AP Physics 1 · Unit 8 · Topic 4 of 4 · 6 min lesson

Bernoulli's Equation

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions Watch the review video ↗

The big idea

Bernoulli’s equation is energy conservation for a flowing fluid: P + ½ρv² + ρgh is the same at every point along the flow. At the same height, faster flow means lower pressure. At the same speed, higher means lower pressure.

1. Faster flow, lower pressure

The water in each tube rises as high as the pressure below it can push it. It stands lowest above the narrow, fast section.

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂Bernoulli

½ρv²kinetic energy per volume

ρghgravitational energy per volume

v = √(2gh)fluid leaving a hole a depth h below the surface

2. How to use it

Pick two points. Cross out any term that’s the same at both (same height? drop ρgh; same speed? drop ½ρv²). Use continuity to find a missing speed.

The fluid speeds up only if something pushes it, so the pressure must be higher where it’s slow and lower where it’s fast.

Hole in a tank: both the water surface and the stream are open to air, and the surface barely moves, so ½ρv² = ρgh and v = √(2gh). Water 1.8 m below the surface leaves at √(2 × 10 × 1.8) = 6.0 m/s, as fast as a rock dropped 1.8 m.

3. Example, then try it

Example

Water in a horizontal pipe moves at 1.0 m/s with pressure 150 kPa. The pipe narrows and the water speeds up to 3.0 m/s. What is the pressure in the narrow part?

Same height, so P₂ = P₁ + ½ρ(v₁² − v₂²).

P₂ = 150,000 + ½(1000)(1 − 9) = 150,000 − 4000 = 146 kPa.

Try it · work it on paper first

Water flows up a pipe. At point 1 (height 0), the area is 0.020 m², the speed is 2.0 m/s, and the pressure is 200 kPa. Point 2 is 5.0 m higher, where the area is 0.010 m². Find the pressure at point 2. (g = 10 m/s²)

Continuity: v₂ = (0.020 ÷ 0.010)(2.0) = 4.0 m/s.

Speed term: ½ρ(v₁² − v₂²) = ½(1000)(4 − 16) = −6000 Pa.

Height term: ρg(h₁ − h₂) = (1000)(10)(0 − 5.0) = −50,000 Pa.

Add up: P₂ = 200,000 − 6000 − 50,000 = 144 kPa.

Answer: 144 kPa. Most of the drop comes from climbing 5.0 m.

4. Common mistakes

“Faster fluid pushes harder.”At the same height, faster flow means lower pressure.

Dropping the height term.If the pipe rises, include ρgh.

Forgetting continuity.If the area changes, the speed changes too. Find v₂ first.

Check yourself

You blow air quickly across the top of a sheet of paper that hangs down from your lips. The paper lifts up. Why?

By Bernoulli’s equation, the fast-moving air above the paper has lower pressure. The still air below pushes harder, so the paper rises.

Ready to practice?

5 questions on Bernoulli's Equation, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 8