Question

A muon produced in the Earth’s atmosphere is travelling with a speed of 0.90c. As

measured in the muon’s frame of reference, it has a lifetime of 1.6 us. What is its

lifetime as measured by an observer on Earth?

Answer 1

The lifetime as measured by an observer on Earth can be found using the relativistic time dilation (see https://en.wikipedia.org/wiki/Time_dilation#Simple_inference):

$$\Delta t' = \dfrac{\Delta t}{\sqrt{1-v^2/c^2}}$$

where $\Delta t'$ is the lifetime as measured by an obsever, $\Delta t = 1.6\mu s = 1.6\times 10^{-6}s$ is the lifetime in the frame of the muon, $v = 0.9c$ is the speed of the muon, and $c$ is the speed of light. Thus, obtain:

$$\Delta t' = \dfrac{1.6\times 10^{-6}}{\sqrt{1-0.9^2c^2/c^2}} \approx 3.67\times 10^{-6}s$$

Answer. 3.67 us.