A cylindrical tank of height $H$ is completely filled with water. On its vertical side, there are two tiny holes, one above the middle at a height of $h_{1}$ and other below the middle at a depth of $h_{2}$ . If the jets of water from the holes meet at the same point on the ground, then the ratio $\frac{h_{1}}{h_{2}}$ is

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Solution

According to question -

Given, range of efflux from both the holes are same.
$\Rightarrow R_{A} = R_{B}$
$\Rightarrow 2 \sqrt{h_{A} (H - h_{A})} = 2 \sqrt{h_{B} (H - h_{B})}$
$[R = 2 \sqrt{h (H - h)}]$
$\Rightarrow h_{A} H - h_{A}^{2} = h_{B} H - h_{B}^{2}$
$\Rightarrow h_{A} H - h_{B} H = h_{A}^{2} - h_{B}^{2}$
$\Rightarrow H (h_{A} - h_{B}) = (h_{A} + h_{B}) (h_{A} - h_{B})$
$\Rightarrow H = h_{A} + h_{B}$
$\Rightarrow H = \frac{H}{2} - h_{1} + \frac{H}{2} + h_{2}$
$[$ from figure $]$
$\Rightarrow h_{1} = h_{2}$
$\Rightarrow \frac{h_{1}}{h_{2}} = 1$

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