A narrower pipe means faster flow at the same rate
You will be able to: Use mass conservation to relate flow areas and speeds, including radius changes.
Where does the same water go when a pipe narrows?
In one second, 3 L of water enters a filled pipe and 3 L must leave if the steady incompressible flow has no leaks or storage. At a narrower section, that volume must travel farther along the pipe in the same second, so speed increases.
A useful starting point: Flow rate: how much fluid passes each second? →
Words and symbols before equations
- Continuity
- Conservation of mass through the selected flow region.
- A_1v_1=A_2v_2
- Equal volume rates for the same steady incompressible stream without branches.
- Circular area
- A=πr², where r is pipe radius.
- Steady, filled-pipe assumption
- The fluid fills the pipe and does not accumulate in the selected segment.
What this picture assumes
Steady incompressible flow, filled unbranched pipe, no leaks or accumulation. Q maintained at 3 L/s; inlet area 30 cm² and inlet speed 1 m/s. Pipe diameters scale with square root of area; longitudinal dimensions are schematic. Arrows: 30 drawing units per m/s.
Connect the picture to the physics
The same density enters and leaves, so equal mass rates imply equal volume rates. Set A_1v_1=A_2v_2 and solve for v_2=(A_1/A_2)v_1. Halving area doubles speed for the specified fixed flow rate.
A radius change is not an area change of the same factor. Halving a circular radius makes area one quarter, so speed becomes four times as large if Q is unchanged. Diameters have the same squared area scaling.
Continuity constrains the velocities but does not alone determine the pressure or the flow rate a real pump delivers. A nozzle can change the overall Q if the source conditions change. When a problem says Q is fixed, compare speeds under that condition; do not assume every garden-hose adjustment keeps Q unchanged.
A worked example, step by step
A pipe narrows from area 0.004 m² to 0.001 m². At the wide section v_1=1 m/s. Find v_2 and verify volume rates.
- Assume steady incompressible flow, filled pipe and no branches.
- v_2=(0.004/0.001)(1)=4 m/s.
- Q_1=0.004(1)=0.004 m³/s.
- Q_2=0.001(4)=0.004 m³/s. Equal flow rates accompany different speeds.
Halving radius quarters area. Continuity preserves flow rate, not speed.
Does a narrower pipe by itself prove the fluid pressure is lower in every situation?
Compare with an explanation
No. Continuity relates speed and area; pressure also depends on height, energy transfers and the applicable flow model.
Predict. Change one thing. Explain.
Hold Q=3 L/s and the inlet area at 30 cm². Change the outlet area and compare the velocity arrows. The pipe drawing uses diameters proportional to √area; its length is schematic.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Inlet: 30 cm² at 1 m/s. Outlet: 15 cm² at 2 m/s. Both give Q=0.003 m³/s=3 L/s. Cross-sectional area, not radius, enters Av.
Steady incompressible flow, filled unbranched pipe, no leaks or accumulation. Q maintained at 3 L/s; inlet area 30 cm² and inlet speed 1 m/s. Pipe diameters scale with square root of area; longitudinal dimensions are schematic. Arrows: 30 drawing units per m/s.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, motion or energy relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA filled pipe has inlet radius 2 cm and outlet radius 1 cm. Inlet speed is 0.5 m/s. (a) Find the inlet/outlet area ratio. (b) Find outlet speed. (c) State the conservation law used. (d) State one condition required for the simple equal-Q form.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Area ratio=2²/1²=4.
- 1 point: v_out=2 m/s.
- 1 point: Conservation of mass.
- 1 point: Steady incompressible flow with no leak/branch or accumulation in the filled segment; any one stated condition earns the point.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does continuity conserve?
Mass flow; for one incompressible fluid, volume flow rate too.
RECALL 2What happens to speed if area halves at fixed Q?
It doubles.
RECALL 3Can continuity alone calculate pressure?
No; an additional force or energy relation is needed.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A narrower pipe means faster flow at the same rate
- A_1v_1=A_2v_2 for the stated steady incompressible flow.
- Circular pipe: A=πr², so at fixed Q, v∝1/r².
Remember: Halving radius quarters area. Continuity preserves flow rate, not speed.
Conditions: Steady incompressible flow, filled unbranched pipe, no leaks or accumulation. Q maintained at 3 L/s; inlet area 30 cm² and inlet speed 1 m/s. Pipe diameters scale with square root of area; longitudinal dimensions are schematic. Arrows: 30 drawing units per m/s.
Refresh Kid · Unit 8 · Objectives 8.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.4, objectives 8.4.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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