Question

The mean lifetime of stationary muons is measured to be $\mathbf{2}\mathbf{.}\mathbf{2000}\mathit{\mu }\mathit{s}$. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be $\mathbf{16}\mathbf{.}\mathbf{000}\mathit{\mu }\mathit{s}$. To five significant figures, what is the speed parameter $\mathit{\beta }$of these cosmic-ray muons relative to Earth?

Short answer

The speed parameter of cosmic ray muons is $0.99050$.

Step-by-step solution

Identification of given data 

The mean life time for stationary muons is $t_0=2.20\mu s$

The mean life time for high-speed muons is $t=16\mu s$

The speed parameter in the above problem is found by considering the Lorentz factor as the ration of mean life time for stationary muons to mean life time of high-speed muons.

Determination of speed parameter of cosmic ray muon relative to Earth 

The speed parameter of cosmic ray muons is given as:

$\beta =\sqrt{1-{\left(\frac{t_0}{t}\right)}^2}$

Substitute all the values in equation.

$$\begin{gathered} \beta =\sqrt{1-{\left(\frac{2.20\mu s}{16\mu s}\right)}^2} \\ \beta =0.99050 \end{gathered}$$

Therefore, the speed parameter of cosmic ray muons is $0.99050$