Write down the formulae for the one-dimensional Lorentz transformation $(x, t) \rightarrow$ $\left(x^{\prime}, t^{\prime}\right)$ for frames moving with relative velocity $v$ along the $x$-direction. Derive the relativistic formula for the addition of velocities $v$ and $u$.

A train, of proper length $L$, travels past a station at velocity $v>0$. The origin of the inertial frame $S$, with coordinates $(x, t)$, in which the train is stationary, is located at the mid-point of the train. The origin of the inertial frame $S^{\prime}$, with coordinates $\left(x^{\prime}, t^{\prime}\right)$, in which the station is stationary, is located at the mid-point of the platform. Coordinates are chosen such that when the origins coincide then $t=t^{\prime}=0$.

Observers A and B, stationary in $S$, are located, respectively, at the front and rear of the train. Observer C, stationary in $S^{\prime}$, is located at the origin of $S^{\prime}$. At $t^{\prime}=0$, C sends two signals, which both travel at speed $u$, where $v<u \leq c$, one directed towards $\mathrm{A}$ and the other towards $\mathrm{B}$, who receive the signals at respective times $t_{A}$ and $t_{B}$. $\mathrm{C}$ observes these events to occur, respectively, at times $t_{A}^{\prime}$ and $t_{B}^{\prime}$. At $t^{\prime}=0, \mathrm{C}$ also observes that the two ends of the platform coincide with the positions of $A$ and $B$.

(a) Draw two space-time diagrams, one for $S$ and the other for $S^{\prime}$, showing the trajectories of the observers and the events that take place.

(b) What is the length of the platform in terms of $L$ ? Briefly illustrate your answer by reference to the space-time diagrams.

(c) Calculate the time differences $t_{B}-t_{A}$ and $t_{B}^{\prime}-t_{A}^{\prime}$.

(d) Setting $u=c$, use this example to discuss briefly the fact that two events observed to be simultaneous in one frame need not be observed to be simultaneous in another.