The Cosmic Ray Detector Network · AI Generator #GEN-CCFA5766

Physics

Problem

A high-energy cosmic ray proton enters Earth's atmosphere and creates a muon through particle interactions. This muon has an initial speed of

v = 0.95c

and a proper lifetime of

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

. The muon is created at an altitude of

h = 15\;\text{km}

above sea level. Two cosmic ray detectors are positioned to study this muon: Detector A is located at sea level directly below the muon's creation point, and Detector B is located on a mountain at an altitude of

5\;\text{km}

, also directly below the muon's path. (a) According to an observer on Earth, what is the dilated lifetime of the muon? (b) What is the probability that the muon will reach Detector B before decaying? Assume the muon travels in a straight line downward and that radioactive decay follows the law

N(t) = N_0 e^{-t/\tau}

, where

\tau

is the dilated lifetime. (c) In the muon's rest frame, what distance must it travel to reach Detector B?

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