Energy of proton moving fast (special relativity)

[FatheadVT]

Homework Statement

Given: in its own reference frame a proton takes 5 minutes to cross the Milky Way (10^5 meters).
(a) What is the approximate energy of the proton?
(b) About how long would the proton take to cross the galaxy as measured by an observer in the galaxy's reference frame?

Homework Equations

All those equations for special relativity I imagine.
L = L₀ Sqrt[1-(v/c)^2]
t = t₀ Sqrt[1-(v/c)^2]

The Attempt at a Solution

So I keep doing this over and over again, getting a wrong answer. The way I approach it is this: I think L is the distance as measured by the proton when crossing the galaxy, and t is the time the proton measures (5 minutes). Since v is the same in both reference frames (how fast the galaxy thinks the proton is moving and how fast the proton thinks the galaxy is moving) I can say that L/t = v. Using those equations I do it all out and find that v = c. Not very useful when I plug it into my equations for energy and find the proton has infinite energy. Where am I going wrong?

Also, I think this should be doable without the Lorentz transformations.

 

[Delphi51]

I'm not very confident about relativity, but I think you may have an error in assuming the galaxy is 10^5 meters wide from the point of view of the proton. I expect you have to use one of the formulas to transform this to the proton's point of view. Or else transform the 5 minutes to the galaxy's point of view. Before calculating the velocity. Of course you can't do that numerically, so you'll have an expression with a v in it rather than a number. Hopefully after the next step - find v - you will be able to get the two v's together somehow and solve for v.