🌊 Continuity Equation

For an incompressible fluid in steady flow, the same mass passes every cross-section per second.

Volume Flow Rate

$Q = A v$

• $Q$ = volume flow rate (m³/s)
• $A$ = cross-sectional area (m²)
• $v$ = average flow speed (m/s)

Continuity Equation

$A_1 v_1 = A_2 v_2$

💡 Key Idea: Where the pipe is narrow, the fluid moves faster. Where the pipe is wide, it moves slower. (Think of squeezing a hose.)

For compressible flow: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$ (mass flow rate). AP Physics 1 only deals with the incompressible case.

Where Continuity Comes From

It's just conservation of mass for an incompressible fluid: in a time $\Delta t$, the volume entering at section 1 ($A_1 v_1 \Delta t$) must equal the volume leaving at section 2 ($A_2 v_2 \Delta t$).

⚠️ Common Mistakes

1. Using diameter instead of area. Area scales as the square of radius (or diameter), so halving the diameter quadruples the speed.
2. Adding speeds at a junction. When two pipes merge, flow rates ($Av$) add, not speeds.
3. Confusing flow speed with mass flow. Speed depends on cross-section; mass flow rate is constant along a streamline.

Key Formulas