# Relativistic momentum and gamma factor - differential equation

1. Apr 12, 2012

### kapitan90

1. The problem statement, all variables and given/known data
I am supposed to show that the force on a relativistic particle when
a) it's perpendicular to particle's velocity is $$F=γm_0\frac{dv}{dt}$$
b) it's parallel to particle's velocity is $$F_x=m_0γ^3\frac{dv_x}{dt}$$

I may make use of the fact that $$\frac{dγ}{dt}=\frac{v_xγ^3}{c^2}*\frac{dv_x}{dt}$$
2. Relevant equation
I don't understand why $$\frac{dγ}{dt}$$is 0 in case a) (this follows from the last formula, but I don't understand it either), why do we use $$v_x$$ instead of v which γ depends on ?

Last edited: Apr 12, 2012
2. Apr 12, 2012

### kapitan90

I am also supposed to show that the kinetic energy of a particle accelerated from rest to $$v_x$$is$$W =∫F_xdx=m_0c^2(γ-1)$$ but I am stuck with the integral$$∫ (1-v^2/c^2)^{-0.5} dx$$ I tried to integrate it by parts and to use Wolfram, but it couldn't solve it either. Any ideas?

3. Apr 12, 2012

### Mindscrape

One problem at a time. First problem
Your formula for d(gamma)/dt is not quite right. Well, it's right for your scenario, but it's also confusing you. Try going over the time derivative again. Maybe first try to take the time derivative of the full momentum vector, and then analyze the two cases when it comes time. Then to predict a future complication, don't be worried if you have a v^2/c^2, you can rewrite as some common terms.

Okay, because I don't know when I'll check back, I'll give you a clue to the second problem, it starts with hyper and ends with trig.

Last edited: Apr 12, 2012