--- title: "Unit 8.4 | Bernoulli’s Equation" description: "Learn Bernoulli's equation for AP Physics 1: fast flow and low pressure, horizontal and rising pipes, and Torricelli's theorem, with practice." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/133455.jpg" url: "https://nerd-notes.com/unit-8-4-bernoullis-equation/" date_modified: "2026-10-31T09:35:29+00:00" --- # Unit 8.4 | Bernoulli’s 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-8-4-bernoullis-equation/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-4-bernoullis-equation/ Faster fluid, lower pressure. Bernoulli’s equation is conservation of energy for flowing fluids, and it explains wings, trains and draining tanks. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 8 Breakdown You are on Lesson 4 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:  - Bernoulli’s equation as conservation of energy for a flowing fluid - Why faster fluid has lower pressure, and where that shows up: wings, trains, shower curtains - Torricelli’s theorem for a draining tank: [katex] v = \sqrt{2gh} [/katex] - The 4 problem types: fast flow and low pressure, horizontal pipes, pipes that change height, and draining tanks [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-8-4-bernoullis-equation&utm_content=top) ### Introduction This lesson covers **Bernoulli’s equation**, the last big idea of AP Physics 1. It ties fluid pressure, speed and height together, and it is just conservation of energy in a new form. In [Unit 8.3](https://nerd-notes.com/unit-8-3-fluid-flow-and-the-continuity-equation/) the water sped up in a narrow pipe. Faster water has more kinetic energy, and that energy has to come from somewhere. It comes from the pressure: the fluid is pushed from a region of high pressure into a region of low pressure, and that push does work on it. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Bernoulli’s Equation Follow a small parcel of fluid of volume [katex] V [/katex] along the pipe. Use [conservation of energy](https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/), with every term divided by the volume: 1. Kinetic energy per volume: [katex] \frac{\tfrac{1}{2}mv^2}{V} = \tfrac{1}{2}\rho v^2 [/katex]. 2. Gravitational potential energy per volume: [katex] \frac{mgy}{V} = \rho g y [/katex]. 3. The surrounding fluid pushes the parcel from high pressure to low. The work it does, per volume, is the pressure difference [katex] P_1 – P_2 [/katex]. [katex] P_1 + \rho g y_1 + \tfrac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \tfrac{1}{2}\rho v_2^2 [/katex] [Remove Ads](https://nerd-notes.com/pricing/?for=students) Along an ideal fluid’s flow, the sum of pressure, [katex] \rho g y [/katex] and [katex] \tfrac{1}{2}\rho v^2 [/katex] stays the same. Raise one term and another must drop. If the fluid is not moving ([katex] v = 0 [/katex]), it gives back [katex] P = P_0 + \rho g h [/katex] from Unit 8.1. Bernoulli’s equation is conservation of energy, and the continuity equation is conservation of mass. Most flow problems use both. ▶ **Watch:** Bernoulli’s equation, worked from scratch (starts at 20:46). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=1246&rel=0) Every Bernoulli question on the AP exam is one of the four types below. | **Problem type** | **The clue** | **The setup** | | --- | --- | --- | | 1. Fast flow, low pressure | wings, trains, curtains, “which point has the lower pressure” | Same height: faster means lower [katex] P [/katex] | | 2. Horizontal pipes | a level pipe that narrows, a Venturi tube | Continuity for [katex] v_2 [/katex], then [katex] P_1 – P_2 = \tfrac{1}{2}\rho(v_2^2 – v_1^2) [/katex] | | 3. Pipes that change height | “on the second floor”, “a height h higher” | Full Bernoulli, all three terms | | 4. Draining tanks | a hole in a tank, a spigot, “efflux speed” | [katex] v = \sqrt{2gh} [/katex], then projectile motion | [Remove Ads](https://nerd-notes.com/pricing/?for=students) Understanding Bernoulli’s Principle Official course · step-by-step video walkthrough [Watch · 21 