Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 2 / ii / Solution

In polar coordinates in the orbital plane, $h=r^2\dot{\theta }$. Setting $u=1/r$, the Binet equation for the effective inverse-square force isIts general solution isand henceThis is a branch of a hyperbolic Kepler orbit; because $1-\beta <0$, its physical branch has a negative denominator.