A satellite of mass is in a circular orbit of radius 2R around the earth. By mistake another satellite 2m of mass m is put in the same orbit and having opposite sense of rotation. The collision between the two satellites is perfectly inelastic and combined mass strikes the earth’s surface at an angle $θ$ with horizontal. Value of $\cos θ$ is $\frac{2}{\sqrt{n}}$ , where n is ______.

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answer is 19.

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Detailed Solution

By cons of momentum $2 m u - m u = 3 m v$
$V = \frac{1}{3} \sqrt{\frac{G M}{2 R}}$
Total energy $= \frac{P^{2}}{6 m} - \frac{G M (3 m)}{R}$
Where P is linear momentum when it hits the earth
$\frac{P^{2}}{6 m} - \frac{3 G M m}{R} = \frac{- 17 G M m}{12 r} p = \frac{19}{2} m \sqrt{\frac{G M}{R}}$
By conservation of angular momentum
$R_{P} \cos θ = 3 m v (2 R) \cos θ = \frac{2}{\sqrt{2}} \times \frac{2}{19} = \frac{2 \sqrt{2}}{19}; \sec θ = \frac{19}{2 \times 1.414} = 6.72$