Energy conservation in Special Relativity

[AllRelative]

Homework Statement

A train is traveling in the x direction at a speed of v. On the train, a passenger is playing darts and fires a dart in the y direction with a speed of Uy. The dart hits the target and stops abruptly. What is the difference in the mass of the dart as observed by a person on the train versus an observer on the ground observing the dart as it hits the dart board?

E = γu * m*c^2
Kinetic energy= energy moving - energy at rest = γu*m*c^2 - m*c^2

Homework Equations

Is the initial mass of the dart (mass while flying) the same for the two observers?

The Attempt at a Solution

Let S be the reference frame of the observer on the ground.
S' is the reference frame of the observer in the train moving at v with the dart.

Since the total energy of the flying dart is the same as when it stops in a reference frame, we can find the final mass in each frames.

I get:
final mass in S = [1-(Uy^2 + v^2)/c^2]^-1/2 * initial mass in S

Final mass in s' = [1-Uy^2/c^2]^-1/2 * initial mass in S'

I'm not sure if we can say that Initial mass in S ans initial mass in S' is the same.

After that I guess I would get the answer by subtracting the final mass of S' with the final mass in S.

 

[mfb]

The mass is always the same for all observers because the concept of a relativistic (velocity-dependent) mass is not used in physics.
We can look at the change in energy. I don't understand your calculations.

What is the energy of a dart, resting in the frame of the train? What is its energy as seen from the train if it is moving slowly (!) towards the board? What is the difference?
You can repeat the same for the other observer.