Continuity, Bernoulli, and energy connections

Intro

The course enters playoffs here, but the content is still real: continuity and Bernoulli ask students to connect fluid motion back to earlier energy reasoning.

Core Lesson

Continuity helps students reason about how fluid speed changes when the geometry of the flow changes. Bernoulli then connects fluid pressure and speed through an energy-style lens. These ideas make the most sense when treated as extensions of earlier Unit 3 thinking rather than as a totally separate fluid trick.

Students should avoid memorizing Bernoulli as a magic sentence about "fast fluid means low pressure" without conditions. The real task is to connect pressure, speed, height, and energy reasoning inside an appropriate fluid model.

This is a good moment to reinforce cross-unit coherence. Fluids are not a fresh language; they reuse conservation habits, representation habits, and qualitative comparison habits built earlier in the year.

AP Lift

The v2 map explicitly frames Bernoulli as Unit 3 energy logic in a fluid context. Students who recognize that continuity can reason across units instead of rebuilding from scratch.

Must-Master Objectives

• Explain continuity qualitatively in changing-flow situations.
• Connect Bernoulli reasoning to earlier energy logic.
• Avoid using fluid shortcuts without model awareness.
• Compare pressure and speed changes within a coherent fluid story.

Problem Set Prompts

1. Why can fluid speed change when the cross-sectional area changes?
2. How is Bernoulli reasoning similar to earlier energy reasoning?
3. Why is "faster fluid means lower pressure" incomplete without context?
4. What role does height play in a Bernoulli-style comparison?
5. Why should continuity and Bernoulli be explained with a system story before formulas?
6. How do these ideas show that fluids are not separate from the rest of mechanics?
7. What mistake appears when a student uses Bernoulli as a slogan instead of a model?
8. Stretch: Describe how water flow in a narrowing pipe can be interpreted with both continuity and energy logic.
9. Stretch: What evidence would show that a student remembers Unit 3 but fails to transfer it to fluids?

Reflection Prompt

• Does Bernoulli feel like a new formula set or like energy reasoning in disguise?
• Which part feels more natural to you right now: continuity or pressure-speed comparisons?