Prove that for an earth satellite,the ratio of its velocity at apogee to its velocity at perigee is the inverse ratio of its distances from apogee and perigee.

Answer 1

The path of a sattelite around the Earth is elliptical. The perigee is the point where the satellite is nearest to the earth, and the apogee is the point where it is farthest from the earth. Let R(p) and R(a) be the respective distances of these points from the centre of the earth, and V(p) and V(a) be the magnitudes of the velocities at these points. Since vectors of velocities V(a) and V(p) are nornal to the position vectors to these points, the angular momenta are
L(p) = m•V(p) •R(p); L(a)=m•V(a) •R(a). Using the law of conservation of angular momentum L(p) = L(a) => m•V(p) •R(p)= m•V(a) •R(a) =>
V(a)/V(p)= R(p)/R(a).