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

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-316/1/c/solution