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Gravitational potential and gravitational force

The gravitational force is the negative of the gradient (or the derivative) of the gravitational potential energy.

The gradient tells you by how much the gravitational potential energy changes per unit of distance that you move towards or away from the gravitational force.

The proof for the equation for the potential around a unit mass $$M$$ is as follows:

The potential is the work done by a gravitational field on a unit mass in bringing the mass from infinity to its current position $$R$$.

If the unit mass is at distance $$S$$ from the planet (the source of the gravitational field), the work done to move it infinitesimally from $$S$$ to $$S+dS$$ is $$F(S) dS = -GM/S^{2}$$ (the force is opposite to the movement).

Integrating between the current position $$R$$ to infinity and using the fact that the integral of $$S^{-2}$$ is $$-S^{-1}$$, we get:$$$V =\int_R^\infty F(S) dS= -\int_R^\infty \frac{GM}{S^2}\, dS = \left[\frac{GM}{S}\right]_R^\infty = -\frac{GM}{R}$$$