What is the Correct Method for Deriving p = γmv?

[physics user1]

I thought this mental experiment: consider an inertial frame of reference solidal to a particle moving with velocity v respect to another inertial frame of reference, i got to find the momentum in the second frame of reference (in the first is 0 since is solidal with the particle)

p= m* dx/dt

And then i plug instead of x and t the lorentz tranformation thwt connects them with t' and x'...

But it doesn't work, where am i wrong? Is that method correct? If not why?

 

[haushofer]

That doesn't work, because the (spatial component of the!) relativistic momentum is p = ymv to start with. You should transform the 4-momentum. This 4-momentum is defined by the relativistic action of a point particle.

 

[SiennaTheGr8]

The idea is to find an expression that:

• is conserved in closed systems (and additive) even when $v \rightarrow c$,
• increases in magnitude without bound as $v \rightarrow c$ (because otherwise it would have a maximum limit like $v$ does, and this would contradict the first criterion), and
  
• reduces to $\mathbf{p} \approx m \mathbf{v}$ in the classical limit.

The famous Tolman/Lewis thought experiment demonstrates that if such a quantity exists, then it must be $\mathbf{p} = \gamma m \mathbf{v}$, which obviously meets the second and third criteria. See here: https://books.google.com/books?id=FrgVDAAAQBAJ&pg=PA76.

As for the first criterion, experiment verifies that this vector is indeed conserved at all possible values of $v$. (Think particle accelerators.)