Bernoulli's Equation

Picture water gliding along a pipe with no friction. Bernoulli's equation tracks three forms of energy per unit volume along that flow: pressure energy, kinetic energy, and gravitational potential energy. Their sum stays constant from point 1 to point 2: p + one-half rho v² + rho g h = constant. That is just energy conservation, rewritten so every term has units of energy per volume instead of plain energy.

Match the pieces to ideas you already know. The one-half rho v² term is kinetic energy per volume, since kinetic energy is one-half m v² and rho is mass per volume. The rho g h term is gravitational potential energy per volume, the same swap applied to m g h. The pressure term p is less obvious: it represents the work done per unit volume by the fluid behind a parcel pushing it forward, and the work done per unit volume as that parcel pushes fluid ahead of it out of the way. So the whole equation reads: work done on the fluid plus its kinetic energy plus its potential energy stays constant as it flows, exactly like the work-energy theorem applied to a chunk of fluid instead of a block on a ramp.

This is why a narrowing pipe at constant height forces pressure down as speed goes up: the fluid converts pressure energy into kinetic energy to conserve the total. It connects back to the continuity equation from the previous topic, which tells you how much the speed must increase in a narrower pipe, and Bernoulli's equation then tells you what that speed change costs in pressure.