Question

3. A small body of mass m falls to the earth from infinite distance away. What will be its velocity on reaching the earth? (Radius of the earth = R, acceleration due to gravity on the surface of the earth is g) :- (1) OR (2) 2gR (3) VgR (4) 2gR​

Answer 1

Answer:

$\dfrac{\sqrt{2GM}}{R}$

Explanation:

Assume:

G = universal gravitational constant

R = radius of the earth

M = mass of the earth

V = velocity of the boy when it reaches the earth's surface

Given:

m = mass of the boy

Assuming the potential energy at infinite distance from the earth to be zero. Since the boy falls from the infinite distance from the earth. So, the boy was initially at rest causing the initial kinetic energy of the boy to be zero.

In this case, the gain in kinetic energy of the boy will be equal to the loss in potential energy of the boy.

$\therefore \textrm{Gain in kinetic energy}=\textrm{Loss in potential energy}\\\Rightarrow \dfrac{1}{2}mV^2=\dfrac{GMm}{R^2}\\\Rightarrow \dfrac{1}{2}V^2=\dfrac{GM}{R^2}\\\Rightarrow V^2=\dfrac{2GM}{R^2}\\\Rightarrow V=\sqrt{\dfrac{2GM}{R^2}}\\\Rightarrow V=\dfrac{\sqrt{2GM}}{R}$

This means there is no option in the question which is right.