Bernoulli's Equation — Lift on an Airplane Wing (Worked Example)

Hard

Problem

In simplified level flight, air moves over a wing's top surface at 85 m/s and under the bottom at 75 m/s. Air density is 1.225 kg/m³. (a) Estimate the pressure difference. (b) Estimate the lift on a 20 m² wing.

Given

v_top = 85 m/s
v_bottom = 75 m/s
ρ = 1.225 kg/m³
A = 20 m²
same height (elevation terms cancel)

Find

Pressure difference ΔP and lift force.

Principle

Bernoulli's equation: where a fluid flows faster, its pressure is lower. The pressure gap across the wing pushes it upward.

Formula

\Delta P = \tfrac{1}{2}\rho\,(v_{top}^2 - v_{bottom}^2), \qquad \text{Lift} = \Delta P \times A

ΔP = ½ρ(v_top² − v_bottom²); Lift = ΔP × A

Solution

Square the speeds: $85^2 = 7225$ ; $75^2 = 5625$
Difference: $7225 - 5625 = 1600\ \mathrm{m^2/s^2}$
Pressure gap: $\Delta P = 0.5 \times 1.225 \times 1600 = 980\ \mathrm{Pa}$
Lift: $980 \times 20 = 19{,}600\ \mathrm{N}$

ΔP ≈ 980 Pa; Lift ≈ 1.96 × 10⁴ N

Unit check

(kg/m³)(m²/s²) = kg/(m·s²) = N/m² = Pa; Pa × m² = N. Consistent.

What this means physically

Just 980 Pa — less than 1% of atmospheric pressure — yet across 20 m² it lifts nearly 2000 kg. Real wings add angle-of-attack effects, but this pressure gap is the core of flight.

Common mistake

Thinking faster air 'pushes harder.' It is the opposite: faster flow means lower pressure, and the slower, higher-pressure air below does the pushing.