Gravitational energy beyond the surface
You will be able to: Use inverse-distance gravitational potential energy and sum distinct interacting pairs.
Why is gravitational potential energy negative when zero is far away?
Far from a planet, imagine assigning zero gravitational potential energy. Bring a small object closer: gravity pulls it inward and can add kinetic energy, while the gravitational potential energy becomes negative.
A useful starting point: Gravitational energy and zero →
Words and symbols before equations
- Separation r
- Center-to-center distance in m.
- G
- Gravitational constant, approximately 6.67×10⁻¹¹ N·m²/kg².
- Zero at infinity
- U approaches zero as the separation grows without bound.
- Pair energy
- Potential energy assigned to one distinct interacting pair of masses.
What this picture assumes
At r₀ the pair energy is −80 J. Source and test masses remain fixed; zero potential energy is at infinity. Point masses or nonoverlapping spherical distributions are assumed.
Connect the picture to the physics
For point masses or nonoverlapping spherically symmetric bodies, U_g=−GMm/r when zero is chosen at infinite separation. The minus sign follows that reference choice. Increasing r makes U less negative: the system gains potential energy.
Gravity’s force decreases as 1/r², while this potential energy depends on 1/r. Do not exchange these relationships. Use ΔU=U_f−U_i, keeping the negative signs before subtracting.
For more than two gravitating objects, sum the energy of each distinct pair once. Three objects have pairs AB, AC and BC. This framework does not require inventing an extra energy for the collection after those pair energies are already included.
A worked example, step by step
A planet–satellite pair has U=−80 J at a reference separation r₀. Find U at 2r₀, the energy change, and gravity’s work during that outward change.
- U scales as −1/r, so at 2r₀ the value is −40 J.
- ΔU=(−40)−(−80)=+40 J.
- Gravity’s work is −ΔU=−40 J.
- The potential energy increased, even though both its initial and final values are negative.
Moving farther away raises U toward zero; it does not make U more negative.
Three distinct pair energies are −6 J, −3 J and −2 J. What is the system’s total U?
Compare with an explanation
−11 J. Add each pair once; do not double-count AB and BA.
Predict. Change one thing. Explain.
Use U₀=−80 J at r₀. Increase r/r₀ from 1 to 2 and 4. Explain why the curve approaches zero from below and why outward motion has positive ΔU.
U=-40 J; ΔU from r₀=40 J; gravity work=-40 J. Larger r gives less-negative U.
At r₀ the pair energy is −80 J. Source and test masses remain fixed; zero potential energy is at infinity. Point masses or nonoverlapping spherical distributions are assumed.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use a work, energy or power 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 pair has U=−60 J at radius r₀. (a) Find U at 3r₀. (b) Find ΔU for the outward move. (c) Find gravity’s work. (d) State why mgΔy would generally be inappropriate for such a large radial change.
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Compare with the answer and four-point rubric
- 1 point: −20 J.
- 1 point: +40 J.
- 1 point: −40 J.
- 1 point: The gravitational field changes substantially; uniform g is not a suitable approximation.
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 1How does universal U depend on distance?
As −1/r with zero at infinity.
RECALL 2Can increasing U leave it negative?
Yes; for example −80 J to −40 J.
RECALL 3How many distinct pairs among three objects?
Three: AB, AC and BC.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Gravitational energy beyond the surface
- U_g=−GMm/r with zero at infinity; W_g=−ΔU_g.
- For multiple masses, sum each distinct pair once.
- Point masses or nonoverlapping spherical distributions; use center distances.
Remember: Moving farther away raises U toward zero; it does not make U more negative.
Conditions: At r₀ the pair energy is −80 J. Source and test masses remain fixed; zero potential energy is at infinity. Point masses or nonoverlapping spherical distributions are assumed.
Refresh Kid · 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. 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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