An inertial frame $S$ and another reference frame $S^{\prime}$ have a common origin $O$, and $S^{\prime}$ rotates with angular velocity vector $\omega(t)$ with respect to $S$. Derive the results (a) and (b) below, where dot denotes a derivative with respect to time $t$ :

(a) The rates of change of an arbitrary vector $\mathbf{a}(t)$ in frames $S$ and $S^{\prime}$ are related by

$$(\dot{\mathbf{a}})_{S}=(\dot{\mathbf{a}})_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{a} .$$

(b) The accelerations in $S$ and $S^{\prime}$ are related by

$$(\ddot{\mathbf{r}})_{S}=(\ddot{\mathbf{r}})_{S^{\prime}}+2 \boldsymbol{\omega} \times(\dot{\mathbf{r}})_{S^{\prime}}+(\dot{\boldsymbol{\omega}})_{S^{\prime}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}),$$

where $\mathbf{r}(t)$ is the position vector relative to $O$.

Just after passing the South Pole, a ski-doo of mass $m$ is travelling on a constant longitude with speed $v$. Find the magnitude and direction of the sideways component of apparent force experienced by the ski-doo. [The sideways component is locally along the surface of the Earth and perpendicular to the motion of the ski-doo.]