Question

Muons are unstable subatomic particles with a mean lifetime of 2.2 μs that decay to electrons. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth’s surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth’s surface.What is the greatest distance a muon could travel during its 2.2 μs lifetime?According to your answer in part A, it would seem that muons could never make it to the ground. But the 2.2 μs lifetime is measured in the frame of the muon, and they are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth?

Answer 1

Muons are either positively charged ore negatively charged. Following are their decay equations:

The life time of the muons and they move with a velocity

So, the distance travelled by the muon should be

But, the muons are observed at the ground level almost below 10 km from stratosphere where they are created by the impact of cosmic rays onto the atmospheric particles.

this is because the life time used in the calculation is the life time in muon's frame of reference (t_o) while in the frame of reference of the observer at rest on the earth, the muon is moving with a speed 0.999c thus, special theory of relativity acts and the life time must be dilated in case of the observer. So, the dilated life time (t) would be,

Distance travelled by the muons (as measured by the observer)

That is why the muons are observed at the ground level by the observer at rest on the ground.

Regards