Question

The proper lifetime of an unstable particle is equal to $\Delta t_0=10ns$. Find the distance this particle will traverse till its decay in the laboratory frame of reference, where its lifetime is equal to $\Delta t=20ns$.

Answer 1

The distance travelled in the laboratory frame of reference is $v\Delta t$ where $v$ is the velocity of the particle. But by time dilation
$\Delta t=\frac{\Delta t_0}{\sqrt{1-\frac{v^2}{c^2}}}$ So
$v=c\sqrt{1-{\left(\frac{\Delta t_0}{\Delta t}\right)}^2}$
Thus the distance traversed is
$c\Delta t\sqrt{1-{\left(\frac{\Delta t_0}{\Delta t}\right)}^2}$