--- title: "Unit 8.3 | Fluid Flow and the Continuity Equation" description: "Learn the continuity equation for AP Physics 1: flow rate Q = Av, A₁v₁ = A₂v₂, why half the diameter means four times the speed, with practice." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133454.jpg" url: "https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/" date_modified: "2026-10-31T09:35:28+00:00" --- # Unit 8.3 | Fluid Flow and the Continuity Equation ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/ Squeeze a hose and the water shoots out faster. The continuity equation, A₁v₁ = A₂v₂, explains why, and solves every flow rate problem on the exam. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 8 Breakdown You are on Lesson 3 of 4 1. [Unit 8.1 | Density and Pressure](https://nerd-notes.com/unit-8-1-density-and-pressure/) 2. [Unit 8.2 | Buoyancy and Archimedes’ Principle](https://nerd-notes.com/unit-8-2-buoyancy-and-archimedes-principle/) 3. [Unit 8.3 | Fluid Flow and the Continuity Equation](https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/) 4. [Unit 8.4 | Bernoulli’s Equation](https://nerd-notes.com/unit-8-4-bernoullis-equation/) #### In this lesson:  - Volume flow rate, [katex] Q = Av [/katex], and mass flow rate, [katex] \rho A v [/katex] - The continuity equation, [katex] A_1 v_1 = A_2 v_2 [/katex]: what flows in must flow out - Why halving a pipe’s diameter makes the fluid four times faster - The 4 problem types: pipes that narrow or widen, flow rate and filling time, branching pipes, and falling streams [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-3-fluid-flow-and-the-continuity-equation&utm_content=top) ### Introduction This lesson covers **fluid flow** and the **continuity equation**. It is the first of two lessons on moving fluids, and it is conservation of mass written for a pipe. In [Unit 8.1](https://nerd-notes.com/unit-8-1-density-and-pressure/) and [Unit 8.2](https://nerd-notes.com/unit-8-2-buoyancy-and-archimedes-principle/) the fluid sat still. Now it moves. Put your thumb over the end of a garden hose and the water shoots out faster. Nothing pushed harder; the same amount of water just has less room to get through. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Flow Rate The **volume flow rate** [katex] Q [/katex] is the volume of fluid passing a point each second. In one second the fluid moves a distance [katex] v [/katex], so a slice of volume [katex] Av [/katex] goes by: [katex] Q = \frac{V}{t} = Av [/katex] Its unit is m³/s. Multiply by the density and you get the **mass flow rate**, [katex] \rho A v [/katex], in kg/s. Watch the letters: here [katex] V [/katex] is a volume and [katex] v [/katex] is a speed. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ▶ **Watch:** what changes when a fluid starts to flow (starts at 16:26). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=986&rel=0) ### The Continuity Equation An ideal fluid is incompressible, so it cannot pile up anywhere in the pipe. Every kilogram that enters one end must leave the other, every second: 1. Mass in per second = mass out per second: [katex] \rho A_1 v_1 = \rho A_2 v_2 [/katex]. 2. The density is the same everywhere (incompressible), so it cancels. [katex] A_1 v_1 = A_2 v_2 [/katex] \(A_1\) \(A_2\) \(v_1\) \(v_2\) \(Q = A_1 v_1 = A_2 v_2\) *Each shaded slice is one second of flow. They hold the same volume, so the slice in the narrow pipe is longer: the fluid moves faster.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) Where the pipe is narrow, the fluid is fast. Where it is wide, the fluid is slow. For a round pipe [katex] A = \pi r^2 [/katex], so the speed goes as one over the radius *squared*: [katex] \frac{v_2}{v_1} = \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 [/katex] ▶ **Watch:** the continuity equation, worked from scratch (starts at 18:13). