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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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As in case of elliptic orbit with semi major axes 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)` …(1)
Now, if at position r, v is the orbital speed of satellite, we have
`KE = (1)/(2) mv^(2)` and `PE = -(GMm)/(r )` ..(2)
So from equation (1) and (2), we have
`(1)/(2) mv^(2) - (GMm)/(r ) = -(GM m)/(2a)`
i.e., `v^(2) = GM[(2)/(r ) - (1)/(a)]`
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