A cylindrical tank of cross-section…

A cylindrical tank of cross-section 1.0 m^2 stands on level ground and is filled with water to a depth H=1.8 m. Small circular holes of area 100 cm^2…

One or more options correct

A cylindrical tank of cross-section

1.0\ \mathrm{m^2}

stands on level ground and is filled with water to a depth

H=1.8\ \mathrm{m}

. Small circular holes of area

100\ \mathrm{cm^2}

can be opened in its side wall, one at a time, at different heights above the ground. Take

g=10\ \mathrm{m\,s^{-2}}

, treat the flow as steady at each instant and ignore viscosity. Which of the following statements is/are correct?

A. The largest horizontal range on the ground (from the wall) that any hole can give is about $1.8\ \mathrm{m}$ .
B. Holes at heights $0.3\ \mathrm{m}$ and $1.5\ \mathrm{m}$ above the ground give jets that land at the same distance from the wall.
C. The efflux speed from a hole at height $0.9\ \mathrm{m}$ is about $3.0\ \mathrm{m\,s^{-1}}$ .
D. If the hole were at the very bottom of the wall, the tank would drain completely in about $60\ \mathrm{s}$ .

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Answer: A, B, D

For a hole at height

y

above the ground the water above it has depth

H-y

, so by Bernoulli the efflux speed is

v=\sqrt{2g(H-y)}

(the surface speed is only

1\%

of this). The jet falls a height

y

t=\sqrt{2y/g}

, so the range is

x=v\,t=2\sqrt{y(H-y)}

. This is maximal at

y=H/2=0.9\ \mathrm{m}

, where

x=H=1.8\ \mathrm{m}

, so (A) is correct. Since

x

is symmetric under

y\leftrightarrow H-y

and

0.3+1.5=1.8

, both holes give

x=2\sqrt{0.3\times1.5}\approx1.34\ \mathrm{m}

; (B) is correct. At

y=0.9\ \mathrm{m}

the speed is

\sqrt{2\times10\times0.9}=4.24\ \mathrm{m\,s^{-1}}

, not

3.0

, so (C) is wrong. For a base hole the level

h

obeys

A\,\dot h=-a\sqrt{2gh}

, giving

T=\dfrac{A}{a}\sqrt{\dfrac{2H}{g}}=100\times0.6=60\ \mathrm{s}

, so (D) is correct. Correct options: A, B and D.

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Original question written in the JEE Advanced 2017 Paper 1 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake