Question

A body starts from rest from a point distance $R_0$ from the centre of the earth of mass M, radius R.. The velocity acquired by the body when it reaches the surface of the earth will be

• A. $2GM(\frac{1}{R}-\frac{1}{R_0})$
• B. $\sqrt{2GM(\frac{1}{R_0}-\frac{1}{R})}$
• C. $GM(\frac{1}{R}-\frac{1}{R_0})$
• D. $2GM\sqrt{(\frac{1}{R}-\frac{1}{R_0})}$

Answer 1

The correct option is B $\sqrt{2GM(\frac{1}{R_0}-\frac{1}{R})}$

$P.E.={\int }_{R_0}^R\frac{GMm}{r^2}dr=-GMm[\frac{1}{R}-\frac{1}{R_0}]$
The K.E. acquired by the body at the surface $=\frac{1}{2}m{v}^2$
$\therefore \frac{1}{2}m{v}^2=-GMm[\frac{1}{R}-\frac{1}{R_0}]$
$v=\sqrt{2GM(\frac{1}{R_0}-\frac{1}{R})}$