Question

The starting point for our discussion of relativity was the observation that
velocities do not combine in the naive way by just adding together. Clearly
then, a good question is “just how do they combine?” In this problem, you
will derive the formula for the ‘relativistic composition of velocities.’ Let
me point out that you have all of the tools with which to do this, since you
know how to translate both distances and times between reference frames.
This is just a quantitative version of problem (3-8), but we now work in
the more general case where vAB is arbitrary. Feel free to choose units so
that c = 1. Then you can ignore all the factors of c to make the algebra
easier.

Answer 1

suppose

$$v_x' =\frac{dx'}{dt'}; v_y'=\frac{dy'}{dt'};v_z'\frac{dz'}{dt'}$$

Form Lorentz transformation

$$dx=\frac{dx'+v_0dt'}{\sqrt{1-\frac{v_0^2}{c^2}}}; dy=dy'; dz=dz'; dt=\frac{dt'+\frac{v_0}{c^2}dx'}{\sqrt{1-\frac{v_0^2}{c^2}}}$$

And we have it

$$v_x=\frac{v_x'+v_0}{1+\frac{v_0v_x}{c^2}}; v_y=\frac{v_y'\sqrt{1-\frac{v_0^2}{c^2}}}{1+\frac{v_0v_x}{c^2}}; v_z=\frac{v_z'\sqrt{1-\frac{v_0^2}{c^2}}}{1+\frac{v_0v_x}{c^2}}$$

If the body moves parallel to the x axis in this case, the law of addition of velocities takes the form

$$v_x=\frac{v_x'+v_0}{1+\frac{v_0v_x}{c^2}}$$