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=1093&rel=0) Every continuity question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Pipes that narrow or widen | two diameters or radii, a nozzle, an artery | [katex] A_1 v_1 = A_2 v_2 [/katex] with [katex] A = \pi r^2 [/katex] | | 2. Flow rate and filling time | “how long to fill”, liters per second, “volume per second” | [katex] Q = Av = \frac{V}{t} [/katex] | | 3. Branching pipes | one pipe splits into two or more | [katex] Av = \sum A_i v_i [/katex] | | 4. Falling streams | a faucet stream that gets thinner as it falls | Kinematics for [katex] v [/katex], then [katex] A_1 v_1 = A_2 v_2 [/katex] | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Fluid Flow and Flow Rate Official course · step-by-step video walkthrough [Watch · 14 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-3-fluid-flow-and-the-continuity-equation&utm_content=video-mid#course=94339&module=98826&lesson=106568) ### Problem Type 1: Pipes that narrow or widen **How to spot it:** A pipe, hose or blood vessel changes size, given by diameters, radii or areas: “what is the speed in the narrow section”. **Setup:** Write [katex] A_1 v_1 = A_2 v_2 [/katex]. With radii or diameters, use the ratio squared: [katex] v_2 = v_1\left(\frac{d_1}{d_2}\right)^2 [/katex]. Water flows at [katex] 1.5\,\text{m/s} [/katex] through a garden hose [katex] 2.0\,\text{cm} [/katex] across. The nozzle is [katex] 0.50\,\text{cm} [/katex] across. How fast does the water leave the nozzle? The diameter shrinks by a factor of 4, so the area shrinks by [katex] 4^2 = 16 [/katex]. [katex] v_2 = (1.5)\left(\frac{2.0}{0.50}\right)^2 = (1.5)(16) = 24 \text{ m/s} [/katex]. The units of the diameters cancel, so you do not even need to convert to meters. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Scaling the speed with the diameter instead of the area. Half the diameter means a quarter of the area, so four times the speed, not twice. [Open in UBQ](https://nerd-notes.com/ubq/81524/) Question 10 A fluid flows through the two sections of cylindrical pipe shown in the figure. The narrow section of the pipe has radius R and the wide section has… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/79285/) Question 11 A liquid flows at a constant flow rate through a pipe with circular cross-sections of varying diameters. At one point in the pipe, the diameter is 2… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107018/) Question 12 An ideal fluid flows through a horizontal pipe that smoothly narrows to (1/2) of its original diameter. How does the speed of the fluid in the narrow… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/87397/) Question 13 The radius of the aorta is about 1 cm and the blood flowing through it has a speed of about 30 cm/s. Calculate the average speed of the blood in the… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-3-fluid-flow-and-the-continuity-equation&utm_content=mid) ### Problem Type 2: Flow rate and filling time **How to spot it:** A tank, pool, room or bucket fills or drains: “how long”, “liters per second”, “how large must the pipe be”. **Setup:** [katex] Q = Av = \frac{V}{t} [/katex]. Convert first: [katex] 1 \text{ L} = 10^{-3} \text{ m}^3 [/katex] and [katex] 1 \text{ cm}^2 = 10^{-4} \text{ m}^2 [/katex]. A garden hose with an inside radius of [katex] 1.0\,\text{cm} [/katex] fills a [katex] 20\,\text{L} [/katex] bucket in [katex] 50\,\text{s} [/katex]. Find the flow rate and the speed of the water in the hose. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] Q = \frac{0.020 \text{ m}^3}{50 \text{ s}} = 4.0 \times 10^{-4} \text{ m}^3\text{/s} [/katex]. The hose area is [katex] \pi(0.010)^2 \approx 3.14 \times 10^{-4} \text{ m}^2 [/katex], so [katex] v = \frac{Q}{A} = \frac{4.0 \times 10^{-4}}{3.14 \times 10^{-4}} \approx 1.3 \text{ m/s} [/katex]. **AP trap:** Mixing up the volume [katex] V [/katex] with the speed [katex] v [/katex], or leaving liters and cm² unconverted. Most wrong answers here are off by a power of ten. [Open in UBQ](https://nerd-notes.com/ubq/87398/) Question 14 How large must a heating duct be if air moving 3 m/s along it can replenish the air in a room of 300 m³ volume every 15 minutes? Assume the air's… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Problem Type 3: Branching pipes **How to spot it:** One pipe splits into two or more smaller pipes (or several join into one). **Setup:** The flow in equals the total flow out: [katex] A v = A_1 v_1 + A_2 v_2 + \dots [/katex]. For identical branches, [katex] Av = n\,A_b v_b [/katex]. \(v\) \(A\) \(A_1, v_1\) \(A_2, v_2\) \(A v = A_1 v_1 + A_2 v_2\) *The water going in each second leaves through the two branches together.