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)]`

Explanation

total energy of the system is `E=-(GMm)/(2a)`
Which is conserved. So, `KE+PE=-(GMm)/(2a)`
At position `r` orbital speed of the statellite is `V`. Then `KE=1/2 mV^(2)` and `PE=-(GMm)/r`
So, `1/2mV^(2)-(GMm)/r=-(GMm)/(2a)`
`rArr V=sqrt(GM(2/r-1/a))`