Escape speed: enough energy to reach infinity
You will be able to: Derive escape speed from energy conservation and distinguish it from circular-orbit speed.
How fast must an object start to escape without more thrust?
Throw an object away from a planet and gravity slows it as it climbs. At the threshold for escape, it can approach infinitely large separation while its speed approaches zero. The needed launch energy exactly balances the negative gravitational potential energy.
A useful starting point: An elliptical orbit: speed changes, E and L do not →
Words and symbols before equations
- Escape speed v_esc
- Minimum initial speed for an ideal outward escape trajectory with no further thrust.
- r_0
- Initial center-to-center separation.
- Threshold energy
- E=0 when potential energy is zero at infinity.
- Ideal escape assumptions
- Only central gravity after launch, negligible satellite mass, no atmosphere and no collision with the central body.
What this picture assumes
Scaled model: GM=100 m³/s², m=1 kg, launch radius 2 m outside a compact central body. Radially outward launch, gravity only afterward. Initial-energy comparisons, not a trajectory. Atmosphere and other bodies ignored.
Connect the picture to the physics
At the threshold, the distant kinetic and potential energies both approach zero. Conservation gives ½mv_esc²−GMm/r_0=0. Cancel m and solve: v_esc=√(2GM/r_0). A heavier test object needs more energy but the same ideal escape speed.
At the same radius, circular speed is √(GM/r_0), so escape speed is √2 times circular speed. Circular motion is bound with negative total energy; escape at the threshold has zero total energy. Above the threshold, E>0 and an outward nonintersecting trajectory retains some speed far away.
Escape does not mean gravity suddenly switches off or becomes zero at a finite boundary. It weakens with distance while still changing speed. The model here treats an outward launch; energy alone does not prevent a poorly directed trajectory from colliding with the central body. Atmosphere, planetary rotation and other bodies require additional modeling.
A worked example, step by step
Use GM=100 m³/s² and launch radius r_0=2 m outside a compact model body. Find escape speed and compare it with circular speed.
- At threshold, ½mv²−GMm/r_0=0.
- v_esc=√[2(100)/2]=10 m/s.
- v_circular=√(100/2)=√50≈7.07 m/s.
- The ratio is √2. At escape threshold the total mechanical energy is zero, whereas the circular orbit has negative energy.
Use center-to-center launch radius, and do not set gravity to zero at a finite escape boundary.
If the satellite mass doubles, does escape speed double?
Compare with an explanation
No. Mass cancels in the energy equation, though the required kinetic energy doubles.
Predict. Change one thing. Explain.
For a fixed scaled planet and launch radius, compare speeds below, at and above escape. Predict the sign of total energy. This is an initial-energy comparison for outward launches, not a simulated atmospheric rocket flight.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Initial v=10 m/s; escape threshold=10 m/s. E=0 J: At escape threshold. Speed approaches zero as separation approaches infinity.
Scaled model: GM=100 m³/s², m=1 kg, launch radius 2 m outside a compact central body. Radially outward launch, gravity only afterward. Initial-energy comparisons, not a trajectory. Atmosphere and other bodies ignored.
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 questionUse GM=72 m³/s², m=2 kg and launch radius r_0=4 m. (a) Find threshold escape speed. (b) Find initial K and U at that speed. (c) Explain the limiting speed at infinity. (d) Predict the effect of doubling satellite mass.
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Compare with the answer and four-point rubric
- 1 point: v_esc=√[2(72)/4]=6 m/s.
- 1 point: K=½(2)(36)=36 J; U=−72(2)/4=−36 J.
- 1 point: E=0 remains constant and U approaches zero, so speed approaches zero.
- 1 point: Escape speed stays 6 m/s; both required K and the magnitude of U double.
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 1Energy condition for threshold escape?
E=0 with U=0 at infinity.
RECALL 2Escape-speed formula?
v_esc=√(2GM/r_0).
RECALL 3Does gravity vanish once escape speed is reached?
No. It continues to act as the object moves away.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Escape speed: enough energy to reach infinity
- v_esc=√(2GM/r_0), derived from E=0.
- At the same radius: v_esc=√2 v_circular.
- Outward gravity-only launch: E<0 cannot reach infinity; E=0 is threshold; E>0 can retain nonzero distant speed.
Remember: Use center-to-center launch radius, and do not set gravity to zero at a finite escape boundary.
Conditions: Scaled model: GM=100 m³/s², m=1 kg, launch radius 2 m outside a compact central body. Radially outward launch, gravity only afterward. Initial-energy comparisons, not a trajectory. Atmosphere and other bodies ignored.
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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