Velocity Addition for Free

Ever go through the traditional derivation of the relativistic formula for addition of velocities? Painful, because there are a bunch of things you have to hold in your head for the duration of a calculation. Modern tools make this easy, almost obvious. First, recall this sketch:

in which Geometry Expressions calculates the Lorentz transformation for free. Now, suppose the point B really represents the endpoint of a particle path, the beginning-point being the origin (0,0). In the blue frame -- our lab frame -- the (average) velocity of the particle over this path is x/t, which we can write as u. What's the velocity of the same particle path, this time measured in the red frame, moving with velocity v w.r.t. our blue frame? Why, let's let Mathematica compute it for us. Use the "Edit / Copy As / Mathematica" menu item in Geometry Expressions on the yellow expressions in the sketch above, and just paste into Mathematica:

Beautiful simplicity itself, eh? Once again, I've been lazy with absolute-value expressions, just sidestepping them. This means the signs -- but only the signs -- may turn out wrong and we would have to use common sense to correct them.