Answer:
The moun lives 2.198*10^-6 s as measured by its own frame of reference
The Earth moved 648 m as measured by the moun's frame of reference
Explanation:
From the point of view of the observer on Earth the muon traveled 3.53 km at 0.983c
0.983 * 3*10^8 = 2.949*10^8 m/s
Δt = d/v = 3530 / 2.949*10^8 = 1.197*10^-5 s
The muon lived 1.197*10^-5 s from the point of view of the observer.
The equation for time dilation is:
![\Delta t' = \Delta t * \sqrt{1 - \frac{v^2}{c^2}}](https://tex.z-dn.net/?f=%5CDelta%20t%27%20%3D%20%5CDelta%20t%20%2A%20%5Csqrt%7B1%20-%20%5Cfrac%7Bv%5E2%7D%7Bc%5E2%7D%7D)
Then:
![\Delta t' = 1.197*10^-5 * \sqrt{1 - \frac{(0.983c)^2}{c^2}} = 2.198*10^-6 s](https://tex.z-dn.net/?f=%5CDelta%20t%27%20%3D%201.197%2A10%5E-5%20%2A%20%5Csqrt%7B1%20-%20%5Cfrac%7B%280.983c%29%5E2%7D%7Bc%5E2%7D%7D%20%3D%202.198%2A10%5E-6%20s)
From the point of view of the moun Earth moved at 0.983c (2.949*10^8 m/s) during a time of 2.198*10^-6, so it moved
d = v*t = 2.949*10^8 * 2.198*10^-6 = 648 m