Question

A particle falls towards earth from infinity. The velocity with which it reaches earths surface is

• A. $v=2gR$
• B. $v=\sqrt{2gR}$
• C. $v=\sqrt{gR}$
• D. $v=\frac{R}{g}$

Answer 1

The correct option is B $v=\sqrt{2gR}$

For a body falling from infinity, there will be no gravitational potential energy on it due to Earth. And if we assume, it's initial speed is zero.
Then the body's initial energy is zero.
As it reaches the Earth's surface it gains speed and also gravitational potential energy.

i.e $E=\frac{1}{2}m{v}^2-\frac{GMm}{R}$

From the conservation of energy, this should be equal to the initial total energy of the body.

$\therefore \frac{1}{2}m{v}^2-\frac{GMm}{R}=0$
$\Rightarrow \frac{1}{2}m{v}^2=\frac{GMm}{R}$
$\Rightarrow v=\sqrt{\frac{2GM}{R}}$

But $GM=g{R}^2$

$\Rightarrow v=\sqrt{2gR}$