A wide open tank stands on a… | JEE Advanced 2019 P1

A wide open tank stands on a horizontal table. It contains water to a depth H=1.0 m and is kept full by a feed from above so that the level stays…

One or more options correct

A wide open tank stands on a horizontal table. It contains water to a depth

H=1.0\ \mathrm{m}

and is kept full by a feed from above so that the level stays constant. Small holes (area much smaller than the tank cross-section) can be opened in the vertical side wall at various depths

h

below the free surface. Take

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

and ignore viscosity. The tank rests on the table surface and the bottom of the tank is at the table level. Which of the following statements is/are correct?

A. The water level in the tank must fall at a speed equal to the efflux speed of the jet from any hole.
B. A jet from a hole at depth $0.20\ \mathrm{m}$ strikes the table at a horizontal distance $0.80\ \mathrm{m}$ from the wall.
C. Holes at depths $0.20\ \mathrm{m}$ and $0.80\ \mathrm{m}$ give jets with the same horizontal range on the table.
D. The maximum possible range of a jet on the table is $1.2\ \mathrm{m}$ .

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Answer: B, C

Efflux speed at depth

h

v=\sqrt{2gh}

(Torricelli). The hole is at height

H-h

above the table, so the fall time is

t=\sqrt{2(H-h)/g}

and the range is

x=vt=2\sqrt{h(H-h)}

. (A) is false: the level stays constant here, and even if it were not, the surface speed would be

v\,a/A

, far smaller than

v

. (B) For

h=0.2

x=2\sqrt{0.2\times0.8}=0.80\ \mathrm{m}

, so it is true. (C)

h(H-h)

is symmetric under

h\to H-h

, so

h=0.2

and

0.8

give the same range; true. (D)

h(H-h)

is maximum at

h=H/2

, giving

x_{max}=H=1.0\ \mathrm{m}

, not

1.2\ \mathrm{m}

; false. Correct options: (B) and (C).

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