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Dimensional Analysis

An asteroid o...

Question

An asteroid of mass M is approaching earth, initially at a distance of $10 R_{e}$ with speed $V_{i}$ . It hits the earth with a speed $V_{f}$ (Consider $R_{e}$ and $M_{e}$ as radius and mass of the earth), then:

$V_{f}^{2} = V_{i}^{2} + \frac{G M_{e}}{R} [1 - \frac{1}{10}]$

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$V_{f}^{2} = V_{i}^{2} + \frac{2 G M}{R_{e}} [1 + \frac{1}{10}]$

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$V_{f}^{2} = V_{i}^{2} + \frac{2 G M_{e}}{R_{e}} [1 - \frac{1}{10}]$

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$V_{f}^{2} = V_{i}^{2} + \frac{2 G M}{R_{e}} [1 - \frac{1}{10}]$

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Solution

The correct option is C
$V_{f}^{2} = V_{i}^{2} + \frac{2 G M_{e}}{R_{e}} [1 - \frac{1}{10}]$

Kinetic energy of asteroid $= \frac{1}{2} m v_{i}^{2}$
P.E. of asteroid $= \frac{- G M_{e} m}{10 R_{e}}$
Apply law of energy conservation
$K_{1} + P_{1} = K_{2} + P_{2} \frac{1}{2} m v_{i}^{2} - \frac{G M_{e} m}{10 R_{e}} = \frac{1}{2} m v_{f}^{2} - \frac{G M_{e} m}{R_{e}} \frac{v_{i}^{2}}{2} - \frac{G M_{e}}{10 R_{e}} + \frac{G m_{e}}{R_{e}} = \frac{v_{f}^{2}}{2} \frac{v_{i}^{2}}{2} + \frac{G M_{e}}{R_{e}} [1 - \frac{1}{10}] = \frac{v_{f}^{2}}{2} v_{f}^{2} = v_{i}^{2} + \frac{2 G M_{e}}{R_{e}} [1 - \frac{1}{10}]$

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