# Kilometers Across the Galaxy from POV of Proton

**1. The problem statement, all variables and given/known data**

The cosmic rays of highest energy are protons, which have kinetic energy on the order of 10^13 MeV.

(a) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter on the order of ~10^5 light-years, as measured in the proton’s frame?

*(b) From the point of view of the proton, how many kilometers across is the galaxy?*

The answer to part (a) is 295 s, I need help with part (b)

**2. Relevant equations**

[itex]\gamma[/itex] = sqrt(1-v^2/c^2)

Δt = [itex]\gamma[/itex] tp

ΔL = [itex]\gamma[/itex] Lp

KE = ([itex]\gamma[/itex] -1)mc^2

**3. The attempt at a solution**

Since I have the answer to part (a), obtained through manipulation of the above equations, I tried simply ΔL = v(tp), which gives 8.88×10^7 km. This answer is apparently off by more than 10%.

Any help is appreciated! Thank you!

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