Bernoulli's Principle — Why Planes Fly
Bernoulli's Principle
In 1738, Daniel Bernoulli published an equation that explains why airplanes fly, why curveballs curve, and why shower curtains get sucked inward.
Along a streamline in steady, incompressible, frictionless flow:
$$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$$
or equivalently, between two points:
$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$
This is an algebraic equation — no calculus needed to use it. It says: when fluid speeds up, its pressure drops.
How Wings Create Lift
An airplane wing (airfoil) is shaped so air flows faster over the top and slower under the bottom. By Bernoulli's equation:
- Faster flow above → lower pressure on top
- Slower flow below → higher pressure on bottom
- Pressure difference → net upward force = LIFT
Air flows over a wing at $v_{\text{top}} = 70$ m/s and under at $v_{\text{bottom}} = 60$ m/s. Air density is $\rho = 1.225$ kg/m³. What is the pressure difference?
Using Bernoulli (at the same height, so $\rho g h$ cancels):
$$\Delta P = \frac{1}{2}\rho(v_{\text{top}}^2 - v_{\text{bottom}}^2)$$
$$= \frac{1}{2}(1.225)(70^2 - 60^2)$$
$$= \frac{1}{2}(1.225)(4900 - 3600) = \frac{1}{2}(1.225)(1300) \approx 796 \text{ Pa}$$
For a wing area of 120 m², the total lift force is:
$$F = \Delta P \times A = 796 \times 120 \approx 95{,}520 \text{ N} \approx 9{,}740 \text{ kg-force}$$
That's nearly 10 tonnes of lift — from substituting values into an equation with squares and fractions!
The Math You Need
To use Bernoulli's equation, you need:
- Exponents: $v^2$ (squaring velocities)
- Fractions: $\frac{1}{2}$ (coefficients)
- Algebra: solving for unknowns
- Arithmetic: plugging in real values
These are the exact skills you're building right now in this course.
Bernoulli's equation is taught in first-year engineering — and it uses nothing more than algebra and arithmetic. The foundations you're learning literally explain flight.