(B) Let the total height of the water surface from the ground be $Y = H + h$.The velocity of efflux at a depth $x$ below the water surface is $v = \sq

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(B) Let the total height of the water surface from the ground be $Y = H + h$.The velocity of efflux at a depth $x$ below the water surface is $v = \sqrt{2gx}$.The height of the hole from the ground is $y = H + (h - x)$.The time taken for the water to reach the ground is $t = \sqrt{\frac{2y}{g}} = \sqrt{\frac{2(H + h - x)}{g}}$.The horizontal range $R$ is given by $R = v \cdot t = \sqrt{2gx} \cdot \sqrt{\frac{2(H + h - x)}{g}} = 2\sqrt{x(H + h - x)}$.To maximize $R$,we maximize the term $f(x) = x(H + h - x) = (H + h)x - x^2$.Taking the derivative with respect to $x$ and setting it to zero: $f'(x) = (H + h) - 2x = 0$.Thus,$x = \frac{H + h}{2}$.

Class 11 Physics Fluid Mechanics and Surface Tension Velocity of Efflux and Torricelli's law

Medium

$A$ tank with vertical walls is mounted so that its base is at a height $H$ above the horizontal ground. The tank is filled with water to a depth $h$. $A$ hole is punched in the side wall of the tank at a depth $x$ below the water surface. To have maximum range of the emerging stream,the value of $x$ is

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A
$\frac{H+h}{4}$
B
$\frac{H+h}{2}$
C
$\frac{H+h}{3}$
D
$\frac{3(H+h)}{4}$

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Class 11

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Fluid Mechanics and Surface Tension

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Velocity of Efflux and Torricelli's law

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If a small orifice is made at a height of $0.25\, m$ from the ground in a tank filled with water up to a height of $1\, m$,the horizontal range of the water stream will be (in $cm$):

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