A wide cylindrical vessel 50 cm in height is | vedclass

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(C) Let $H = 50 \ cm$ be the total height of the vessel. Let the hole be made at a height $y$ from the bottom. Then the depth of the hole from the fre

Vedclass · Accepted Answer

$25$

Vedclass · Answer

(C) Let $H = 50 \ cm$ be the total height of the vessel. Let the hole be made at a height $y$ from the bottom. Then the depth of the hole from the free surface is $h = H - y = 50 - y$.The velocity of the water jet is $v = \sqrt{2gh} = \sqrt{2g(50-y)}$.The time taken for the water to reach the table is $t = \sqrt{\frac{2y}{g}}$.The horizontal range $R$ is given by $R = v 	imes t = \sqrt{2g(50-y)} 	imes \sqrt{\frac{2y}{g}} = 2\sqrt{y(50-y)}$.To maximize the range $R$, we differentiate $R$ with respect to $y$ and set it to zero:$\frac{dR}{dy} = 2 	imes \frac{1}{2\sqrt{y(50-y)}} 	imes (50 - 2y) = 0$.This gives $50 - 2y = 0$, so $y = 25 \ cm$.Thus, the hole should be made at a height of $25 \ cm$ from the bottom.

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