Gravitational potential measures the work per kilogram to escape to infinity. Setting the zero at infinity and remembering that gravity attracts makes the sign, and the shape of the potential well, fall into place.
Gravitational potential is the work done per unit mass to bring a test mass from infinity, ϕ = −GM / r, always negative. The potential energy of a mass m is Eₙ = mϕ = −GMm / r.
Gravitational potential is the work done per kilogram to bring a mass from infinity to a point. Because the zero is set at infinity and gravity pulls inward, the potential is always negative, a well that deepens as you approach the mass. Move the test mass in the simulation and watch the potential rise toward zero as it moves away.
For a point mass the potential is ϕ = −GM / r, and the potential energy of a mass m placed there is Eₙ = mϕ = −GMm / r. Both are negative and both rise toward zero as r increases. Moving a mass outward increases its potential energy (work is done against the field); moving it inward makes the energy more negative. The familiar Eₙ = mgh is only the small-height approximation of this, valid where g is effectively constant.
Four quick checks on the definition, the negative sign and potential energy. Each correct answer earns XP and lights this skill on your star map.
The gravitational potential at a point is the work done per unit mass in bringing a small test mass:
Gravitational potential ϕ = −GM / r is negative because:
The gravitational potential energy of a mass m at a point where the potential is ϕ is:
As a probe moves to a larger distance r from a planet, its gravitational potential energy:
Mind the sign and the reference. Potential and potential energy are negative because zero is defined at infinity and gravity is attractive. Moving away from the mass increases ϕ and Eₙ toward zero (work done against the field); moving closer makes them more negative. Do not use Eₙ = mgh except as the small-height approximation where g is constant; the general result is Eₙ = −GMm / r.
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