--- title: "Unit 3.3 | Conservation of Energy, Systems, and Problem Solving" description: "Master conservation of energy for AP Physics 1: open vs closed systems, conservative vs nonconservative forces, and a 3 step framework with worked examples." featured_image: "https://nerd-notes.com/wp-content/uploads/2026/10/skier-high-above-a-snow-jump-conservation-of-energy.jpg" url: "https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/" date_modified: "2026-10-31T09:35:12+00:00" --- # Unit 3.3 | Conservation of Energy, Systems, and Problem Solving ## 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-3-3-conservation-of-energy-systems-and-problem-solving/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/ Conservation of energy turns pages of force and kinematics math into three lines. This lesson shows you how to choose a system and solve any energy problem. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Unit 3 Breakdown You are on Lesson 3 of 4 1. [Unit 3.1 | Powering the Universe – Understanding Energy in Physics](https://nerd-notes.com/unit-3-1-understanding-energy-in-physics/) 2. [Unit 3.2 | Work and the Work-Energy Theorem](https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/) 3. [Unit 3.3 | Conservation of Energy, Systems, and Problem Solving](https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/) 4. [Unit 3.4 | Understanding and Applying Power](https://nerd-notes.com/unit-3-4-understanding-and-applying-power/) #### In this lesson:  - Explain the law of conservation of energy - Choose a system (open or closed) and know why it matters - Tell conservative forces from nonconservative forces - Solve almost any energy problem with a 3 step framework - The 4 problem types: no friction, friction, springs, and an outside push [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-3-3-conservation-of-energy-systems-and-problem-solving&utm_content=top) ### Introduction This lesson covers **conservation of energy**: the law itself, how to choose a system, and a 3 step framework that uses conservation of energy to solve almost any problem. You know the types of energy ([Unit 3.1](https://nerd-notes.com/unit-3-1-understanding-energy-in-physics/)) and how work moves energy in and out ([Unit 3.2](https://nerd-notes.com/unit-3-2-work-and-the-work-energy-theorem/)). This lesson ties them into one framework. This is the most important lesson in the unit. Once it clicks, problems that would take a full page of FBDs and kinematics take three lines. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### The Law of Conservation of Energy Energy cannot be created or destroyed. It can only transfer between objects or transform from one type into another. A falling ball trades potential energy for kinetic energy. A sliding box trades kinetic energy for thermal energy. In every case the **total** stays the same. Watch it happen for a [katex] 1\,\text{kg} [/katex] ball dropped from [katex] 10\,\text{m} [/katex]: A 1 kg ball dropped from 10 m nerd-notes.com *Every joule of potential energy lost becomes a joule of kinetic energy. The total line never moves.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) Written as an equation, the energy at the start equals the energy at the end: [katex] E_i = E_f [/katex] That only holds if no energy enters or leaves what you are looking at. Which brings us to systems. ### Systems: Open vs. Closed A **system** is the object (or group of objects) you choose to analyze. Everything else is the surroundings. - A **closed (isolated) system**: no outside force does work on it, so its total energy stays constant: [katex] E_i = E_f [/katex]. - An **open system**: an outside force does work on it, adding or removing energy. For an open system, the outside work makes up the difference. This is the equation the whole unit has been building to: [katex] E_i + W = E_f [/katex] [Remove Ads](https://nerd-notes.com/pricing/?for=students) Positive outside work (a person pushing, a rocket engine) adds energy. Negative outside work (friction, air resistance) removes it. #### Choosing a system The system is your choice, and the choice changes how you describe the problem. Take a falling ball: \(\text{stores } PE_g\) *Ball only: gravity is an outside force doing work. Ball + Earth: gravity is inside, the pair stores potential energy, and the system is closed.* Both descriptions give the same answer. You’ll almost always include Earth so you can use [katex] PE_g [/katex], but the AP exam likes to ask you to explain the difference. ▶ **Watch:** how to pick the system on AP questions (starts at 4:20). [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Embedded content](https://www.youtube-nocookie.com/embed/6ItRgoQqVNM?start=260&rel=0) ### Conservative vs. Nonconservative Forces | **Conservative forces** | **Nonconservative forces** | | --- | --- | | Gravity, spring force | Friction, air resistance, applied forces (pushes, pulls, engines) | | Energy is stored as potential energy and can be fully recovered | Energy is added from outside or turned into thermal energy | | Work depends only on the start and end points | Work depends on the path (a longer path means more friction) | If only conservative forces do work, **mechanical energy is conserved**: [katex] KE_i + PE_i = KE_f + PE_f [/katex]. As soon as friction or air resistance does work, some mechanical energy becomes thermal energy and you account for it with [katex] W [/katex]. [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Conceptual Practice Check your understanding of systems and conserved energy before we start calculating. [Open in UBQ](https://nerd-notes.com/ubq/21432/) Question 11 A ball is thrown straight up. At what point does the ball have the most energy? 