Volume of water discharged from a tank

[nemixus]

Homework Statement

A rectangular opening is cut into the side of a large open-topped water tank. The opening has width w and height h2-h1, where h1 and h2 are distances of the opening below the water surface as identified in the figure. Determine the volume V of water that emerges from the opening per unit time (i.e. per second). You may assume that the surface area of the tank is extremely large compared to the area of the opening, but you should not assume that the water emerges from the opening with a single, uniform velocity. (Sorry for not able to show figure. Water is filled almost to the top. Opening on the side is about 1/9 of the tank.)

Homework Equations

Let Q = volume/second, A2 = Area of the hole, V2 = velocity of the hole

The Attempt at a Solution

Q = A2*V2
Torricelli's theorem:
v2 = sqrt(2*g*(h2 - h1))
Q/A2 = sqrt(2*g*(h2 - h1))
Q= w(h2 - h1)*sqrt(2*g*(h2 - h1))

It is after this point when I realized that I cannot assume the velocity is the same throughout the hole and I'm at lost how to approach this problem. Can I still solve the problem this way or do I have to do it a different way since the velocity is not constant throughout?

 

[aim1732]

The velocity varies with height so you should assume a differential cross sectional element of height dh and find the volume flow rate for that element(which would be a function of height). Then integrate under limits of the heights given for volume flowing out per unit time.