A spaceship is sent to investigate a planet of mass M and radius R, while hanging motionless in space at a distance 5R from the center of the planet. It fires a package with speed $v_{0}$ as shown in the figure. The package has a mass m, which is much smaller than the mass of the spaceship. The angle $θ$ at which the package will just graze the surface of the planet at the top is

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$\sin^{- 1} (\frac{1}{5} \sqrt{1 + \frac{8 G M}{3 v_{0}^{2} R}})$

$\sin^{- 1} (\frac{1}{2} \sqrt{1 + \frac{8 G M}{5 v_{0}^{2} R}})$

$\sin^{- 1} (\frac{1}{5} \sqrt{1 + \frac{8 G M}{5 v_{0}^{2} R}})$

$\sin^{- 1} (\frac{1}{5} \sqrt{1 + \frac{2 G M}{5 v_{0}^{2} R}})$

answer is B.

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Detailed Solution

a) $(m v_{o} \sin θ) 5 R = m v_{1} R .... (1)$
b) $\begin{array}{l} \frac{1}{2} m v_{0}^{2} - \frac{G M m}{5 R} = \frac{1}{2} m v_{1}^{2} - \frac{G M m}{R} .... (2) \end{array}$
Solving $θ = \sin^{- 1} [\frac{1}{5} \sqrt{1 + \frac{8 G M}{5 v_{0}^{2} R}}]$