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/109309/) Question 12 A ball of mass m is thrown vertically upward. The ball moves from an initial height yᵢ to a higher final height y_f while in free fall. Consider two… 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/109207/) Question 13 A block is attached to a cable and is being pulled up a frictionless incline at a constant speed. The system is defined to include the block and… 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/109176/) Question 14 A crate of mass m is sliding along a horizontal warehouse floor with an initial speed v. The floor is rough, providing a constant kinetic friction… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) 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-3-3-conservation-of-energy-systems-and-problem-solving&utm_content=mid) ### The 3 Step Energy Framework Just like the [framework for force problems](https://nerd-notes.com/how-to-solving-any-force-problem-framework/), here is one for energy (the full write-up is [here](https://nerd-notes.com/how-to-solve-any-ap-physics-energy-question/)): [Remove Ads](https://nerd-notes.com/pricing/?for=students) 1. **Identify** the energies. Moving ([katex] KE [/katex])? Changing height ([katex] PE_g [/katex])? Touching a spring ([katex] PE_s [/katex])? Is friction or a push doing work ([katex] W [/katex])? 2. **Mark** two points: a start A and an end B. Pick points where you know the most, or the point the question asks about. 3. **Apply** conservation of energy: [katex] E_A + W = E_B [/katex]. Substitute each equation, then solve for the unknown. Before solving, sketch a quick **energy bar chart** for points A and B. If the bars don’t add up to the same total, an outside force must have done work. Every energy problem on the AP exam runs through the same framework. What changes is what is in it. These are the four types you will see: [Remove Ads](https://nerd-notes.com/pricing/?for=students) | **Problem type** | **The clue** | **The equation** | | --- | --- | --- | | 1. No friction | “frictionless”, “negligible air resistance” | [katex] KE_A + PE_A = KE_B + PE_B [/katex] | | 2. Friction takes energy | “rough”, “friction”, “lost to air resistance” | [katex] E_A + W_f = E_B [/katex], [katex] W_f \lt 0 [/katex] | | 3. Springs | a spring launches or stops something | [katex] \frac{1}{2}kx^2 [/katex] at the squeezed point | | 4. An outside push | an engine, thrust, a person pushing, a pulley system | [katex] E_A + W = E_B [/katex], [katex] W \gt 0 [/katex] | Solving Hard Energy Problems With \(E_i = E_f\) Official course · step-by-step video walkthrough [Watch · 28 min](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-3-3-conservation-of-energy-systems-and-problem-solving&utm_content=video-mid#course=94339&module=94419&lesson=98965) ### Problem Type 1: No friction: find a speed or a height **How to spot it:** “Frictionless”, “negligible air resistance”. Ramps, drops, pendulums, roller coasters. **Setup:** Mechanical energy is conserved: [katex] KE_A + PE_A = KE_B + PE_B [/katex]. Mass usually cancels. [Remove Ads](https://nerd-notes.com/pricing/?for=students) A skateboarder starts from rest at the top of a frictionless ramp [katex] 3\,\text{m} [/katex] tall. How fast is the skateboarder moving at the bottom? \(h\) \(KE\) \(PE_g\) \(KE\) \(PE_g\) *Point A: all potential energy. Point B: all kinetic energy. No friction, so the bars match.* **Identify:** [katex] PE_g [/katex] at the top, [katex] KE [/katex] at the bottom, no outside work. **Mark:** A at the top, B at the bottom. **Apply:** [katex] PE_A = KE_B \rightarrow mgh = \frac{1}{2}mv^2 \rightarrow v = \sqrt{2gh} = \sqrt{2(9.8)(3)} \approx 7.7 \, m/s [/katex] The mass canceled out, and the ramp’s shape never mattered: straight, curved, or loopy, only the height drop counts. Try doing that with Newton’s laws! [Remove Ads](https://nerd-notes.com/pricing/?for=students) **AP trap:** Reaching for kinematics on a curved track. It only works on straight paths. Energy only needs the height drop. [Open in UBQ](https://nerd-notes.com/ubq/111375/) Question 15 A small block is released from rest at the top of a frictionless ramp of height H. When the block reaches the bottom of the ramp, its speed 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/22681/) Question 16 How does the speed v₁ of a block m reaching the bottom of slide 1 compare with v₂, the speed of a block 2m reaching the end of slide 2? The blocks… 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 2: Friction takes energy **How to spot it:** “Rough surface”, “coefficient of friction”, “energy lost to air resistance”, or a final speed lower than no-friction would give. **Setup:** [katex] E_A + W_f = E_B [/katex], where [katex] W_f = -F_f d [/katex] is negative: friction removes energy as heat. A [katex] 50\,\text{kg} [/katex] sled starts from rest at the top of a [katex] 10\,\text{m} [/katex] tall hill. It reaches the bottom moving at [katex] 12\,\text{m/s} [/katex]. How much energy did friction remove? \(KE\) \(PE_g\) \(KE\) \(PE_g\) \(E_{th}\) *The bars at B are shorter than at A by exactly the energy friction turned into heat.