Potential energy belongs to an interaction
You will be able to: Choose a gravitational system and a zero level, then calculate changes in potential energy.
Where is the energy stored when you lift an object?
Lift a 2 kg book from a desk to a shelf 1 m higher. The book–Earth system gains 20 J of gravitational potential energy when g = 10 m/s². Calling the desk or floor zero changes the labels, but not that 20 J increase.
A useful starting point: Same endpoints, different routes →
Words and symbols before equations
- Potential energy U
- Energy associated with the configuration of interacting parts of a system, in J.
- Gravitational system
- The object and Earth together; gravitational potential energy belongs to their interaction.
- Reference level y₀
- Height assigned U = 0, chosen consistently throughout a calculation.
- ΔU
- U_final − U_initial; this difference determines energy changes.
What this picture assumes
Object + Earth gravitational system near the surface; upward positive and g = 10 m/s². U = mg(y − y₀). A zero-level change adds a constant and does not alter forces.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Zero at y₀ = 0 m: U_i = 20 J, U_f = 80 J. ΔU = 60 J; gravity work = -60 J.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
A conservative force permits a potential-energy function: ΔU = −W_conservative. Lifting slowly against gravity requires positive external work and increases U. When gravity later does positive work during a fall, U decreases.
Near Earth where g is effectively constant, U(y) = mg(y − y₀). Since W_g = −mgΔy, the sign relation follows directly. y is a height in an upward-positive coordinate system; it is not the sloping distance up a ramp.
Changing y₀ adds the same constant to every U value. ΔU and the force F_y = −dU/dy are unchanged. If the system is just the book, account for gravity as external work instead of also storing the same gravitational interaction inside the system.
A worked example, step by step
A 3 kg bag moves from y = 1 m to y = 4 m with g = 10 m/s². Compare potential energies using y₀ = 0 m and y₀ = 2 m.
- Select bag + Earth as the system, with upward positive and nearly constant g.
- With zero at 0 m, U_i = 30 J and U_f = 120 J, giving ΔU = 90 J.
- With zero at 2 m, U_i = −30 J and U_f = 60 J, again giving ΔU = 90 J.
- Gravity does −90 J in both descriptions. The arbitrary zero has no effect on the predicted motion.
Negative U is allowed. Unlike kinetic energy, a potential-energy value depends on the chosen zero.
Does a negative potential energy imply negative kinetic energy?
Compare with an explanation
No. The zero of U is arbitrary, while K = ½mv² remains nonnegative.
Predict. Change one thing. Explain.
Keep both heights fixed and move only the zero level. Compare U_i, U_f and ΔU. Then change the final height and explain the sign of gravity’s work.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Zero at y₀ = 0 m: U_i = 20 J, U_f = 80 J. ΔU = 60 J; gravity work = -60 J.
Object + Earth gravitational system near the surface; upward positive and g = 10 m/s². U = mg(y − y₀). A zero-level change adds a constant and does not alter forces.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus 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 questionFor a 1 kg object moving from y = 2 m to y = 5 m, use g = 10 m/s² and zero potential energy at y = 3 m. Find both U values, their change and gravity’s work.
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Compare with the answer and four-point rubric
- 1 point: U_i = (1)(10)(2 − 3) = −10 J.
- 1 point: U_f = (1)(10)(5 − 3) = 20 J.
- 1 point: ΔU = 30 J.
- 1 point: W_g = −ΔU = −30 J; choose object + Earth for the stored U description.
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 1Why does gravitational U belong to a system?
It depends on the relative configuration of an object and the gravitational source.
RECALL 2What is physically relevant about the zero of U?
Use it consistently; its absolute placement is arbitrary.
RECALL 3When is mgΔy appropriate?
When the gravitational field is approximately uniform across the height change.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Potential energy belongs to an interaction
- ΔU = −W_conservative.
- Near Earth: ΔU_g = mgΔy; U_g = mg(y − y₀).
- Adding a constant to U changes neither forces nor energy differences.
Remember: Negative U is allowed. Unlike kinetic energy, a potential-energy value depends on the chosen zero.
Conditions: Object + Earth gravitational system near the surface; upward positive and g = 10 m/s². U = mg(y − y₀). A zero-level change adds a constant and does not alter forces.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.3, objectives 3.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. 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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