Let $S$ be an inertial frame with coordinates $(t, x)$ in two-dimensional spacetime. Write down the Lorentz transformation giving the coordinates $\left(t^{\prime}, x^{\prime}\right)$ in a second inertial frame $S^{\prime}$ moving with velocity $v$ relative to $S$. If a particle has constant velocity $u$ in $S$, find its velocity $u^{\prime}$ in $S^{\prime}$. Given that $|u|<c$ and $|v|<c$, show that $\left|u^{\prime}\right|<c$.