Bernoulli's equation

Also called Bernoulli's principle

Bernoulli's equation says that for an ideal fluid in flow, the sum of pressure, the term rho g y and the term one half rho v squared is the same at any two points. It is conservation of mechanical energy per unit volume.

The AP Physics 1 and AP Physics 2 sheets both print it as

P_1 + \rho g y_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \tfrac{1}{2}\rho v_2^2

Essential knowledge 8.4.B.2 names what it is: a description of the conservation of mechanical energy in fluid flow. The bookkeeping is per unit volume, which is why every term carries units of $\text{J/m}^3$ , and why $\text{J/m}^3$ is the same thing as $\text{Pa}$ . Read $\rho g y$ as gravitational potential energy per unit volume and $\tfrac{1}{2}\rho v^2$ as kinetic energy per unit volume. Use $g = 9.8\ \text{m/s}^2$ .

It applies to an ideal fluid, and the AP exam conventions say fluids are assumed to be ideal, and pipes assumed completely filled by fluid, unless a question states otherwise.

The trap is the popular summary "faster flow means lower pressure". That only follows when $y_1 = y_2$ , so the height terms cancel. In a pipe that also climbs, pressure can rise and speed rise together if the height falls enough to pay for both. Solve the equation rather than reciting the slogan.

Torricelli's theorem, EK 8.4.B.3, is the special case of a tank draining through a small hole.

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