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JEE Advanced 2022 Paper 2 · Physics · Fluid Mechanics, Elasticity and Surface Tension · hard

A cylindrical tank of cross-sectional area 0.05 m^2 is filled with water to a height 0.8 m above a small circular hole of area 5 cm^2 in its vertical…

One or more options correct

A cylindrical tank of cross-sectional area 0.05 m20.05\ \mathrm{m^2} is filled with water to a height 0.8 m0.8\ \mathrm m above a small circular hole of area 5 cm25\ \mathrm{cm^2} in its vertical side wall. The hole is 0.45 m0.45\ \mathrm m above the horizontal ground. The water level falls as the water leaks out. Take g=10 m s−2g=10\ \mathrm{m\,s^{-2}} and treat the flow as ideal and quasi-steady. Which of the following is/are correct?
  1. A. The initial speed of efflux is 4 m s−14\ \mathrm{m\,s^{-1}}.
  2. B. The tank takes 20 s20\ \mathrm s to drain completely down to the level of the hole.
  3. C. The level takes 20 s20\ \mathrm s to fall from 0.8 m0.8\ \mathrm m to 0.2 m0.2\ \mathrm m above the hole.
  4. D. Initially the water jet strikes the ground 2.4 m2.4\ \mathrm m horizontally from the wall.
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Answer: A, C

Torricelli: v=2ghv=\sqrt{2gh}, initially 2×10×0.8=4 m s−1\sqrt{2\times10\times0.8}=4\ \mathrm{m\,s^{-1}}, so (A) is correct. Continuity: −Adhdt=a2gh-A\dfrac{dh}{dt}=a\sqrt{2gh}, hence the time to fall from h1h_1 to h2h_2 is t=Aa2g(h1−h2)t=\dfrac{A}{a}\sqrt{\dfrac2g}\left(\sqrt{h_1}-\sqrt{h_2}\right) with A/a=100A/a=100. For complete drainage, t=100×0.2×0.8=40 st=100\times\sqrt{0.2}\times\sqrt{0.8}=40\ \mathrm s, so (B) is incorrect (20 s20\ \mathrm s is the wrong estimate Ah/(av0)Ah/(av_0) obtained by assuming the efflux speed stays at its initial value). For 0.8→0.20.8\to0.2: 100×0.4472×(0.8944−0.4472)=20 s100\times0.4472\times(0.8944-0.4472)=20\ \mathrm s, so (C) is correct. The jet leaves horizontally from 0.45 m0.45\ \mathrm m height: fall time 2(0.45)/10=0.3 s\sqrt{2(0.45)/10}=0.3\ \mathrm s, range =4×0.3=1.2 m=4\times0.3=1.2\ \mathrm m, so (D) is incorrect.
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Original question written in the JEE Advanced 2022 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake