 Solution

The signed rapidity corresponding to velocity $u$ isEquivalently $u/c=\tanh \varphi$, $\gamma =\cosh \varphi$, and $\gamma u/c=\sinh \varphi$. Thus the Lorentz boost from part (i) becomesMultiplying two such matrices gives diagonal entries $\cosh \varphi \cosh \psi +\sinh \varphi \sinh \psi =\cosh (\varphi +\psi )$ and off-diagonal entries $\cosh \varphi \sinh \psi +\sinh \varphi \cosh \psi =\sinh (\varphi +\psi )$, by the addition identities for hyperbolic functions. HenceThe resulting rapidity is $\varphi +\psi$. In velocity variables this is the relativistic velocity-addition formula $u\oplus v=(u+v)/(1+uv/c^2)$.