Question

A particle falls towards earth from infinity. It's velocity on reaching the earth would be

• A. Infinity
• B. $\sqrt{2gR}$
• C. $2\sqrt{gR}$
• D. Zero

Answer 1

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

Since, there are not any external force acting on the

object. We can apply consecration of mechanical

energy

$\therefore$ conservation of Mechanical Energy :-

$K.E(kineticenergy{)}_{\alpha }+(PotentialEnergy{)}_{\alpha }P.E$

$=(k.E+P.E{)}_{surface}$

$K.E_{\alpha }$ and $P.E_{\alpha }=0$ (since, the object falls from infinite)

$\therefore (K.E+P.E{)}_{\alpha }=(K.E+P.E{)}_s$

$\Rightarrow 0+0=\frac{1}{2}=\frac{1}{2}m{v}^2+\left(\frac{-GMm}{R}\right)$

$\Rightarrow \frac{1}{2}n{v}^2=\frac{GMm}{R}$

$\Rightarrow v^2=\frac{2GM}{R}$

$\Rightarrow v^2=2g^R$ [$\because g=\frac{GM}{R^2}\Rightarrow g^R=\frac{GM}{R}]$

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