Two particles of masses $m_{1}$ and $m_{2}$ have position vectors $\mathbf{r}_{1}(t)$ and $\mathbf{r}_{2}(t)$ at time $t$. The particle of mass $m_{1}$ experiences a force $\mathbf{f}$ and the particle of mass $m_{2}$ experiences a force $-\mathbf{f}$. Show that the centre of mass moves at a constant velocity, and derive an equation of motion for the relative separation $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$.

Now suppose that $\mathbf{f}=-k \mathbf{r}$, where $k$ is a positive constant. The particles are initially at rest a distance $d$ apart. Calculate how long it takes before they collide.