Isn't the figure given below (taken from a slide in the lecture, at time 37:50) referring to the case where there is an independent object moving relative to the inertial frames S and S'? In the example of the twins that he discusses, there is no object independent of the frames of reference. We simply have the twins, each attached to her respective reference frame. Why, then, does he use the formula for Lorentz velocity addition? What object is he referring to?

-
I don't quite understand how the formula for Lorentz velocity addition applies to the situation of the twins. –  Joebevo Oct 8 '12 at 4:17