Answer to Question #257947 in Physics for Anand

Question

The mean lifetime of a particle is 2.5 μs when it is produced in the top layers of the Earth’s atmosphere. If it travels at the speed of 0.9999c, what is its life time as measured by an observer on the Earth? Calculate the distances travelled by the particle in its own frame of reference and the Earth’s frame of reference.

Answer 1

The time for observer on the Earth:

$$t=t_0/\sqrt{1-v^2/c^2}=177\space\mu\text{s}.$$

The distance for the muon itself:

$$L_0=0.9999c·2.5·10^{-6}=750\text{ m}.$$

The distance measured by the observer:

$$L=0.9999c·177·10^{-6}=53\text{ km}.$$