Adding Velocities: Formulas & Calculations

[lavster]

When adding velocities how do you know whether to use formula of the form$$w=\frac{u+v}{1+\frac{uv}{c^2}}$$, derived from using lorentz transformation, and when to use w = c sqrt[c^2(u-v)^2-u^2v^2+(u.v)^2]/(c^2-u.v) derived using four velocities?

thanks

 

[henry_m]

The second equation is the more general one, and $\vec{u}$ and $\vec{v}$ (vectors) can be arbitrary. The first is the special case of parallel vectors of magnitudes u, v, which you can derive fairly straightforwardly from the second (though the sign conventions here are different: you must be careful about who is moving relative to whom and in what directions).