* A pipe with a cross-sectional area of [katex] 4.0\,\text{cm}^2 [/katex] carries water at [katex] 2.0\,\text{m/s} [/katex]. It splits into two identical branches, each [katex] 1.0\,\text{cm}^2 [/katex]. How fast is the water in each branch? [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] (4.0)(2.0) = 2(1.0)\,v_b [/katex], so [katex] v_b = 4.0 \text{ m/s} [/katex]. The two branches together have only half the area of the main pipe, so the water moves twice as fast. **AP trap:** Comparing the main pipe to one branch, [katex] A v = A_1 v_1 [/katex]. The flow divides between the branches; compare the main pipe to all of them together. [Open in UBQ](https://nerd-notes.com/ubq/79293/) Question 15 An ideal fluid flows through a pipe with radius Q and flow speed V. If the pipe splits up into three separate paths, each with radius (Q/2), what is… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107161/) Question 16 Water flows through a horizontal pipe that branches into two identical pipes, each with cross-sectional area equal to (1/2) the cross-sectional area… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Open in UBQ](https://nerd-notes.com/ubq/107170/) Question 17 A horizontal water pipe branches into three smaller identical pipes. The main pipe has a cross-sectional area A and the water flows at a speed v. If… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Problem Type 4: Falling streams **How to spot it:** A stream of water pours from a faucet and gets narrower as it falls. **Setup:** The water speeds up as it falls, so the stream must narrow to keep [katex] Av [/katex] the same. Find the speed lower down with [kinematics](https://nerd-notes.com/unit-1-3-mastering-motion-applying-kinematic-equations-in-1-dimension/), [katex] v^2 = v_0^2 + 2gh [/katex], then use continuity for the area. Water leaves a faucet at [katex] 0.50\,\text{m/s} [/katex] through an opening of [katex] 1.0\,\text{cm}^2 [/katex]. What is the area of the stream [katex] 10\,\text{cm} [/katex] below the faucet? Speed: [katex] v^2 = (0.50)^2 + 2(9.8)(0.10) = 2.21 [/katex], so [katex] v \approx 1.49 \text{ m/s} [/katex]. Area: [katex] A = \frac{(1.0)(0.50)}{1.49} \approx 0.34 \text{ cm}^2 [/katex]. The stream is a third of its starting area. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Thinking the stream narrows because of pressure or surface tension. It narrows because it speeds up, and continuity makes the faster flow thinner. [Open in UBQ](https://nerd-notes.com/ubq/110508/) Question 18 Why does a water stream from a faucet narrow as it falls? Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) Continuity Equation Practice Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-3-fluid-flow-and-the-continuity-equation&utm_content=practice-practice#course=94339&module=98826&lesson=110496) Learn Physics from Scratch The only course that teaches you to solve problems with dead-simple frameworks that make physics look easy. - Self-paced - About 12 weeks - ★ 4.9 · 12,000+ students [Enroll here](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-3-fluid-flow-and-the-continuity-equation&utm_content=end) ### Lesson 8.4 Preview When the fluid speeds up in a narrow pipe, its kinetic energy grows. In the [final lesson](https://nerd-notes.com/unit-8-4-bernoullis-equation/) we find where that energy comes from: **Bernoulli’s equation**, conservation of energy for a flowing fluid. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/