Gravitation - Result Question 44

47. A particle of mass $m$ is thrown upwards from the surface of the earth, with a velocity $u$. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and $g$ is acceleration due to gravity on the surface of the earth. The minimum value of $u$ so that the particle does not return back to earth, is

[2011M]

(a) $\sqrt{\frac{2 GM}{R}}$

(b) $\sqrt{\frac{2 GM}{R^{2}}}$

(c) $\sqrt{2 gR^{2}}$

(d) $\sqrt{\frac{2 GM}{R^{2}}}$

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Answer:

Correct Answer: 47. (a)

Solution:

  1. (a) The velocity $u$ should be equal to the escape velocity. That is,

$u=\sqrt{2 gR}$ But $g=\frac{G M}{R^{2}}$

$\therefore u=\sqrt{2 \cdot \frac{GM}{R^{2}} \cdot R} \Rightarrow \sqrt{\frac{2 G M}{R}}$



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