Question

a person is moving in same direction as muon measures lifetime of muon to be 1 micro second if speed of muon and person with respect to lab are c/2 and c/3 .calculate proper life time of muon and distance moved by muon with respect to lab.

Answer 1

solution:-

given data

speed of muon with respect to lab(v)=$\frac{c}{2}$

speed of person with respect to lab(v')=$\frac{c}{3}$

relative velocity can be written as

$v_{rel}=\frac{v-v'}{1-\frac{vv'}{c^2}}$

$v_{rel}=\frac{\frac{c}{2}-\frac{c}{3}}{1-\frac{c^2}{6c^2}}=\frac{c}{5}$

proper life time for muon can be written

$\Delta t'=\Delta t \sqrt{1-\frac{v_{rel}^2}{c^2}}$

by putting the value

$\Delta t=1\sqrt{1-\frac{c^2}{25c^2}}=0.9797\mu s$

distance moved by muon with respect to lab is

$d=v_{rel} \Delta t'=\frac{c}{5}\times(0.9797)\times10^{-6}$

$=\frac{3\times10^8}{5}\times(0.9797)\times10^{-6}$

$d=58.78m$

therefore distance moved by muon is 58.78m.