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If a satellite revolves around a planet of mass $M$ in an elliptical orbit of semi-major axis $a$ , find the orbital speed of that satellite when its distance is $r$ from the focus.

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v^{2} = GM [\frac{2}{r} - \frac{1}{a}]

v^{2} = GM [\frac{2}{r^{2}} - \frac{1}{a}]

v^{2} = GM [\frac{1}{r} - \frac{1}{a}]

None of these

answer is A.

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Detailed Solution

Orbital speed of that satellite is

v^{2} = GM [\frac{2}{r} - \frac{1}{a}]

.
Total energy planet revolving in an elliptical orbit of semi-major axis

a

E_{1} = - \frac{GMm}{2 a}

… (1)
When a planet is at a distance

r

from the focus, let

v

is its speed.
Then energy of a planet,

E_{2} = \frac{1}{2} m v^{2}

⇒

E_{2} = - \frac{GMm}{r}

… (2)
By energy conservation, we get,

E_{1} = E_{2}

\Rightarrow

- \frac{GMm}{2 a} = \frac{1}{2} m v^{2} - \frac{GMm}{r}

\Rightarrow

GM [\frac{1}{r} - \frac{1}{2 a}] = \frac{1}{2} v^{2}

\Rightarrow

v^{2} = 2 GM [\frac{2 a - r}{2 ra}]

\Rightarrow

v^{2} = GM [\frac{2}{r} - \frac{1}{a}]

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