(i) An inertial frame $S$ has orthonormal coordinate basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. A second frame $S^{\prime}$ rotates with angular velocity $\boldsymbol{\omega}$ relative to $S$ and has coordinate basis vectors $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$. The motion of $S^{\prime}$ is characterised by the equations $d \mathbf{e}_{i}^{\prime} / d t=\boldsymbol{\omega} \times \mathbf{e}_{i}^{\prime}$ and at $t=0$ the two coordinate frames coincide.

If a particle $P$ has position vector $\mathbf{r}$ show that $\mathbf{v}=\mathbf{v}^{\prime}+\boldsymbol{\omega} \times \mathbf{r}$ where $\mathbf{v}$ and $\mathbf{v}^{\prime}$ are the velocity vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$.

(ii) For the remainder of this question you may assume that $\mathbf{a}=\mathbf{a}^{\prime}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$ where $\mathbf{a}$ and $\mathbf{a}^{\prime}$ are the acceleration vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$, and that $\omega$ is constant.

Consider again the frames $S$ and $S^{\prime}$ in (i). Suppose that $\omega=\omega \mathbf{e}_{3}$ with $\omega$ constant. A particle of mass $m$ moves under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{r}$. When viewed in $S^{\prime}$ its position and velocity at time $t=0$ are $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(1,0,0)$ and $\left(\dot{x}^{\prime}, \dot{y}^{\prime}, \dot{z}^{\prime}\right)=(0,0,0)$. Find the motion of the particle in the coordinates of $S^{\prime}$. Show that for an observer fixed in $S^{\prime}$, the particle achieves its maximum speed at time $t=\pi /(4 \omega)$ and determine that speed. [Hint: you may find it useful to consider the combination $\zeta=x^{\prime}+i y^{\prime}$.]