A particle is released from height S abvove the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively

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Q.
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively [2025]

1 $\frac{S}{2}, \sqrt{\frac{3 g S}{2}}$

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

3 $\frac{S}{4}, \frac{3 g S}{2}$

4 $\frac{S}{4}, \sqrt{\frac{3 g S}{2}}$

Ans.
(4)

Let height above the surface of earth be x,

$v^{2} = 0 + 2 g (S - x) = 2 g (S - x)$ ... (i)

The potential energy at a height x from the surface of earth,

U = mgx ... (ii)

$3 m g x = \frac{1}{2} m v^{2} \Rightarrow 3 g x = \frac{1}{2} \times 2 g (S - x)$

$4 x = S \Rightarrow x = \frac{S}{4}$

$\Rightarrow V = \sqrt{2 g \times \frac{3 S}{4}} = \sqrt{\frac{3 g S}{2}}$

Solution

Q.
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively [2025]

1 $\frac{S}{2}, \sqrt{\frac{3 g S}{2}}$

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

3 $\frac{S}{4}, \frac{3 g S}{2}$

4 $\frac{S}{4}, \sqrt{\frac{3 g S}{2}}$

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Solution

Q. A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively [2025]

1 S2, 3gS2

2 S2, 3gS2

3 S4, 3gS2

4 S4, 3gS2

Ans. (4) Let height above the surface of earth be x, v2=0+2g(S–x)=2g(S–x) ... (i) The potential energy at a height x from the surface of earth, U = mgx ... (ii) 3mgx=12mv2 ⇒ 3gx=12×2g(S–x) 4x=S ⇒ x=S4 ⇒ V=2g×3S4=3gS2

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Q.
A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively [2025]

1 $\frac{S}{2}, \sqrt{\frac{3 g S}{2}}$

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

3 $\frac{S}{4}, \frac{3 g S}{2}$

4 $\frac{S}{4}, \sqrt{\frac{3 g S}{2}}$