An asteroid was fast approaching the earth. Scientists fired a rocket which hit the asteroid at a distance of 5 R from the centre of the earth (R = radius of the earth). Immediately after the hit the asteroid’s velocity (V₀) was making an angle of $θ$ = 30° with the line joining the centre of the earth to the asteroid. The asteroid just grazed past the surface of the earth. Find V₀[Mass of the earth = M]

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$\sqrt{\frac{32 G M}{105 R}}$

$\sqrt{\frac{45 G M}{108 R}}$

$\sqrt{\frac{62 G M}{109 R}}$

$\sqrt{\frac{24 G M}{65 R}}$

answer is B.

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Detailed Solution

$Conserving angular momentum, {mV}_{0} . 5 R = mvR \Rightarrow V = 5 V_{0} . . . . . . . . (i) conserving energy, \frac{1}{2} {mV}_{0}^{2} + (- \frac{GMm}{5 R}) = \frac{1}{2} {mV}^{2} + (- \frac{GMm}{R}) . . . . . . . . (ii) Solving, V_{0} = \sqrt{\frac{32}{105} \frac{GM}{R}}$

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