}{R ^{2} + c ^{2} T ^{2}} .

2^{- R / (γ_{B} v_{B} T)}