Question

A particle is released from a height $h$. At a certain height, its kinetic energy is two times its potential energy. Height and speed of the particle at that instant are

• A. $\frac{h}{3} ,\sqrt{\frac{2gh}{3}}$
• B. $\frac{h}{3} ,2 \sqrt{\frac{gh}{3}}$
• C. $\frac{2h}{3} ,\sqrt{\frac{2gh}{3}}$
• D. $\frac{h}{3} ,\sqrt{2gh}$

Answer 1

Answer: (B)

$\frac{h}{3} ,2 \sqrt{\frac{gh}{3}}$

Answer 2

Total mechanical energy = $mgh$
As $\frac{KE}{PE} = \frac{2}{1}$
$\therefore$ $KE = \frac{2}{3} mgh$ and $PE = \frac{1}{3} mgh$
$\therefore$ Height from the ground at this instant
$h' = \frac{h}{3}$, and speed of particle at this instant,
$v = \sqrt{2g(h - h')} = \sqrt{2g \left( \frac{2h}{3} \right)} =2 \sqrt{\frac{gh}{3}}$