Relativistic Velocity Addition

We have already discussed the motion of a system S' with respect to another system S and derived the Lorentz equations to transform the coordinates of one system into the other one. Now we go one step further and think about an object that moves within the system S' with the speed u'. If an observer in the system S wants to measure the speed $u$ in his system, he will use the formula $$u = \frac{\Delta x}{\Delta t}$$ For both quantities $\Delta x$ and $\Delta t$ we can now insert the previously derived Lorentz transformation formulas (more precisely the back-transformation) $$\Delta x = \frac{\Delta x' + v\Delta t'}{\sqrt{1-\frac{v^2}{c^2}}}$$ for the distance $\Delta x$ and $$\Delta t = \frac{\Delta t' + \frac{v}{c^2}\Delta x'}{\sqrt{1-\frac{v^2}{c^2}}}$$ for the time difference $\Delta t$. Inserting these two equations into $u$ leads to $$u = \frac{\Delta x' + v\Delta t'}{\Delta t' + \frac{v}{c^2}\Delta x'}$$ Now we can rearrange this equation according to $$u = \frac{\frac{\Delta x'}{\Delta t'} + v}{1+\frac{v}{c^2}\frac{\Delta x'}{\Delta t'}}$$ Replacing now $\Delta x'/\Delta t'$ with u' leads to the following very important formula for adding up two velocities within the theory of relativity: $$\boxed{u = \frac{u' + v}{1+\frac{vu'}{c^2}}}$$