A particle falls towards the Earth from infin | vedclass

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(B) The velocity of a particle falling from infinity to the Earth's surface is equal to the escape velocity of the Earth. By the principle of conserva

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$\sqrt{2gR}$

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(B) The velocity of a particle falling from infinity to the Earth's surface is equal to the escape velocity of the Earth. By the principle of conservation of energy,the total energy at infinity is zero (kinetic energy + potential energy = $0 + 0 = 0$). At the surface of the Earth,the total energy is $\frac{1}{2}mv^2 - \frac{GMm}{R} = 0$. Solving for $v$,we get $v = \sqrt{\frac{2GM}{R}}$. Since $g = \frac{GM}{R^2}$,we have $GM = gR^2$. Substituting this,$v = \sqrt{\frac{2gR^2}{R}} = \sqrt{2gR}$.

$A$ particle falls towards the Earth from infinity. Its velocity on reaching the Earth would be:

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