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.

 

[thomate1]

Since t0 is the time measured by the proton, I can just use that to find the velocity (v = L/t0). Then use that to find the energy. But I still get an infinite answer.I would approach this problem by first clarifying the given information and assumptions. The statement says that the proton takes 5 minutes to cross the Milky Way in its own reference frame. This implies that the proton is moving at a constant speed in its own reference frame. However, in special relativity, the speed of an object is relative and can be different in different reference frames. So, is the given time based on the speed of the proton in its own reference frame or in the reference frame of the galaxy?

Assuming that the given time is based on the speed of the proton in its own reference frame, we can use the Lorentz transformations to find the speed of the proton in the reference frame of the galaxy. Using the equation t = t0 Sqrt[1-(v/c)^2], we can solve for v to find that the speed of the proton in the reference frame of the galaxy is v = 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999