Question

Prove that for an earth satellite, the ratio of its velocity at apogee (when farthest from earth) to its velocity at perigee (when closer to earth) is in the inverse ratio of its distance at apogee and perigee.

Answer 1

Let distances of satellites from earth at apogee be `r_(1)` and at perigee be `r_(2)`.
Let `v_(1) and v_(2)` be the respective velocities of the satellite.
According to the principle of conservation of angular momentum,
`I_(1)omega_(1) = I_(2)omega_(2) or mr_(1)^(2) ((v_(1))/(r_(1))) = mr_(2)^(2) ((v_(2))/(r_(2)))`
or`r_(1) v_(1) = r_(2) v_(2) :. (v_(1))/(v_(2)) = (r_(2))/(r_(1))`
which was to be proved.