Past exam of the physics 1 paper of the University of Oxford / 2019 resit / 3 / Solution

Treat the asteroid as a point mass with negligible influence on the planet. Outside a spherical planet, Newton's law of universal gravitation gives the specific gravitational potential $-GM/r$. The conserved specific energy and specific angular momentum areAt the periapsis $r_p$, the radial velocity vanishes, so conservation of energy givesThe right-hand side increases with $r_p>0$. The grazing threshold $r_p=R$ is therefore the grazing impact parameter in gravitational focusing relation. For $d>R$ its equivalent speed isThe threshold is customarily called the minimum avoiding speed, with tangency counted as its limiting boundary. If any contact with the surface counts as a hit, it is an infimum, not an attained minimum. For $d\le R$, no finite incoming speed avoids contact: gravitational focusing bends the path inward, and even its infinite-speed straight-line limit has closest approach $d\le R$. The finite radius of the asteroid has been neglected.