Bernoulli's principle, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Bernoulli's principle says that where a fluid flows faster, its pressure is lower. Together with the continuity equation, it lets you predict how water moves through pipes, out of tanks and around objects, and it is part of the fluids unit in AP Physics 1.

The ideal fluid model

Bernoulli's equation describes an ideal fluid: incompressible (its density does not change), non-viscous (no internal friction), and flowing steadily without turbulence. Real fluids only approximately behave like this, but the model works well for water in pipes and for many air flows at moderate speeds.

Step 1: the continuity equation

If a fluid cannot be compressed, the volume flowing past any point each second must be the same along a pipe. Volume flow rate is cross-sectional area times speed, so

A₁v₁ = A₂v₂

Where the pipe narrows, the fluid speeds up. That is why putting your thumb over the end of a garden hose makes the water spray faster and further.

Step 2: Bernoulli's equation

P₁ + ½ρv₁² + ρgy₁ = P₂ + ½ρv₂² + ρgy₂

• P is the pressure, in pascals;
• ρ is the fluid's density, in kg/m³;
• v is the flow speed and y the height of each point.

Each term is an energy per unit volume: ½ρv² is kinetic energy per unit volume, ρgy is gravitational potential energy per unit volume, and P represents the work done by pressure. Bernoulli's equation is conservation of energy for a flowing fluid. If the fluid speeds up on a level path, the kinetic term increases, so the pressure must fall.

Worked examples

Example 1: continuity

Water flows at 2.0 m/s through a pipe of cross-sectional area 0.030 m². The pipe narrows to 0.010 m². Find the new speed.

• A₁v₁ = A₂v₂: 0.030 × 2.0 = 0.010 × v₂.
• v₂ = 6.0 m/s. A third of the area means three times the speed.

Example 2: pressure drop in a level pipe

In the pipe from Example 1, the pressure in the wide section is 150 kPa. Find the pressure in the narrow section. (ρ = 1000 kg/m³)

• Level pipe, so the height terms cancel: P₂ = P₁ + ½ρ(v₁² − v₂²).
• P₂ = 150,000 + 500 × (4 − 36) = 150,000 − 16,000.
• P₂ = 134 kPa. Faster flow, lower pressure.

Example 3: Torricelli's theorem

Water leaks from a small hole 1.8 m below the surface of an open tank. Find the speed of the water leaving the hole. (g = 9.8 m/s²)

• Both the surface and the hole are open to the air, so P₁ = P₂. The surface moves very slowly, so v₁ ≈ 0.
• ρgh = ½ρv², giving v = √(2gh) = √(2 × 9.8 × 1.8).
• v ≈ 5.9 m/s, the same speed as an object dropped from 1.8 m.

Example 4: flow up a hill

Water at 3.0 m/s and 200 kPa flows in a pipe of constant width up to a point 5.0 m higher. Find the pressure there. (g = 9.8 m/s²)

• Constant width, so by continuity the speed stays 3.0 m/s and the kinetic terms cancel.
• P₂ = P₁ − ρgΔy = 200,000 − 1000 × 9.8 × 5.0 = 151,000 Pa.
• P₂ ≈ 151 kPa.

Example 5: volume flow rate

Water leaves a hose of cross-sectional area 2.0 × 10⁻⁴ m² at 8.0 m/s. How long does it take to fill a 0.040 m³ bucket?

• Flow rate = Av = 2.0 × 10⁻⁴ × 8.0 = 1.6 × 10⁻³ m³/s.
• Time = 0.040/1.6 × 10⁻³ = 25 s.

How to set up a Bernoulli problem

Choose two points along the same flow, such as the surface of a tank and a hole, or the wide and narrow parts of a pipe. Write down everything you know about P, v and y at each point. Use the continuity equation to link the speeds if the area changes. Then cancel any terms that are equal at both points before substituting numbers. Points open to the air are both at atmospheric pressure, so their pressure terms cancel.

Applications

• Venturi meters measure flow speed from the pressure difference between a wide and a narrow section of pipe.
• Atomisers and spray bottles blow fast air over a tube, lowering the pressure so liquid is drawn up.
• Chimneys draw better on windy days, because fast wind over the top lowers the pressure there.
• Roofs can lift off in storms: fast wind over the roof lowers the pressure above, while still air inside pushes up.
• Blood flow through a narrowed artery speeds up and the pressure drops, which matters in medicine.

A note on aeroplane wings

Wings are often explained with Bernoulli's principle alone, sometimes with the claim that air over the top must “meet up” with air underneath. That claim is false; air over the top actually arrives first. A full explanation of lift also uses Newton's third law: the wing deflects air downwards, and the air pushes the wing up. Faster flow and lower pressure above the wing are real, but they are part of the picture rather than the whole story. If an exam asks about lift, link pressure difference to force and avoid the “equal transit time” myth.

Common mistakes

• Forgetting the continuity equation when the pipe changes width.
• Using diameters directly instead of areas; area depends on diameter squared.
• Thinking faster flow means higher pressure.
• Mixing kPa and Pa in the same equation.
• Dropping the height terms when the two points are at different heights.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is Bernoulli's equation on AP Physics 1?

Yes. Fluids, including the continuity equation and Bernoulli's equation, are a unit of AP Physics 1 in the current course framework.

Is Bernoulli's principle in A level Physics?

It is not in most core A level Physics specifications, though it can appear in engineering physics options and is used widely in engineering degrees.

Key takeaways

• Continuity: A₁v₁ = A₂v₂ for an incompressible fluid.
• Bernoulli: P + ½ρv² + ρgy is constant along a flow.
• Faster flow means lower pressure, at the same height.
• Torricelli: v = √(2gh) for a hole below an open surface.
• Bernoulli's equation is energy conservation for fluids.

Related guides

• AP Physics 1 fluids
• AP Physics 1 energy
• Density