Muons are created in the upper atmosphere of the Earth - Scottish Highers Physics - Question 7 - 2023

To find the observed lifetime of the muons, we apply the time dilation formula from the theory of relativity:

$$t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$$

where:

• $t'$ is the dilated time (lifetime measured by the observer on Earth)
• $t$ is the proper time (lifetime in the muon’s rest frame, 2.20 μs)
• $v$ is the speed of the muons (0.9c)
• $c$ is the speed of light.

First, we calculate $\sqrt{1 - \frac{(0.9c)^2}{c^2}}$:

$$\sqrt{1 - 0.81} = \sqrt{0.19} \approx 0.43589$$

Now, substituting back into the time dilation formula:

$$t' = \frac{2.20 \times 10^{-6} \text{ s}}{0.43589} \approx 5.04 \times 10^{-6} \text{ s}$$