Mean decay length
= Mean decay length
{title2=$\ell_d=\beta\gamma c\tau_0$}
For a freely moving unstable <particle> of fixed <speed> $v=\beta c$ and <proper mean lifetime> $\tau_0$, the mean travelled <distance> before decay is $\ell_d=v\gamma\tau_0$. Survival beyond <distance> $L$ has <probability> $e^{-L/\ell_d}$, since elapsed <proper time> is $L/(\gamma v)$. For <rest mass> $m$ and <three-momentum> magnitude $p$, $\ell_d=p\tau_0/m$.