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If a satellite is revolving around a pla...

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)]`

Text Solution

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Total mechanical energy of the system is `E = -(GMm)/(2a)` which remains conserved
`rArr KE + PE = -(GMm)/(2a)`
At a position .r. orbital speed of the satellite is v.
Then `KE = (1)/(2) mv^(2), PE = -(GMm)/(r)`
So, `(1)/(2) mv^(2) -(GMm)/(r) = -(GMm)/(2a)` (or) `v^(2) = GM((2)/(r) - (1)/(a))`.
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