A particle with unit mass moves in a central potential where . Initially is a distance away from the origin moving with speed on a trajectory which, in the absence of any force, would be a straight line whose shortest distance from the origin is . The shortest distance between 's actual trajectory and the origin is , with , at which point it is moving with speed .
(i) Assuming , find in terms of and .
(ii) Assuming , find an expression for 's farthest distance from the origin in the form
where , and depend only on , and the angular momentum .
[You do not need to prove that energy and angular momentum are conserved.]