E  = -GMm/r        (potential energy = negative-one * Mass * mass / distance

dE =  mgh          (change in potential energy = mass of small object *
                    gravitational acceleration of large object * 
                    height of small object)


The gravitational potential energy equation describes the amount of energy
an object must have to stay at a particular location in the gravity field of
another object.  (An example might be the amount of energy a brick must have
to stay 6500 km away from Earth's center without falling inward.)

To determine the amount of gravitational potential energy released when an
object moves in a gravity field, we (1) compute the potential energy it had at its 
previous location before it moved, then (2) compute the potential energy it has 
now at its current location, then (3) take the difference between the two 
energies.  Usually, the direction an object must move to release potential 
energy is toward the center of the gravity field - on the surface of a planet,
for example, that usually means falling downward toward the planet's center.

Conveniently, this gravitational potential energy equation reverts to the 
second, simpler version when the height (or distance to be traveled) is so small 
that the gravitational acceleration on the object doesn't change much.  This 
would be the case, for example, when dropping a brick from shoulder height 
to the floor, or even from a skyscraper to street level.  This approximation
fails for objects moving large distances, such as planets, stars and satellites.

Interestingly, the potential energy
of an object in a gravitational potential field is "sort of" the gravitational
force of the object integrated over the distance.  What does this mean to you?