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-4-bernoullis-equation&utm_content=video-mid#course=94339&module=98826&lesson=110494) ### Problem Type 1: Fast flow, low pressure **How to spot it:** No numbers, just “why”: airplane wings, a train passing a platform, a shower curtain, a roof in a storm, air blown between two papers. Or “which point has the lower pressure”. **Setup:** At the same height, [katex] P + \tfrac{1}{2}\rho v^2 [/katex] is constant: where the fluid is faster, its pressure is lower. Find where the flow is fastest, then ask which way the higher pressure pushes. *The water in each open tube rises until it balances the pressure below it. Over the narrow, fast section the pressure is lowest, so that column is shortest.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) A train speeds past a person standing on the platform. Is the person pushed toward the train or away from it? The air between the person and the train is dragged along and moves fast, so its pressure drops. The still air on the person’s other side is at normal pressure. The pressure difference pushes the person **toward** the train. Wings work the same way: faster air over the top, lower pressure there, a net push up. ▶ **Watch:** the conceptual Bernoulli examples (starts at 24:59). [Embedded content](https://www.youtube-nocookie.com/embed/uubzy_lNCV8?start=1499&rel=0) **AP trap:** Thinking the narrow section has higher pressure because the fluid is “squeezed”. The fluid speeds up there, and its pressure goes down. [Open in UBQ](https://nerd-notes.com/ubq/107171/) Question 1 A horizontal pipe carrying an ideal fluid narrows from a cross-sectional area of A to (1/2)A. How do the speed v and the pressure P of the fluid… 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/84453/) Question 2 A person is standing on a railroad station platform when a high-speed train passes by. The person will tend to be 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/107167/) Question 3 Air flows over the top of a curved airplane wing faster than it flows underneath. Using Bernoulli's principle, which statement best explains the… 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/79274/) Question 4 In a carbonated drink dispenser, bubbles flow through a horizontal tube that gradually narrows in diameter. Assuming the change in height is… 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-8-4-bernoullis-equation&utm_content=mid) ### Problem Type 2: Horizontal pipes: pressure from speed **How to spot it:** A level pipe or Venturi tube changes size, and the question asks for a pressure, a pressure difference or a speed. **Setup:** Use continuity to get the second speed, then Bernoulli without the height terms: [katex] P_1 – P_2 = \tfrac{1}{2}\rho\,(v_2^2 – v_1^2) [/katex]. Water flows at [katex] 2.0\,\text{m/s} [/katex] through a level pipe with a cross-sectional area of [katex] 10\,\text{cm}^2 [/katex], at a pressure of [katex] 1.8 \times 10^5 \text{ Pa} [/katex]. The pipe narrows to [katex] 2.5\,\text{cm}^2 [/katex]. What is the pressure in the narrow section? [Remove Ads](https://nerd-notes.com/pricing/?for=students) Continuity: the area drops by a factor of 4, so [katex] v_2 = 4(2.0) = 8.0 \text{ m/s} [/katex]. Bernoulli: [katex] P_1 – P_2 = \tfrac{1}{2}(1000)(8.0^2 – 2.0^2) = 3.0 \times 10^4 \text{ Pa} [/katex], so [katex] P_2 = 1.8 \times 10^5 – 0.30 \times 10^5 = 1.5 \times 10^5 \text{ Pa} [/katex]. **AP trap:** Writing [katex] \tfrac{1}{2}\rho(v_2 – v_1)^2 [/katex]. Square each speed first, then subtract: [katex] 8^2 – 2^2 = 60 [/katex], not [katex] (8 – 2)^2 = 36 [/katex]. [Open in UBQ](https://nerd-notes.com/ubq/107164/) Question 5 An ideal fluid flows through a horizontal pipe that constricts from a diameter D to a diameter (D/2). If the fluid speed in the wider section is v₁… 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/107036/) Question 6 An ideal fluid flows through a horizontal pipe that narrows from a radius of R to a radius of (R/2). If the initial speed of the fluid is v, 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/91347/) Question 7 A Venturi tube has a pressure difference of 15 000 Pa. The entrance radius is 3 cm, while the exit radius is 1 cm. What are the entrance velocity… 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/79359/) Question 8 The figure shows a horizontal pipe with sections with different cross-sectional areas. Small tubes extend from the top of each section. 