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 at a distance r from the planet will be given by `v^(2)=GM[2/r-1/a]`

Answer 1

As in case of elliptic orbit with semi major axis a, of a satellite total mechanical energy remains constant, at any position of satellite in the orbit, given as
`E = -(GMm)/(2a)`
or `KE+PE= -(GMm)/(2a)` …(i)
Now, if at position r, v is the orbital speed of satellite, we have
`KE = (1)/(2)mV^(2)` and `PE = -(GMm)/(r )` ....(ii)
so from equation (i) and (ii), we have
`(1)/(2)mV^(2) - (GMm)/(r )= -(GMm)/(2a), i.e., V^(2)=GM[(2)/(r )-(1)/(a)]`