We have derived the velocity addition formula using two Lorentz transformations (in the same direction) and rapidity.

We can also compute this in terms of the usual going from one frame in which an object is moving to a frame boosted along the direction. (This should give the same result as above if the particle is moving in the direction.)

Let the velocity in the original system be .

This formula gives the same result as above but is more general since it allows us to add velocities that are in different directions.

Jim Branson 2012-10-21