Question

A body at rest starts from a point at a distance r (>R) from the center of the earth. If M and R stand for the mass and the radius of earth respectively, then the speed of the body when it reaches the Earth surface is

• A. $\sqrt{\frac{2GM}{R}}$
• B. $\sqrt{\frac{2GM}{R-r}}$
• C. $\sqrt{\frac{2GM(r-R)}{Rr}}$
• D. zero

Answer 1

The correct option is C $\sqrt{\frac{2GM(r-R)}{Rr}}$

$V_{potentialatr}=\frac{-GM}{r},V_{atR}=\frac{-GM}{R}$

$energy=\frac{-GMm}{R}+\frac{GMm}{r}$

$=GM-\left(\frac{R-r}{Rr}\right)$

By energy theoram,

$\frac{\left(R-r\right)GMm}{Rr}=\frac{1}{2}m{v}^2$

$v=\sqrt{\frac{2GM\left(r-R\right)}{Rr}}$

So option $C$ is correct.