The gravitational potential energy of a particle of mass at due to a particle of mass at is

where .

To take the gradient,

and

Hence

This gives

If we let , this is the familiar inverse square law

Sometimes we write , where is the gravitational potential

Near the surface of the Earth, take the centre of the Earth, and , where is the radius of the Earth and is the height above the surface. Then

Thus, near the surface of the Earth, we approximate the gravitational potential as

The force is then

which is a constant force near the surface of the Earth.

This force leads to the simplest example of motion due to a force.

Newton’s 2nd law gives

where . Consider the -component,

which gives

where and are the initial position and velocity at .