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Say the dam is Hoover dam for example. The intake is where the water flows in to turn the turbines that generate electricity. I want to know if there is an equation that describes the speed of the water. I plan to use this equation to figure out how much energy any given dam can generate.

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Consider the following image, enter image description here

we have some fluid volume, $V$, having density $\rho$ and traveling at a velocity $v$ along a pipe with some cross-sectional area $A$. The rate at which the water flows through the pipe is called the volumetric flow rate. This is given by, $$ \frac{dV}{dt}\equiv Q=\mathbf v\cdot\mathbf A $$ where $V$ is the volume of the fluid, $Q$ the flow rate, $\mathbf v$ is the velocity of the flow and $\mathbf A$ the cross-sectional area of the flow.

If you know the density of the fluid, then the mass flow rate is $$ \rho\frac{dV}{dt}\equiv\dot m=\rho\mathbf v\cdot\mathbf A $$

Note that while $\mathbf v$ and $\mathbf A$ are both vectors, you probably can assume a one-dimensional flow, i.e., $Q=vA$, to approximate the energy.

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