Relate orbit shape, conserved quantities and escape energy

1. Frame the question

Consider a satellite much less massive than a central body, so central-body recoil is negligible. With gravity as the only interaction, central force exerts zero torque about the central point and the satellite’s angular momentum about it is constant. The satellite–central-body system also conserves total mechanical energy. The gravitational potential reference is zero at infinite separation, giving U=−GMm/r.

2. Build the mechanism

For a circular orbit, gravity supplies centripetal acceleration: GMm/r²=mv²/r, so v²=GM/r. Then K=GMm/(2r), U=−GMm/r and E=−GMm/(2r). Each is constant around that circular orbit. A larger circular orbit has smaller speed and less negative total energy; moving between circular orbits requires an energy-changing interaction rather than simply coasting at a different radius.

In an elliptical orbit, total E and angular momentum remain constant while K and U exchange. Nearer the central body, U is more negative and speed is larger. At closest and farthest points velocity is perpendicular to radius, so rpvp=rava. Away from those turning points, angular momentum uses the transverse velocity component, not simply mr times total speed. Equal conserved quantities do not require constant angular speed.

Escape at the marginal energy threshold satisfies ½mv²−GMm/r=0, giving vesc=√(2GM/r). Under the ideal gravity-only, outward-moving condition, speed approaches zero at infinite distance. Above that energy threshold it remains unbound with nonzero asymptotic speed. This is an ideal point-mass comparison, not a procedure for launching anything or a claim that the minimum energy accounts for atmosphere, surface obstacles or propulsion.

3. Work an example

The virtual central parameter is GM=100 m³/s² and r=20 m. Circular speed is √5≈2.236 m/s; specific kinetic energy is 2.5 J/kg, potential −5 J/kg and total −2.5 J/kg. Escape speed is √10≈3.162 m/s. A separate ellipse with closest radius 10 m and farthest 30 m has speeds √15 and √15/3 m/s at those points: both give the same specific angular momentum and total specific energy −2.5 J/kg.

4. Predict and investigate

Predict first: If a satellite is faster when closer in an ellipse, which quantities remain constant and which exchange?

At r=20 m compare speed ratios 0.75, 1 and 1.5 relative to circular speed. Identify the circular case and the sign of specific total energy. Then compare circular orbits at 10 and 40 m and derive speed, energy and angular-momentum ratios before checking the ledger. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

For a virtual central parameter GM=180 m³/s² at radius 10 m, find the ideal escape speed.

Your numerical result (m/s) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

Why can you not apply the circular formula K=GMm/(2r) at every point of a noncircular ellipse?

Retain your first response, system and axis choices, separate scalar energy and signed angular-momentum ledgers, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 6 topic mapping: 6.6. Checked 30 October 2026. Planar signed analysis without required three-dimensional angular-momentum direction. Slipping evolution is qualitative, in accordance with the framework boundary; no rolling-resistance model. Framework cross-reference, not AP authorization.