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An earth satellite is revolving in a circular orbit of radius a with velocity $v_{0}$ . A gun is in the satellite and is aimed directly towards the earth. A bullet is fired from the gun with muzzle velocity $\frac{v_{0}}{2}$ . Neglect the air resistance and recoil of gun.
M – mass of earth; m = mass of bullet (m << M)
Column – I Column – II
A) The maximum distance of the bullet from centre of earth is n.a. The value of ‘n’ is …….. (P) 0.5
B) The minimum distance of the bullet from the centre of earth is n.a. The value of ‘n’ is ….. (Q) $\frac{4}{3}$
C) The speed of bullet at the maximum distance from the Centre of earth is $k v_{0}$ . The value of ‘k’ is (R) $\frac{2}{3}$
D) The total energy of bullet + earth system is $- \frac{G M m}{2 k a}$ (S) 2
The value of ‘k’

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A – Q; B – S; C – R; D – S

A – S; B – R; C – P; D – Q

A – S; B – P; C – Q; D – Q

A – P; B – R; C – S; D – S

answer is B.

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Detailed Solution

$v_{0} = \sqrt{\frac{G M}{a}}$

Question Image

Conservation of angular momentum
Mass = mass ……… (1)
Conservation of mechanical energy
$\frac{1}{2} m [v_{0}^{2} + \frac{v_{0}^{2}}{4}] - \frac{G M m}{a} = \frac{1}{2} m v^{2} - \frac{G M m}{r} \Rightarrow r = 2 a, \frac{2 a}{3}; s e m i m a j o r a x i s x = (2 a + \frac{2 a}{3}) \frac{1}{2}$

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