How to get final velocity in relativistic mechanics

[Agustin.R]

Hello guys, I am not sure whether this is actually a simple problem but I'm not really a physics student, just someone very passionate for it.

So I'm trying to use Newton's second law in its relativistic version (force is equal to the rate of change in momentum) to find the velocity and acceleration of a particle after a discrete period of time t. My difficulty arises because the velocities are present in the gammas, so I tried solving by substitution but it ended being really messy.

So my problem is to get the final velocity (or the acceleration, of course) of a particle given the force acting on it, its initial velocity, position and acceleration, and its mass, using discrete quantities and in three dimensions. Should I be using four-vectors?
Relevant formulae:
F = dp/dt (should actually be deltas as I'm looking at the discrete case)
p = m*v*gamma
gamma = 1 / sqrt(1 - v²/c²)

I would appreciate any help you can give me, even if it is just pointing in the right direction. Thanks in advance.

 

[jbsteele]

The best way to approach this problem is to use a numerical approach. In other words, you can break down the problem into smaller parts, and use numerical methods such as Runge-Kutta or Euler's method to solve for the velocity and acceleration of the particle at discrete time steps. You could also use an iterative approach, where you start with an initial velocity and then use the formula F = dp/dt to calculate the force, which you can then use to calculate the acceleration. Then you can use the acceleration to calculate the new velocity after a small time step. You can repeat this process until you reach the desired time step. This approach will allow you to solve for the velocity and acceleration without having to solve complex equations.