Question

A muon has a mean lifetime of 2.2$\mathbf{\mu s}$in its rest frame. Suppose muons are travelling at 0.92c relative to Earth. What is the mean distance a muon will travel as measured by an observer on Earth?

Short answer

Themean distance travelled by muon on Earth is 1549 m.

Step-by-step solution

Identification of given data

The mean lifetime of muon is $t=2.2\mathrm{\mu s}$

The speed of muonis $v=0.92c$.

Definition of Time Dilation

The difference in the observed time and actual time of event due to the relative movement between observer and frame is called the time dilation.

Determination of mean distance travelled by muon as measured by an observer on Earth

The mean distance travelled by muon on Earth is given as:

$d=\frac{v.t}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}}$

Here, c is the speed of light and its value is $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Substitute all the values in the above equation.

$$\begin{gathered} d=\frac{\left(0.92\mathrm{c}\right)\left({\displaystyle \frac{3\times {10}^8\mathrm{m}/\mathrm{s}}{\mathrm{c}}}\right)\left(2.2\mathrm{\mu s}\right)\left({\displaystyle \frac{{10}^{-6}\mathrm{s}}{1\mathrm{\mu s}}}\right)}{\sqrt{1-{\left({\displaystyle \frac{0.92\mathrm{c}}{\mathrm{c}}}\right)}^2}} \\ \mathrm{d}=1549\mathrm{m} \end{gathered}$$

Therefore, the mean distance travelled by muon on Earth is $1549\mathrm{m}$.