Continuity and Bernoulli’s Principle
Lesson notes
Overview
Narrow a pipe and the fluid speeds up while its pressure drops. This topic is rated advanced and is number 106 of 227 in Physics. Deeper models, formal results and harder reasoning.
What you are seeing
The panel above shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
For an incompressible fluid, the flow rate is the same everywhere along a pipe, A₁v₁ = A₂v₂, so the fluid speeds up where the pipe narrows. Bernoulli’s principle, which expresses energy conservation, says that where the speed is greater the pressure is lower. It explains how a carburettor draws in fuel, why a shower curtain bows inward, and part of the lift on a wing. It assumes steady, smooth flow without friction.
Reading the relationship
The expression "A₁v₁ = A₂v₂" is a compact summary of the idea. It uses the symbols A₁v, A₂v; name what each one stands for before you start. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.
Try this
Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.
Common pitfalls
Check units and directions first. Many mistakes come from mixing up a scalar with a vector, or from using a formula outside the conditions where it holds.
How it connects
It builds on “Archimedes’ Principle”, the topic before it in Physics. Once you are comfortable, continue with “Hydraulic Systems”.
Why it matters
Physics explains how the world moves, heats, shines and interacts, and the same principles power technology around you.
Where you might use it
- Revising “Continuity and Bernoulli’s Principle” before a class, exam or certification
- Designing vehicles, bridges and devices where motion, forces or waves must be predicted
- Understanding how sensors, lasers and communication links actually work
- Explaining everyday phenomena such as sound, light, heat and orbits
Frequently asked questions
What is Continuity and Bernoulli’s Principle?
Narrow a pipe and the fluid speeds up while its pressure drops. It is an advanced topic in Physics.
How do I use the Continuity and Bernoulli’s Principle simulation?
The panel at the top shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression A₁v₁ = A₂v₂ packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Deeper models, formal results and harder reasoning. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Archimedes’ Principle”, the previous topic in Physics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.