MotionLab

Bernoulli's Equation Calculator

Connects pressure, speed, and height along a flow — the physics behind airplane wings and venturi effects.

Pressure at point 1 (Pa)

Velocity at point 1 (m/s)

Height at point 1 (m)

Velocity at point 2 (m/s)

Height at point 2 (m)

Fluid density (kg/m³)

Pressure at point 2 (Pa)

139,875

P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)

What you entered

Pressure at point 1 (Pa): 101325Velocity at point 1 (m/s): 2Height at point 1 (m): 5Velocity at point 2 (m/s): 5Height at point 2 (m): 0Fluid density (kg/m³): 1000

1
P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)
101325 + 0.5×1000×(2²−5²) + 1000×9.81×(5−0)= 139875 Pa

Result

Pressure at point 2 (Pa): 139,875

Using Bernoulli's equation, pressure at point 2 works out to 139,875 Pa.

Bernoulli's equation: downstream pressure

For water at 1000 kg/m3, pressure plus half rho v squared plus rho g h stays constant along a streamline. These solve for the second pressure.

p1 (Pa)	v1 (m/s)	h1 (m)	v2 (m/s)	h2 (m)	p2 (Pa)
101325	2	0	5	0	90825.00
101325	1	0	3	2	77705.00
200000	3	0	6	0	186500.00
101325	0	10	5	0	186925.00

The first three rows all show pressure falling as the fluid speeds up - the counterintuitive result at the heart of the equation, and the reason a carburettor draws fuel and a spray gun atomises. The last row runs the other way: water starting 10 m up with no velocity converts height into both speed and pressure, ending higher than it started. Bernoulli assumes steady, incompressible, frictionless flow along a single streamline, so it describes ideal pipes well and viscous or turbulent flow poorly - use the Reynolds number to check which regime you are in.

The core trade-off

Bernoulli's equation says that along a streamline, as fluid speeds up, its pressure drops (all else equal) — this is the basic idea behind how an airplane wing generates lift and why a shower curtain gets pulled inward by fast-moving water.

Assumes an idealized fluid

This calculator, like the classic Bernoulli equation, assumes the fluid is incompressible and has no viscosity (no internal friction) — real fluids like air and water depart from this ideal to varying degrees, especially at high speed or in turbulent flow.

Frequently asked questions

Water flows at 3 m/s in a wide pipe and enters a narrow section where it speeds up to 8 m/s. How does pressure change?

Using Bernoulli's equation (same height): ΔP = ½ρ(v₁² − v₂²) = ½ × 1000 × (9 − 64) = −27,500 Pa. Pressure drops by 27.5 kPa in the narrow section — faster flow means lower pressure.

Is Bernoulli's equation how airplane wings really work?

Partially — Bernoulli's principle (faster air over the curved upper surface = lower pressure = net upward force) contributes to lift, but it's an oversimplification. The full picture involves Newton's third law (the wing deflects air downward) and circulation theory. See the lift force calculator.

Why does a shower curtain blow inward?

Fast-moving water creates a high-velocity air stream inside the shower, which lowers the air pressure inside relative to the still air outside. The pressure difference pushes the curtain inward — a direct application of Bernoulli's principle.

How is this related to the flow rate calculator?

Flow rate (Q = Av) tells you volume per second through a pipe. Bernoulli's equation tells you what happens to pressure when that flow speeds up or slows down. Together they fully describe steady incompressible flow through varying cross-sections.