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) ### Problem Type 3: Pipes that change height **How to spot it:** The fluid rises or falls along the pipe: “pumped up to the second floor”, “a point a height h higher”, a hose up a hill. **Setup:** Keep all three terms. Pick the lower point as [katex] y = 0 [/katex]. Use continuity for the speeds. Rising costs pressure ([katex] \rho g\,\Delta y [/katex]), and so does speeding up. Water enters a house through a basement pipe at [katex] 2.0\,\text{m/s} [/katex] and an absolute pressure of [katex] 3.0 \times 10^5 \text{ Pa} [/katex]. A pipe on a floor [katex] 5.0\,\text{m} [/katex] higher has half the cross-sectional area. Find the pressure there. Continuity: [katex] v_2 = 2(2.0) = 4.0 \text{ m/s} [/katex]. Bernoulli: [katex] P_2 = P_1 – \tfrac{1}{2}\rho(v_2^2 – v_1^2) – \rho g\,\Delta y [/katex] [katex] = 3.0 \times 10^5 – \tfrac{1}{2}(1000)(16 – 4) – (1000)(9.8)(5.0) [/katex] [katex] = 3.0 \times 10^5 – 6.0 \times 10^3 – 4.9 \times 10^4 \approx 2.45 \times 10^5 \text{ Pa} [/katex]. The climb costs far more pressure than the speed-up. [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Dropping the height term because the pipe “looks level”, or giving it the wrong sign. Going up, the pressure drops by [katex] \rho g\,\Delta y [/katex]; going down, it rises. [Open in UBQ](https://nerd-notes.com/ubq/112929/) Question 9 Water of density ρ flows through the section of circular pipe shown in the figure. At Point A, where the diameter of the pipe is D, the water has a… 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/107159/) Question 10 A vertical pipe has a constant cross-sectional area. As an ideal fluid flows upward through this pipe, how do the speed and pressure change? 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/107158/) Question 11 An ideal fluid flows through a pipe that both narrows in cross-sectional area and increases in height. Which of the following statements must be true… 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/87399/) Question 12 Water circulates throughout a house in a hot water heating system. If the water is pumped at a speed of 0.5 m/s through a 2 cm diameter pipe 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) ### Problem Type 4: Draining tanks: Torricelli’s theorem **How to spot it:** Water leaves a hole or spigot in an open tank: “efflux speed”, “how fast does it leave”, “how far does the stream land”. **Setup:** Point 1 at the surface, point 2 at the hole. Both are open to the air ([katex] P_1 = P_2 = P_0 [/katex]), and a wide tank’s surface barely moves ([katex] v_1 \approx 0 [/katex]). Bernoulli leaves [katex] v = \sqrt{2gh} [/katex], with [katex] h [/katex] the depth of the hole below the surface. \(v\) \(h\) \(y\) \(v^2 = 2gh\) *The water leaves as fast as a drop falling from the surface to the hole. After that, the stream is a horizontal projectile.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) Deeper hole, faster stream nerd-notes.com *\(v = \sqrt{2gh}\): four times the depth gives twice the speed.* An open tank has a small hole in its side, [katex] 1.8\,\text{m} [/katex] below the water surface and [katex] 1.25\,\text{m} [/katex] above the ground. How fast does the water leave, and how far from the tank does it land? Torricelli: [katex] v = \sqrt{2(9.8)(1.8)} \approx 5.9 \text{ m/s} [/katex], horizontally. Fall time from [katex] 1.25\,\text{m} [/katex]: [katex] t = \sqrt{\frac{2(1.25)}{9.8}} \approx 0.51 \text{ s} [/katex]. Distance: [katex] x = (5.94)(0.505) \approx 3.0 \text{ m} [/katex]. ▶ **Watch:** after it leaves the hole the stream is a horizontal projectile. Here is the projectile method, step by step. [Embedded