Write down the formulae for a Lorentz transformation with velocity $v$ taking one set of co-ordinates $(t, x)$ to another $\left(t^{\prime}, x^{\prime}\right)$.

Imagine you observe a train travelling past Cambridge station at a relativistic speed $u$. Someone standing still on the train throws a ball in the direction the train is moving, with speed $v$. How fast do you observe the ball to be moving? Justify your answer.