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-section 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 height $v^2/2g$ and then stop
• D. The water level will be oscillating

Answer 1

The correct option is C Water level will rise to height $v^2/2g$ and then stop

In the beginning ,the water escaping the vessel at the rate $\sqrt{2gh}$ and filling it at the rate of $vA$

So the rate of change in the water level of the vessel at any instant is

($v-\sqrt{2gh})A$

where 'h' is the instantaneous height

Since h is quite small in the beginning, the water is escaping the vessel at a slow rate but filling at a much larger rate as such ($v-\sqrt{2gh})A$ ,is positive (increase in water level)

soon after the filling of water occurs, the height increases and the rate of escape become larger

But when the rate =rate of filling ,then an equilibrium state is reached and there is no net change in the height of the water

the equilibrium is reached when ($v-\sqrt{2gh})A$ =0 $\to$ $h=\frac{v^2}{2g}$