A fixed cylindrical vessel is filled with water up to height $H$ . A hole is bored in the wall at a depth $h$ from the free surface of water. For maximum horizontal range $h$ is equal to :

H

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3 H / 4

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H / 2

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H / 4

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Solution

The correct option is C $H / 2$
Let velocity at hole be $v$ , density be $ρ$ . Applying Bernoulli theorem for top-most layer and hole,
$P_{a t m} + ρ g H + 1 / 2 ρ 0^{2} = P_{a t m} + ρ g h + 1 / 2 ρ V^{2}$
$V = \sqrt{2 g (H - h)}$

Time taken to reach ground $t = \sqrt{2 h / g}$
Horizontal range
$(H o r i z o n t a l_{r a n g e})$ $= V \times t$
$= \sqrt{2 g H - 2 g h} \times \sqrt{2 h / g}$

Differentiating $H o r i z o n t a l_{r a n g e}$ and equating it to zero for maximum $H o r i z o n t a l_{r a n g e}$ we get maximum $H o r i z o n t a l_{r a n g e}$ when $h = H / 2.$

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