Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 1 / c / Solution

In axes directed toward periapsis and along the motion there, the position is $\mathbf{r}=r(\cos f,\sin f)$. Differentiating it and using $\dot{f}=h/r^2$ givesDifferentiating the orbit equation gives $\dot{r}=(\mu e/h)\sin f$, and substitution simplifies the components to