* [Remove Ads](https://nerd-notes.com/pricing/?for=students) **Identify:** [katex] PE_g [/katex] at the top, [katex] KE [/katex] at the bottom, friction doing work along the way. **Apply:** [katex] PE_A + W_f = KE_B [/katex], so [katex] W_f = KE_B – PE_A [/katex]. - [katex] PE_A = (50)(9.8)(10) = 4900 \, J [/katex] - [katex] KE_B = \frac{1}{2}(50)(12)^2 = 3600 \, J [/katex] - [katex] W_f = 3600 – 4900 = -1300 \, J [/katex] Answer: friction removed [katex] 1300 \, J [/katex], which became thermal energy in the sled and the snow. Show Answer: What was the average friction force if the slope was [katex] 25\,\text{m} [/katex] long? [katex] W_f = -F_f d [/katex], so [katex] -1300 = -F_f(25) [/katex] and [katex] F_f = 52 \, N [/katex]. Your Turn — Applying Conservation of Energy  Official course · graded, with step-by-step explanations [Practice](https://nerd-notes.com/courses/?utm_source=blog&utm_medium=article&utm_campaign=unit-3-3-conservation-of-energy-systems-and-problem-solving&utm_content=practice-practice#course=94339&module=94419&lesson=98933) Quiz Lab Energy with Friction When friction takes a share. Account for the energy it removes. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=169) ▶ **Watch:** a worked example of mechanical energy lost to friction (starts at 10:53). [Remove Ads](https://nerd-notes.com/pricing/?for=students) [Embedded content](https://www.youtube-nocookie.com/embed/6dCiU2bASsU?start=653&rel=0) **AP trap:** Giving the friction work a plus sign. Friction always removes mechanical energy, so [katex] W_f [/katex] is negative. [Open in UBQ](https://nerd-notes.com/ubq/23232/) Question 17 A boulder is raised above the ground so that its potential energy is 550 J. Then it is dropped. Assuming 92 J of energy was lost to air resistance… 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: Springs: launch or stop **How to spot it:** A compressed spring launches something, or something runs into a spring and squeezes it. **Setup:** Put [katex] \frac{1}{2}kx^2 [/katex] in at the squeezed point and [katex] \frac{1}{2}mv^2 [/katex] where it moves: [katex] \frac{1}{2}kx^2 = \frac{1}{2}mv^2 [/katex] (add [katex] mgh [/katex] or friction if the problem has them). A spring with [katex] k = 400 \, N/m [/katex] is compressed [katex] 0.1\,\text{m} [/katex] and launches a [katex] 0.5\,\text{kg} [/katex] block across a frictionless floor. How fast does the block leave the spring? [Remove Ads](https://nerd-notes.com/pricing/?for=students) [katex] \frac{1}{2}(400)(0.1)^2 = \frac{1}{2}(0.5)v^2 \rightarrow 2 = 0.25v^2 \rightarrow v \approx 2.8 \, m/s [/katex] **AP trap:** Using the length of the spring for [katex] x [/katex]. [katex] x [/katex] is how far it is squeezed or stretched from its relaxed length. Try this one before you watch: a spring launch, then a rough incline. [Open in UBQ](https://nerd-notes.com/ubq/23122/) Question 18 A spring launches a 4 kg block across a frictionless horizontal surface. The block then ascends a 30° incline with a kinetic friction coefficient of… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) ▶ **Watch:** the framework solving this exact problem (starts at 16:36). [Embedded content](https://www.youtube-nocookie.com/embed/SmxZyPEXXgo?start=996&rel=0) [Remove Ads](https://nerd-notes.com/pricing/?for=students) ### Problem Type 4: An outside push adds energy **How to spot it:** An engine, a rocket’s thrust, a person pushing, or two objects tied together over a pulley. **Setup:** [katex] E_A + W = E_B [/katex], where [katex] W [/katex] is positive for a push along the motion. In a pulley system, put every moving object’s kinetic energy in the sum. **AP trap:** Leaving out one object in a pulley problem. Both blocks move, so both have kinetic energy. [Remove Ads](https://nerd-notes.com/pricing/?for=students) Try the rocket before you watch: thrust adds energy while air resistance takes some away. [Open in UBQ](https://nerd-notes.com/ubq/23120/) Question 19 A 2 kg model rocket is launched with a thrust force of 275 N and reaches a height of 90 m, at which point the thrust cuts out, but the rocket… Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz) ▶ **Watch:** the framework solving this exact problem (starts at 12:43). [Embedded content](https://www.youtube-nocookie.com/embed/SmxZyPEXXgo?start=763&rel=0) [Open in UBQ](https://nerd-notes.com/ubq/21960/) Question 20 A 0.5 kg cart, on a frictionless 2 m long table, is being pulled by a 0.1 kg mass connected by a string and hanging over a pulley. The system 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) Quiz Lab Conservation of Energy Ramps, pendulums and springs. Track the energy from start to finish. 15 questions · Free [Take the quiz](https://nerd-notes.com/quiz/?id=177) 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-3-3-conservation-of-energy-systems-and-problem-solving&utm_content=end) ### Lesson 3.4 Preview In the [final lesson](https://nerd-notes.com/unit-3-4-understanding-and-applying-power/) of this unit we cover **power**: not just how much energy is transferred, but how *fast*. ## 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-3-3-conservation-of-energy-systems-and-problem-solving/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-3-conservation-of-energy-systems-and-problem-solving/