Derivation of the relativistic velocity addition equations

We wish to derive the equation for addition of a velocity u’ at angle q’ with respect to the x axis to a platform that is already moving at speed v along the x axis.

Begin with the standard Lorentz transformation equations:

(1, 2, 3)

where

Then since u’ is at an angle q’ with respect to the x axis the velocity sum u will be at the angle q with respect to the x axis and the following equation ensue :

(4,5,6,7)

Using equations 4,5,6 and 7 with x’=u’t’cos(q’) and y’= u’t’sin(q’) in equations 1,2 and 3 we have:

(8, 9, 10)

Divide equations 8 and 9 by equation 10 to obtain:

(11, 12)

If q’=0 then q=0 and we have from equation 11:

so that no velocity sum can exceed c.