A frame $S^{\prime}$ moves with constant velocity $v$ along the $x$ axis of an inertial frame $S$ of Minkowski space. A particle $P$ moves with constant velocity $u^{\prime}$ along the $x^{\prime}$ axis of $S^{\prime}$. Find the velocity $u$ of $P$ in $S$.

The rapidity $\varphi$ of any velocity $w$ is defined by $\tanh \varphi=w / c$. Find a relation between the rapidities of $u, u^{\prime}$ and $v$.

Suppose now that $P$ is initially at rest in $S$ and is subsequently given $n$ successive velocity increments of $c / 2$ (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of $P$ in $S$ is

$$c\left(\frac{e^{2 n \alpha}-1}{e^{2 n \alpha}+1}\right)$$

where $\tanh \alpha=1 / 2$.

[You may use without proof the addition formulae $\sinh (a+b)=\sinh a \cosh b+\cosh a \sinh b$ and $\cosh (a+b)=\cosh a \cosh b+\sinh a \sinh b$.]