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Let's assume we have a high energy muon (e.g. 500 GeV) that enters into the water. To calculate what is the distance traveled by the muon before it decays, is it enough to consider that it will stop when it reaches the critical energy of water ($E_c$=74 MeV), traveling ~9 radiation lengths ($X_0$(water)~36 g/cm$^2$)? Or should I take into account that the proper time is also changing with the energy loss?

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