Answer Happy

A satellite is going around the earth in an
elliptical orbit and has maximum and minimum
distance from the centre of earth equal to 10r and
r respectively. It was planned to fire on board
rocket so as to increase the energy of the satellite
by maximum amount. Assume that the rocket
is fired for a small time (almost instantaneous)
and gives an impulse J to the satellite in forward
direction. Take J to be small compared to overall
momentum of the satellite.
(a) Show that firing the rocket when the satellite
is at perigee (nearest to earth) will result in
maximum gain in energy of the satellite.
[The orbit of mass orbiter mission, informally
called Mangalyaan, was raised in five steps
using this principle; before it was given the
escape speed]
(b) Find the impulse J that the rocket must impart
to the satellite at perigee so that its maximum
distance from earth’s centre, during its course
of motion in elliptical path, becomes 12r. Take
mass of satellite as m and mass of earth as M.
Assume that there is negligible change in mas