An elliptical orbit: speed changes, E and L do not
You will be able to: Use angular momentum and energy conservation to compare points on an elliptical orbit.
Why does a satellite move fastest near its planet?
A satellite on an elongated orbit comes close to its planet, speeds through the near part, then slows as it travels farther away. Its speed changes without a rocket burn: gravitational potential energy and kinetic energy trade while total mechanical energy stays constant.
A useful starting point: Circular orbits: gravity, energy and angular momentum →
Words and symbols before equations
- Periapsis and apoapsis
- Nearest and farthest points from the central body.
- Central force
- A force along the line joining the two bodies; gravity has zero torque about the central body.
- rv sinφ
- The radius–velocity product that enters particle angular momentum; φ is the angle between radius and velocity.
- Bound orbit
- An orbit with negative total mechanical energy, for U=0 at infinity.
What this picture assumes
One scaled orbit: GM=12 m³/s², m=1 kg, closest radius 2 m, farthest radius 6 m. Central body at a focus with radius below 2 m. Control is a geometric ellipse parameter, not a uniform clock. Drawing equations are not an additional lesson prerequisite.
Connect the picture to the physics
Gravity points toward the central body, giving zero torque about that point. Therefore the satellite’s angular momentum about it stays constant in the ideal model. Gravity is also conservative, so K+U stays constant for the satellite–central-body system.
At periapsis and apoapsis, velocity is perpendicular to the radius. Thus mr_pv_p=mr_av_a. Farther away at apoapsis, speed is lower. At a general point, velocity may not be perpendicular: use L=mrv sinφ, not simply mrv.
As r increases, U=−GMm/r becomes less negative, so K decreases by the same amount. Total E and L stay constant; K and U generally do not. The central body lies at a focus of the ellipse, not its geometric center. You can explain these changes using conservation without memorizing a new ellipse-speed formula.
A worked example, step by step
In a scaled gravity model, GM=12 m³/s² and m=1 kg. An ellipse has r_p=2 m, v_p=3 m/s and r_a=6 m. Find v_a and verify energy at the two endpoints.
- At both endpoints v is perpendicular to r, so r_pv_p=r_av_a.
- v_a=(2)(3)/6=1 m/s. L=6 kg·m²/s at both points.
- Near: K=4.5 J and U=−12/2=−6 J, so E=−1.5 J.
- Far: K=0.5 J and U=−12/6=−2 J, so E=−1.5 J. The 4 J increase in U matches a 4 J decrease in K.
Do not use circular-orbit speed at arbitrary points of an ellipse. Also, mrv omits a needed sine factor away from perpendicular velocity.
Is gravitational potential energy constant in an ellipse?
Compare with an explanation
No. r changes, so U changes. Its changes are balanced by opposite changes in K.
Predict. Change one thing. Explain.
Move a satellite marker around one ellipse. Compare K, U and E at the nearest, farthest and intermediate locations. Watch the radius and velocity directions; the central body is at a focus.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
r=2 m; speed=3 m/s. K=4.5 J; U=-6 J. E=-1.5 J and L=6 kg·m²/s stay constant. Position control is not elapsed time.
One scaled orbit: GM=12 m³/s², m=1 kg, closest radius 2 m, farthest radius 6 m. Central body at a focus with radius below 2 m. Control is a geometric ellipse parameter, not a uniform clock. Drawing equations are not an additional lesson prerequisite.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition 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 1 kg satellite has r_p=2 m, v_p=4 m/s and r_a=8 m in a scaled model with GM=20 m³/s². (a) Find L at periapsis. (b) Find v_a. (c) Calculate E at both endpoints. (d) Explain why gravity changes K but not L about the central body.
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Compare with the answer and four-point rubric
- 1 point: L=1×2×4=8 kg·m²/s.
- 1 point: v_a=8/(1×8)=1 m/s.
- 1 point: Near E=8−10=−2 J; far E=0.5−2.5=−2 J.
- 1 point: Gravity does work during radial motion, transferring energy between K and U, but its line of action passes through the central reference point, giving zero torque.
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 1What stays constant in an ideal ellipse?
Total mechanical energy and angular momentum about the central body.
RECALL 2Where is speed largest?
At the closest point, periapsis.
RECALL 3When is |L|=mrv valid?
When velocity is perpendicular to the radius, including the near and far endpoints.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
An elliptical orbit: speed changes, E and L do not
- Gravity only: E=K+U and angular momentum about the central body stay constant.
- At near/far endpoints: r_pv_p=r_av_a.
- General particle magnitude: |L|=mrv sinφ.
Remember: Do not use circular-orbit speed at arbitrary points of an ellipse. Also, mrv omits a needed sine factor away from perpendicular velocity.
Conditions: One scaled orbit: GM=12 m³/s², m=1 kg, closest radius 2 m, farthest radius 6 m. Central body at a focus with radius below 2 m. Control is a geometric ellipse parameter, not a uniform clock. Drawing equations are not an additional lesson prerequisite.
Refresh Kid · Unit 6 · Objectives 6.6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.6, objectives 6.6.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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