First slide

Fluid dynamics

Question

A large tank filled with water to a height ‘h’ is to be emptied through a small hole at the bottom. The ratio of time taken for the level of water to fall from h to $\frac{h}{2}$ and from $\frac{h}{2}$ to zero is

Moderate

A

$\sqrt{2}$
B

$\frac{1}{\sqrt{2}}$
C

$\sqrt{2} - 1$
D

$\frac{1}{\sqrt{2} - 1}$

Solution

Time taken for the level to fall from H to $H'$ $t = \frac{A}{A_{0}} \sqrt{\frac{2}{g}} [\sqrt{H} - \sqrt{H'}]$
According to problem- the time taken for the level to fall from h to $\frac{h}{2}$ $t_{1} = \frac{A}{A_{0}} \sqrt{\frac{2}{g}} [\sqrt{h} - \sqrt{\frac{h}{2}}]$
and similarly time taken for the level to fall from $\frac{h}{2}$ to zero $t_{2} = \frac{A}{A_{0}} \sqrt{\frac{2}{g}} [\sqrt{\frac{h}{2}} - 0]$
$∴ \frac{t_{1}}{t_{2}} = \frac{1 - \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} - 0} = \sqrt{2} - 1 .$

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