AP® Physics 1: Algebra-Based · Unit 8 · Topic 8.4
The Continuity Equation
The continuity equation says that for an incompressible fluid flowing through a pipe, A1v1 = A2v2: wherever the pipe narrows, the fluid speeds up, and wherever it widens, the fluid slows down.
Picture water as an unbreakable chain of volume moving through a pipe. If the fluid can't be compressed and none of it is leaking out or piling up anywhere, then the volume passing any cross-section each second has to equal the volume passing any other cross-section each second. That quantity, area times speed, is the volume flow rate: Q = Av. Since Q stays constant along the pipe, A1v1 = A2v2. Shrink the area and the speed must grow to keep the product the same.
This is why covering part of a garden hose opening makes the water shoot out faster and farther. You haven't added energy or pressure from nowhere, you've just forced the same volume per second through a smaller opening, so each bit of water has to move faster to get through. Because area depends on diameter squared (A = π r2), small changes in diameter cause big changes in speed: half the diameter means a quarter the area, which means four times the speed.
Continuity is the volume bookkeeping half of fluid flow. It pairs with Bernoulli's equation, which tracks energy: once you know how speed changes from continuity, Bernoulli tells you how pressure must change to match. A falling stream of water from a tap thins as it drops because gravity speeds it up, and continuity forces the cross-section to shrink to keep the volume flow rate constant.
Key terms
- Continuity equation
- A1v1 = A2v2, the statement that volume flow rate stays constant along a pipe of incompressible fluid.
- Volume flow rate
- The volume of fluid passing a cross-section per unit time, Q = Av, measured in units like m3/s.
- Incompressible fluid
- A fluid whose density does not change, so it cannot be squeezed into less volume.
- Cross-sectional area
- The area of the pipe's interior perpendicular to the flow direction, A = π r2 for a circular pipe.
Watch it explained
- How it worksHow does conservation of volume force a narrower pipe to carry faster fluid?
- Worked exampleFind the speed in a 2 cm pipe when water flows at 1.5 m/s through a 6 cm section.
- Common mistakeWhy isn't the flow rate different at the wide end and the narrow end of one pipe?
- Going furtherWhat happens to a stream of water falling from a tap as it drops and thins out?
Check yourself
End of topic.