Bernoulli's Principle
A moving fluid trades pressure for speed: wherever it flows faster, it pushes less hard sideways, so its pressure drops. This is why air rushing over a curved wing or water squeezing through a narrow pipe pulls at lower pressure than the slower fluid around it.
The formula
P + ½ · ρ · v² = constant
- P — pressure (Pa): the pushing force the fluid exerts per unit area
- ρ — density (kg/m³): how much mass is packed into each cubic meter of the fluid
- v — speed (m/s): how fast the fluid is flowing at that point
Worked example
Water at 1000 kilograms per cubic meter flows at 2 meters per second through a wide pipe, then speeds up to 4 meters per second in a narrow section. How much does the pressure drop?
- ρ = 1000 kg/m³
- v₁ = 2 m/s
- v₂ = 4 m/s
- ΔP = ½ · ρ · (v₂² − v₁²)
- ΔP = ½ · 1000 kg/m³ · (16 − 4) m²/s²
ΔP = 6000 Pa
Test yourself
Water flows from a wide pipe into a narrow section, so its speed increases. What happens to its pressure in the narrow section?
- It increases
- Correct answer: It decreases
- It stays the same
Right! Faster flow means lower pressure — that is Bernoulli's principle in action.
Same water, still 1000 kilograms per cubic meter, this time 1 meter per second in the wide pipe, but now it speeds up to 3 meters per second in the narrow part. What is the pressure drop?
- Correct answer: 4000 Pa
- 8000 Pa
- 1000 Pa
Exactly — ΔP = ½ · 1000 · (9 − 1) = ½ · 1000 · 8 = 4000 Pa.
Where you see this
An airplane wing is the classic case: air flowing over the curved upper surface moves faster than the air beneath, so by this principle its pressure drops, and the wing is pushed up from below harder than it is pressed down from above. The same fast-flow-means-low-pressure pairing pulls a shower curtain inward over running water and draws two sheets of paper together when you blow between them.
Common mistakes
The reflex error reverses the principle: since the fluid in the narrow section is moving faster, surely it must push harder there — in fact faster flow means lower pressure, because the speed was bought with pressure. Watch the algebra too: the drop depends on the difference of squared speeds, v₂² − v₁² — square each speed first, subtract second. For speeds of 2 and 4 m/s the squared difference is 16 − 4 = 12, not (4 − 2)² = 4.
How it connects
This is energy conservation applied to a flowing fluid: the ½ · ρ · v² term is kinetic energy per unit volume, and pressure is the fluid's stored ability to push, trading back and forth along a streamline while their sum stays constant. It completes the fluids unit — pressure at rest, pressure transmitted, pressure as buoyancy, and finally pressure traded for speed.