A particle is released from a height H. At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

Question

A particle is released from a height H. At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively (A) (H/3), √(2gH/3)  (B) (H/3), 2 √(gH/3)  (C) (2H/3), √2 g H  (D) (2H/3), 2 √(2gH/3)

Tardigrade Admin · Accepted Answer

Total mechanical energy of particle, i.e.

PE + K E = m g H

...(i)
Given,

K E = \frac{1}{2} PE

i.e.

\frac{K E}{PE} = \frac{1}{2}

or

PE = 2 K E

Substituting in this Eq. (i), we get

PE + K E = m g H

2 K E + K E = m g H

3 K E = m g H

K E = \frac{m g H}{3}

Similarly,

PE = \frac{2}{3} m g H

So, height from ground at that instant,

h = \frac{2 H}{3}

So, speed of particle,

v = 2 g h = 2 g H /3

Q. A particle is released from a height $H$ . At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

$\frac{H}{3}, \frac{2 g H}{3}$

$\frac{H}{3}, 2 \frac{g H}{3}$

$\frac{2 H}{3}, 2 g H$

$\frac{2 H}{3}, 2 \frac{2 g H}{3}$

Q. A particle is released from a height H. At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

3H​,32gH​​

3H​,23gH​​

32H​,2gH​

32H​,232gH​​

Total mechanical energy of particle, i.e. PE+KE=mgH ...(i) Given, KE=21​PE i.e. PEKE​=21​ or PE=2KE Substituting in this Eq. (i), we get PE+KE=mgH 2KE+KE=mgH 3KE=mgH KE=3mgH​ Similarly, PE=32​mgH So, height from ground at that instant, h=32H​ So, speed of particle, v=2gh​=2gH/3​

Q. A particle is released from a height $H$ . At a certain height its kinetic energy is half of its potential energy with reference to the surface of the earth. Height and speed of the particle at that instant are respectively

$\frac{H}{3}, \frac{2 g H}{3}$

$\frac{H}{3}, 2 \frac{g H}{3}$

$\frac{2 H}{3}, 2 g H$

$\frac{2 H}{3}, 2 \frac{2 g H}{3}$