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Work done by water flowing down a hill
Let a rectangle of area $A$ and height $H$ be filled with a volume of water. Then, suppose that the water is allowed to flow down a hill (which has the same area as the rectangle) of height $h$. Then, how much work do we do if we allow the water to flow down the hill?
I have the following doubt. Consider a volume of water of height $H$. Let $Q$ be the mass of that water (water's mass $= \rho \cdot V$ where $\rho$ is the density of the water and $V$ is the volume). Then, let the water flow down a hill of height $h$ where the area of the rectangle is $A$. Now, the mass of water that flows down is $Q\cdot A$. But if I assume that the water has to flow down the hill at a rate of $Q\cdot A$, then that means that the net force on the water is zero which, in turn, implies the work done by the water to be zero. But, I can also easily solve the problem and get the work done to be $Q\cdot h$. Which of these two is the correct answer?
A:
If I understand the question correctly, we have:
$w = \int_0^\Delta (mg-F)\,dt = \int_0^\Delta m\,\left(g-\frac{dz}{dt}\right)\,dt$
And you wish to know what the work is, for a constant $H$.
The work done against gravity, is obtained by letting $g-\frac{dz}{dt}=0$.
The flow is constant $v=\frac{Q}{A}$ which is the same as $\frac{dz}{dt}=v$ so we are left with the integral
$w = Q\,\Delta\left(\frac{mg}{Q}-v\right)$
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