content](https://www.youtube.com/embed/E6lmxBJ6afg?feature=oembed) **AP trap:** Measuring [katex] h [/katex] from the bottom of the tank, or from the ground. It is the depth of the hole below the water surface. The exit speed also does not depend on the size of the hole, the width of the tank or the liquid’s density. [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Open in UBQ](https://nerd-notes.com/ubq/107136/) Question 13 Two identical open containers are filled with water to the same height H. Container A has a small hole at a depth h below the surface, and Container… 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/84466/) Question 14 You have a giant cask of water with a spigot some height below the water surface. The surface of the water, which is essentially at rest, is exposed… 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/107123/) Question 15 An open container is filled with an ideal fluid. If a small hole is poked in the side of the container at a depth h below the surface, the speed of… 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/87400/) Question 16 The side of an above-ground pool is punctured, and water gushes out through the hole. If the total depth of the pool is 2.5 m, and the puncture is 1… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) Bernoulli Calculations: 11 Quantitative Problems Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-8-4-bernoullis-equation&utm_content=practice-practice#course=94339&module=98826&lesson=106570) Quiz Lab Fluid Flow and Bernoulli Narrow pipes, fast water and draining tanks. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=188) ### Unit 8 Wrap-Up - **Density and pressure:** [katex] \rho = \frac{m}{V} [/katex], [katex] P = \frac{F_\perp}{A} [/katex], and pressure grows with depth, [katex] P = P_0 + \rho g h [/katex] (8.1). - **Buoyancy:** the fluid pushes up with the weight of the fluid displaced, [katex] F_B = \rho_f V_{sub} g [/katex] (8.2). - **Continuity:** what flows in flows out, [katex] A_1 v_1 = A_2 v_2 [/katex] (8.3). - **Bernoulli:** energy per volume is conserved, so faster flow means lower pressure (8.4). [Remove Ads](https://nerd-notes.com/pricing/?for=students) Test the whole unit in one go: Quiz Lab Unit 8: Mastering Fluid Mechanics: Statics and Dynamics A comprehensive set of problems covering fluid pressure, buoyancy, and ideal fluid dynamics, including applications of Bernoulli's and Continuity equations. 10 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=32) ### You Finished AP Physics 1 Unit 8 is the last unit of the course. You now have every tool the AP exam uses: kinematics, forces, energy, momentum, rotation, oscillations and fluids. What is left is practice, mixing the units the way the exam does. - All the free notes, worksheets and practice in one place: the [AP Physics 1 course materials page](https://nerd-notes.com/physics/ap-physics-1-from-scratch-course/ap-physics-1-course-all-materials/). - One page per unit when you review: [Kinematics](https://nerd-notes.com/kinematics-speed-review/), [Forces](https://nerd-notes.com/forces-in-10-minutes/), [Circular motion](https://nerd-notes.com/circular-motion-in-10-minutes/), [Energy, work and power](https://nerd-notes.com/energy-in-10-minutes/), [Momentum](https://nerd-notes.com/linear-momentum-in-10-minutes/), [Torque and rotational kinematics](https://nerd-notes.com/torque-in-10-minutes/), [Rotational energy and angular momentum](https://nerd-notes.com/rotational-motion/), [Simple harmonic motion](https://nerd-notes.com/simple-harmonic-motion-speed-review/), [Fluids](https://nerd-notes.com/ap-physics-1-fluids-dynamics-review-key-equations-concepts-made-easy/). [Remove Ads](https://nerd-notes.com/pricing/?for=students) Review the unit you feel least sure of first, then work mixed practice under time. ## 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-8-4-bernoullis-equation/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-8-4-bernoullis-equation/ 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-8-4-bernoullis-equation&utm_content=end)