A rectangular opening is cut into the side of a large open-topped water tank. Th

Question

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.

Explanation / Answer

let us consider at the depth at a distance x from the top.

let the thickness of that portion be dx.

so area of that portion=w*dx

so speed of water emergin from that point=(2*g*x)^0.5

now volume rate flow=area*speed of water

=(2*g*x)^0.5 * w*dx

now integrating this expression from h1 to h2 ,

we get

V=(1/1.5)*(2*g)^0.5*(h2^1.5 - h1^1.5) * w

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