Gravity beyond the constant-field approximation
You will be able to: Use U = −GMm/r, derive the near-surface limit and determine escape energy.
How do you calculate gravitational energy when height is not small?
Far above a planet, gravity is weaker than at its surface. Adding mg times a very large height would overestimate the required energy if g were held at its surface value. The inverse-distance potential accounts for the weakening field.
A useful starting point: Equilibrium, turning points and allowed motion →
Words and symbols before equations
- G
- Universal gravitational constant, about 6.67 × 10⁻¹¹ N·m²/kg².
- r
- Distance between mass centers, not height above the surface.
- U = 0 at infinity
- The standard reference for an isolated gravitational pair.
- Escape speed
- Minimum initial speed to reach infinity with zero limiting speed in the ideal isolated model.
What this picture assumes
Spherical dominant source, probe outside its surface and zero U at infinity. Energy is in units GMm/R, force in GMm/R², and radius in R. The constant-g comparison uses g_surface = GM/R².
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- At r = 2R: U / (GMm/R) = -0.5. Gain from R = 0.5 energy units; constant-g estimate 1. Signed radial force / (GMm/R²) = -0.25.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Outside a spherical source, radial force is F_r = −GMm/r². Integrate −F_r dr from infinity to r to obtain U(r) = −GMm/r. The negative sign means the pair has less energy than it would infinitely separated.
From radius R to R+h, ΔU = GMm(1/R − 1/(R+h)) = GMm h/[R(R+h)]. For h much smaller than R this is approximately (GM/R²)mh = mgh. The approximation is a limit, not a competing law.
If the central mass stays effectively fixed and other forces are absent, K + U is conserved. Escape with zero limiting speed requires E = 0, so ½mv_esc² − GMm/R = 0 and v_esc = √(2GM/R). This describes an unpowered test-object idealization, not a rocket’s fuel requirement.
A worked example, step by step
A spherical planet has GM = 4 × 10¹⁴ m³/s². A 2 kg probe moves from r = 1 × 10⁷ m to 2 × 10⁷ m. Find ΔU and gravity’s work.
- Use center-to-center radii, outside the planet, with zero U at infinity.
- U_i = −(4×10¹⁴)(2)/(1×10⁷) = −8×10⁷ J.
- U_f = −(4×10¹⁴)(2)/(2×10⁷) = −4×10⁷ J, so ΔU = +4×10⁷ J.
- W_g = −ΔU = −4×10⁷ J. Moving outward increases U toward zero.
U is negative with zero at infinity, but moving outward increases it. Use radius from the center, not altitude.
Is twice the radius twice the altitude?
Compare with an explanation
No. If radius changes from R to 2R, altitude changes from 0 to R. Radius is measured from the center.
Predict. Change one thing. Explain.
Increase r/R from 1 to 4. Compare exact ΔU with the constant-surface-g estimate. Keep source mass and radius fixed and explain why the estimate diverges for large heights.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At r = 2R: U / (GMm/R) = -0.5. Gain from R = 0.5 energy units; constant-g estimate 1. Signed radial force / (GMm/R²) = -0.25.
Spherical dominant source, probe outside its surface and zero U at infinity. Energy is in units GMm/R, force in GMm/R², and radius in R. The constant-g comparison uses g_surface = GM/R².
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 test mass at a planet’s surface R, derive escape speed and compare it with circular-orbit speed at the same radius. State the required assumptions.
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Compare with the answer and four-point rubric
- 1 point: Set the threshold total energy to zero: ½mv_esc² − GMm/R = 0.
- 1 point: Solve v_esc = √(2GM/R).
- 1 point: Circular force balance gives v_circ = √(GM/R), hence v_esc = √2 v_circ.
- 1 point: Assume a spherical dominant source, negligible atmosphere and other bodies, and an unpowered test object with negligible source recoil.
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 choose U = 0 at infinity?
It makes the isolated gravitational pair’s potential −GMm/r and the escape threshold E = 0.
RECALL 2How does mgh emerge?
It is the h ≪ R approximation to the exact gravitational energy difference.
RECALL 3Does escape speed depend on the test mass?
No, because the test mass cancels between kinetic and gravitational energies.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Gravity beyond the constant-field approximation
- U_g = −GMm/r outside separated spherical masses.
- ΔU = GMm(1/r_i − 1/r_f).
- v_esc = √(2GM/R) for negligible other bodies, drag and propulsion after launch.
Remember: U is negative with zero at infinity, but moving outward increases it. Use radius from the center, not altitude.
Conditions: Spherical dominant source, probe outside its surface and zero U at infinity. Energy is in units GMm/R, force in GMm/R², and radius in R. The constant-g comparison uses g_surface = GM/R².
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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