Question

A large cylindrical tank has a hole of area $A$ at its bottom and water is poured into the tank through a tube of cross-sectional area $A$ ejecting water at the speed $v$. Which of the following is true?

• A. Water level in tank keeps on rising.
• B. No water can be stored in the tank.
• C. Water level will rise to a height $\frac{v^2}{2g}$ and then stop.
• D. The water level will be oscillating.

Answer 1

Answer: (C)

Water level will rise to a height $\frac{v^2}{2g}$ and then stop.

Initially, the water flowing out will be less than that flowing into it. Hence, the water level will go on rising. When the water level is h, the velocity of efflux $=\sqrt{2gh}$. When this becomes equal to the velocity of inflow, the level will become steady as the area of cross section of the filling tube and area of cross section of the hole are equal. This height is given by Torricelli's theorem,
$v=\sqrt{2gh}$ or $\frac{v^2}{2g}=h$