Calculating Energy and Time for a Proton Crossing the Milky Way Galaxy

[jk4]

Homework Statement

In its own frame of reference, a proton takes 5 min to cross the Milky Way galaxy, which is about $$10^{5}$$ light-years in diameter.
(a) What is the approximate energy of the proton in electronvolts?
(b) About how long would the proton take to cross the galaxy as measured by an observer in the galaxy's reference frame?

mass of proton = $$1.6726 x 10^{-27} kg$$
$$1eV = 1.602 x 10^{-19} J*s$$

Homework Equations

It says the problem belongs to the "Energy and Momentum" section, so here are those equations:
$$E = \gamma mc^{2}$$
$$p = \gamma mv$$
$$E^{2} = (mc^{2})^{2} + p^{2}c^{2}$$
$$E = pc$$ (Massless particle, probably doesn't apply to this problem)

The Attempt at a Solution

well, I'm pretty confused. I calculated that the distance of $$10^{5}$$ light-years corresponds to about $$9.45 x 10^{15} m$$. But I can't figure out how to proceed from here.
I'm not necessarily looking for an answer but a little help.

 

[Pacopag]

I'm no expert so take this with a grain of salt.
Write down the time dilation equation. Then use
$$v={L\over \Delta t^'}$$ i.e. proper length divided time observed by a stationary observer in the Milky way.
Putting these together you can get the speed. The rest is gravy.
As a guide, I get the order of 10^3 in joules for the energy. That sucker is flyin.

Good luck.

 

[pam]

The galaxies velocity in the proton's rest frame is
v/c=(L/5 min)/gamma (a dimensionless number).
Use this equation to find gamma. Be careful with the numbers
The proton's energy is E=\gamma mc^2. Use mc^2=931 MeV for a proton.
The time to cross the galaxy in its rest frame is T=gamma X 5 min.