Integrate gravitational work with a defensible reference

Integrate force along the radial coordinate

Outside a spherical source, radial gravitational force is −GMm/r², with positive r outward. The work from ri to rf is the integral of this force, GMm(1/rf−1/ri). Since potential change is minus conservative work, choosing U=0 at infinity gives U=−GMm/r. Negative potential does not mean negative kinetic energy.

Compare external work under a stated process

For a quasistatic outward displacement, an external agent supplies a force balancing gravity and transfers work ΔU. Gravity supplies the opposite signed work. If the object instead accelerates, external work also changes kinetic energy; the quasistatic result cannot be reused without that qualification. The workbench fixes the source and neglects recoil.

Test the local approximation

With GM=20, ri=2 and test mass m, initial g=5 m/s². A constant-initial-g estimate gives ΔU≈5m(rf−2), while the exact result is 20m(1/2−1/rf). The linear estimate overstates the required outward energy because the field decreases with radius. For a small height h, expanding 1/(ri+h) recovers the leading m g_i h term.

Count interaction pairs once

For several separated spherical bodies in the Newtonian point approximation, total gravitational potential sums −G mi mj/rij over distinct unordered pairs. Three objects have three pairs, not six; summing each object’s interaction with every other object and then adding without a one-half correction double counts. This potential describes the configuration, not one object’s isolated property.

Use escape energy without claiming a finite endpoint

For fixed-source radial escape, marginal total energy zero gives initial K=GMm/ri. The speed approaches zero only as r tends to infinity, not at a finite stop. Positive total energy permits a nonzero limiting speed under this isolated model; negative energy is bound. Real launch dynamics, atmosphere and source motion are outside this calculation.

Worked original case

For m=2,ri=2,rf=4, potential rises from −20 to −10 J; gravity work is −10 J and quasistatic external work +10 J. The initial-g linear estimate gives 20 J, twice the exact difference.

Investigation

Compare final radii 3 and 8 for m=2. Reconstruct the definite integral and both work signs, then express the relative change of g. Propose how a small-height comparison would test the approximation without claiming a launch experiment.

Public numerical transfer practice

Try a new practice case

With GM=20 m³/s², mass 3 kg and initial radius 2 m, what marginal escape kinetic energy is required in joules?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching, values and calculations, checked 30 October 2026 against the official calculus-based Mechanics framework and OpenStax work and kinetic energy. The numerical packets are supplied or generated, not measured apparatus records. No physical launch, machinery, loading or field intervention is requested. Public feedback and virtual captures do not establish independent mastery or supervised laboratory participation.