# A rectangular tank is filled to the brim with water. When a hole at its bottom is unplugged, the tank is emptied in time T. How long will it take to empty the tank if it is half filled?

##
I have to five the answer in terms of T.

I have to five the answer in terms of T.

##### 2 Answers

Let the actual Height of the tank be

Now let us consider that at certain moment the height of water level be

Let

Again the rate of flow of water at this moment through the orifice is given by

These two rate must be same by the principle of cotinuity.

Hence

If the tank is filled to the brim then height of water level will be

If the tank be half filled with water and the time to empty it be

So

Bernoulli's theorem states that the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure due to the gravitational potential energy associated with height

In the instant case we need to apply Torricelli's law, which is a special case of Bernoulli's theorem.

For a water particle of mass

Assuming that when tank is completely empty the velocity of water coming out of hole is

The average velocity of the emerging water for tank full up to height

Similarly, for the tank when half full, the average velocity is given by

Let **completely empty the tank** in respective cases. Also let

For case of full tank we have

Volume discharged

Similarly in case of half full tank we have

Volume discharged

We know that Volume discharged in case of equation (4) is twice the volume discharged in case of equation (5)

Hence we have the equality from (4) and (5)

Using (2) and (3) and solving for