Physics · Fluid Mechanics

JEE Advanced 2024 — Paper 1 — Question 29

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 $H=\left(\frac{16}{9}\right) \mathrm{h}$ , then the ratio $t_{1} / t_{2}$ is $\qquad$

Answer: 3

Numerical answer — enter this value.

Step-by-step solution

For tank-1 $\begin{gathered} -A \frac{d y}{d t}=a \sqrt{2 g y} \end{gathered}$

For tank-2 $\begin{gathered} -A \frac{d y}{d t}=a \sqrt{2 g(y+H)} \end{gathered}$

From - 1 $-\int_{h}^{0} \frac{d y}{\sqrt{y}}=a \sqrt{2 g} \int_{0}^{t_{1}} d t \Rightarrow t_{1}=\frac{2 A}{a} \frac{\sqrt{h}}{\sqrt{2 g}}$

From-2 $-\int_{h}^{0} \frac{d y}{\sqrt{y+H}}=a \sqrt{2 g} \int_{0}^{t_{2}} d t \Rightarrow t_{2}=\frac{2 A \sqrt{h+H}}{a \sqrt{2 g}}-\frac{2 A \sqrt{H}}{a \sqrt{2 g}}$ $\frac{t_{1}}{t_{2}}=3$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2024
Paper: Paper 1
Subject: Physics
Chapter: Fluid Mechanics
Topic: Bernoulli's Equation and its Applications