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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 focus will be given by 
$$\displaystyle v^{2}=GM \left[ \frac{2}{r}-\frac{1}{a}\right]$$


Solution

Total mechanical energy of a satellite in an elliptical I orbit of semi major axis a is $$\displaystyle -\frac{GMm}{2a}$$.
$$E=K+U$$
$$\therefore $$   $$\displaystyle -\frac{GMm}{2a}=\frac{1}{2}mv^{2}-\frac{GMm}{r}$$
or $$\displaystyle v^{2}=GM\left [ \frac{2}{r}-\frac{1}{a} \right ]$$
Hence proved.

Physics

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