JEE Advanced 2024 · Paper 1 · Physics Q12

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PhysicsInteger+4Question 29 of 51

Two large, identical water tanks, 1 and 2, kept on the top of a building of height $H$ , are filled with water up to height $h$ in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are $t_1$ and $t_2$ , respectively. If $H = \left(\dfrac{16}{9}\right)h$ , then the ratio $t_1/t_2$ is _____.

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Answer and worked solution

Official answer: 3

Let $A$ be the tank area and $a$ the hole area, with water height $y$ .

Tank 1: efflux speed $\sqrt{2gy}$ , so $A\,\dfrac{dy}{dt} = -a\sqrt{2gy}$ , giving
$t_1 = \frac{A}{a}\cdot\frac{2\sqrt h}{\sqrt{2g}}.$

Tank 2: the pipe drops a further height $H$ , so the exit speed is $\sqrt{2g(y+H)}$ :
$t_2 = \frac{A}{a}\cdot\frac{2\left(\sqrt{h+H} - \sqrt H\right)}{\sqrt{2g}}.$

With $H = \frac{16}{9}h$ : $\sqrt{h+H} - \sqrt H = \frac{5}{3}\sqrt h - \frac{4}{3}\sqrt h = \frac{\sqrt h}{3}$ .

$\frac{t_1}{t_2} = 3.$

Topics:Bernoulli's principle Continuity equation

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JEE Advanced 2024 · Paper 1, Physics question 12