A small satellite of mass ' m ' is revolving around earth in a circular orbit of radius r 0 with speed v 0 . At certain point of its orbit, the direction of motion of satellite is sudden

Question

A small satellite of mass ' m ' is revolving around earth in a circular orbit of radius r 0 ​ with speed v 0 ​ . At certain point of its orbit, the direction of motion of satellite is suddenly changed by angle θ = cos − 1 ( 3/5 ) by turning its velocity vector in the same plane of motion, such that speed remains constant. The satellite, consequently goes to elliptical orbit around earth. The ratio of speed at perigee to speed at apogee is :-

TestHub · Accepted Answer

Using conservation of angular momentum about O . $mV_{P} r_{P} = mV_{A} r_{A} = mV_{0} r_{0} cos θ$ $V_{A} r_{A} = V_{P} r_{P} = \frac{3 v _{0} r _{0}}{5}$ Using conservation of energy

$\frac{1}{2} m V_{A}^{2} + \frac{- GM m}{r _{A}} = \frac{- GM m}{r _{0}} + \frac{1}{2} m V_{0}^{2} \Rightarrow \frac{9 V _{0}^{2} r _{0}^{2}}{50 r _{A}^{2}} - \frac{V _{0}^{2} r _{0}}{r _{A}} + \frac{V _{0}^{2}}{2} = 0 [Let \frac{r _{0}}{r _{A}} = x] \Rightarrow 9 x^{2} - 50 x + 25 = 0 \Rightarrow x = 5 or (5/9) \Rightarrow r_{A} = \frac{9 r _{0}}{5}; r_{P} = \frac{r _{0}}{5} \Rightarrow \frac{V _{P}}{V _{A}} = \frac{r _{A}}{r _{P}} = 9$

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JEE Advanced (2028)