The Cosmic Ray Muon Detector · AI Generator #GEN-945009EF

Physics · modern-physics

Problem

A cosmic ray muon detector on a mountaintop records muons with an average energy of

E = 4.0 \times 10^{-11}\;\text{J}

. These muons have a rest mass of

m_0 = 1.88 \times 10^{-28}\;\text{kg}

and a proper lifetime of

\tau_0 = 2.2 \times 10^{-6}\;\text{s}

(the lifetime measured in the muon's rest frame). Due to time dilation, these high-energy muons live longer in the laboratory frame than their proper lifetime would suggest, allowing them to reach the detector before decaying. If a muon travels

h = 15\;\text{km}

vertically downward through the atmosphere at constant speed before being detected, what is the probability that it survives the journey without decaying? Assume the muon maintains constant energy during its travel and use

c = 3.0 \times 10^8\;\text{m/s}

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