If a satellite is revolving around a planet of mass M in an elliptical orbit of semi-major axis a , find the orbital speed of the satellite when it is at a distance r from the focus

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TestHub · Accepted Answer

As in case of elliptic orbit of a satellite mechanical energy

remains constant, at any position of satellite in the orbit,

$K E + P E = - \frac{G M m}{2 a}$ ...(i)

Now, if at position $r, v$ is the orbital speed of satellite

$K E = \frac{1}{2} m v^{2}$ and $P E = - \frac{G M m}{r}$ ...(ii)

So, from Eqs. (i) and (ii), we have

$\frac{1}{2} m v^{2} - \frac{G M m}{r} = - \frac{G M m}{2 a}, i . e ., v^{2} = G M [\frac{2}{r} - \frac{1}{a}]$

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