Calculating Proton Momentum, Energy, Time Stretch & Lab Time

[Blueshift5]

A proton emits a flash of light every meter of its own (proper) time. Between successive flash emissions, each proton travels a distance specified as recorded by a lab observer. Find the momentum, Energy, Time stretch (gamma), and lab time between flashes for the specified values.

Homework Equations

(proper time)^2 = (lab time)^2 - (lab distance)^2

Energy = rest mass + kinetic energy

Energy = restmass * lab time/proper time = restmass * gamma

The Attempt at a Solution

I am asking this question because I get the same values for Energy, Gamma, and Lab Time. I also get that the momentum should be equal to the lab distance. This doesn't feel right and am looking for a double check. Thanks!

 

[Jhero]

First, we need to convert the given distance into proper time using the equation (proper time)^2 = (lab time)^2 - (lab distance)^2. We can rearrange this equation to solve for proper time:
(proper time)^2 = (lab time)^2 - (lab distance)^2
proper time = sqrt((lab time)^2 - (lab distance)^2)

Next, we can use the equation for Energy to find the momentum. The equation for Energy is: Energy = rest mass + kinetic energy. We can substitute the value for kinetic energy with the equation restmass * lab time/proper time = restmass * gamma. This gives us:
Energy = rest mass + (restmass * lab time/proper time)
Energy = rest mass + (restmass * lab time/sqrt((lab time)^2 - (lab distance)^2))

To find the momentum, we simply substitute the value for lab distance into the equation for momentum:
Momentum = lab distance

Finally, we can use the equation for Time stretch (gamma) to find the value for gamma. The equation for Time stretch (gamma) is:
gamma = lab time/proper time
gamma = lab time/sqrt((lab time)^2 - (lab distance)^2)

To find the lab time between flashes, we simply subtract the proper time from the lab time:
Lab time between flashes = lab time - proper time
Lab time between flashes = lab time - sqrt((lab time)^2 - (lab distance)^2)

Overall, the values for Energy, Gamma, and Lab Time seem correct. However, the value for momentum should not be equal to the lab distance. This is because momentum is a vector quantity and the direction of the proton's motion must also be taken into account. We would need to know the direction of the proton's motion in order to calculate its momentum accurately.