Question

If a satellite is revolving around a planet of mass `M` in an elliptical orbit of semi-major axis `a`. Show that the orbital speed of the satellite when it is a distance `r` from the focus will be given by
`upsilon^(2) = GM[(2)/(r ) - (1)/(a)]`

Answer 1

When a satellite is orbiting in ellipic orbit, its mechanical energy, i.e., sum of `P.E.` amd
`K.E. ( = - (GM m)/(2a))` remains constant, at every
position of satellite in the orbit, i.e.,
`P.E. + K.E. = - (G M m)/(2 a)`
If `upsilon` is the orbital speed of satellite, at position `r`,
then `P.E. = - (GM m)/(r )`
and `K.E. = (1)/(2) m upsilon^(2)`
So `- (GM m)/(r ) + (1)/(2) m upsilon^(2) = - (GM m)/(2a)`
or `upsilon^(2) = GM[(2)/(r ) - (1)/(a)]`