A satellite moves around the earth in an elliptical orbit. When it is at a distance which is average of its maximum and minimum distance from earth (at point P or at point Q on the orbit), angle between its velocity and acceleration is $60^{0}$ . Which of the following options is/are correct ?

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Maximum separation is thrice the minimum separation from earth

Maximum speed is thrice the minimum speed in the orbital motion

The speed at P or Q is geometric mean of maximum and minimum speed

Time to go from P to Q and back from Q to P is equal

answer is A, B, C.

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Detailed Solution

From conservation of energy :
$- \frac{G M m}{a} + \frac{1}{2} m V_{P}^{2} = - \frac{G M m}{2 a} \Rightarrow V_{P} = \sqrt{\frac{G M}{a}}$

At minimum or maximum separation $r ⊥ v$

From conservation of angular momentum

$r v = a V_{P} \sin θ$
$\Rightarrow r v = \sqrt{\frac{3 G M a}{4}} \Rightarrow v^{2} = \frac{3 G M a}{4 r^{2}}$
Also, $- \frac{G M m}{r} + \frac{3 G M m a}{8 x^{2}} = - \frac{G M m}{2 a} \Rightarrow r = \frac{a}{2}, \frac{3 a}{2}$
Also, $v_{1} v_{2} = \frac{3 G M a}{r_{1} r_{2}} = \frac{G M}{a}$