The muons created by cosmic rays in the upper atmosphere rain down uniformly on

Question

The muons created by cosmic rays in the upper atmosphere rain down uniformly on the earth's surface, although some of them decay on the way down, with a half-life of about 1.5?s (measured in their rest frame). A muon detector is carried in a balloon to an altitude of 2000m, and in the course of one hour detects 650 muons traveling at 0.99c towards the earth. If an identical detector remains at sea level, how many muons will it register in one hour? (Remember that after n half-lives, 2^(-n) of the original particles survives).

Explanation / Answer

Time taken by the muons to reach sea level = D'/0.99c

= Relativistically shortened distance / Speed of the muons

= [2000 m * {(1 - 0.99*0.99)}/0.99c]

= 285/(3*10^8) s

= 0.95*10^(-6) s

= 0.95 s.

Now, half life of a muon in its rest frame = 1.5 s.

So, N' = N*2^(-n)

n = t/t(1/2) = 0.95/1.5 = 0.6333

So, N' = 650*2^(-0.6333) = 419.

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