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Bernoulli's theorem is conservation of energy for ideal fluid flow: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$ along a streamline. The three terms represent pressure energy, kinetic energy, and potential energy per unit volume. Assumptions: ideal fluid (incompressible, non-viscous), steady flow, along a streamline.

Combined with the equation of continuity ($A_1 v_1 = A_2 v_2$), Bernoulli explains many phenomena. The Venturi effect: at a constriction, velocity increases and pressure decreases. This is the principle behind Venturi meters (flow measurement), atomizers (sprayers), and carburetors.

Torricelli's theorem ($v = \sqrt{2gh}$) gives the speed of efflux from a hole at depth $h$. The exiting stream follows projectile motion with horizontal range $R = 2\sqrt{h(H-h)}$, maximized when the hole is at mid-height ($h = H/2$, $R_{\max} = H$). Time to empty a tank requires integration: $t = (A/a)\sqrt{2H/g}$.

Aerodynamic lift on wings arises from faster airflow over the curved top surface (lower pressure) versus the flatter bottom (higher pressure). The Magnus effect explains why spinning balls curve — asymmetric airflow creates a lateral pressure difference.