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(D) Let $H = 12\, 	ext{m}$ be the total height of the water column.Let $h$ be the depth of the hole from the water surface.The velocity of efflux is

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(D) Let $H = 12\, 	ext{m}$ be the total height of the water column.Let $h$ be the depth of the hole from the water surface.The velocity of efflux is given by Torricelli's law: $v = \sqrt{2gh}$.The time taken for the water to reach the ground from a height $(H - h)$ is given by $t = \sqrt{\frac{2(H - h)}{g}}$.The horizontal range $R$ is given by $R = v 	imes t = \sqrt{2gh} 	imes \sqrt{\frac{2(H - h)}{g}}$.Simplifying the expression: $R = \sqrt{4h(H - h)} = 2\sqrt{hH - h^2}$.To find the maximum range, we differentiate $R$ with respect to $h$ and set it to zero: $\frac{dR}{dh} = 0$.$\frac{d}{dh}(2\sqrt{hH - h^2}) = 2 \cdot \frac{1}{2\sqrt{hH - h^2}} \cdot (H - 2h) = 0$.This implies $H - 2h = 0$, so $h = \frac{H}{2}$.Given $H = 12\, 	ext{m}$, we get $h = \frac{12}{2} = 6\, 	ext{m}$.

The water is filled up to a height of $12\, \text{m}$ in a tank having vertical sidewalls. $A$ hole is made in one of the walls at a depth $h$ below the water level. The value of $h$ for which the emerging stream of water strikes the ground at the maximum range is $.....\, \text{m}$.

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