# Solve relativistic velocity in terms of momentum (vector equation)

1. Dec 3, 2012

### winstonyin

Given the formula $\vec{p}=\frac{m_0}{\sqrt{1-\frac{|\vec{v}|^2}{c^2}}}\vec{v}$, I'd like to make $\vec{v}$ the subject, so I can do a numerical approximation for some relativistic motion problem. I want to treat it as a vector equation, but since it is non-linear, the only way I can think of is to split it into 3 equations with $|\vec{v}|^2=v_x^2+v_y^2+v_z^2$. This is however very complicated, though Mathematica gave me the answer analogous to the equation with only magnitudes of the vectors. Is there a simple way I can solve such kind of vector equations?

Edited 3 Dec:
Solution: $\vec{v}=\frac{\vec{p}}{\sqrt{m_0^2+\frac{|\vec{p}|^2}{c^2}}}$

Last edited: Dec 3, 2012
2. Dec 5, 2012

### Meir Achuz

1. Square the equation to get p^2=f(v^2).
2. Solve this for v^2=g(p^2).
3. Put this nto the square root to replace the v^2 by p^2.
4. Rewrite the original equation with the new square root.