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Q.

A particle falls on earth :
(i) from infinity ; (ii) from a height $10$ times the radius of earth. The ratio of the velocities gained on reaching at the earth's surface is :

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a

$\sqrt{11} : \sqrt{10}$

b

$10 : 11$

c

$11 : 10$

d

$\sqrt{10} : \sqrt{11}$

answer is A.

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Detailed Solution

When particle falls from infinity

$\begin{array}{rcl} P E_{i} + K E_{i} & = & P E_{f} + K E_{f} \\ 0 + 0 & = & \frac{- G M m}{R} + \frac{1}{2} m v_{1}^{2} \\ v_{1} & = & \sqrt{\frac{2 G M}{R}} \end{array}$

When particle falls from a height

$\begin{array}{rcl} P E_{i} + K E_{i} & = & P E_{f} + K E_{f} \\ \frac{- G M m}{11 R} + 0 & = & \frac{- G M m}{R} + \frac{1}{2} m v_{2}^{2} \\ v_{2} & = & \sqrt{\frac{10}{11} \times \frac{2 G M}{R}} \end{array}$

$∴ \frac{v_{1}}{v_{2}} = \sqrt{\frac{11}{10}}$

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A particle falls on earth :(i) from infinity ; (ii) from a height 10 times the radius of earth. The ratio of the velocities gained on reaching at the earth's surface is :