[5] The term "potential energy" was coined by William Rankine a Scottish engineer and physicist in 1853 as part of a specific effort to develop terminology.

[3] He chose the term as part of the pair "actual" vs "potential" going back to work by Aristotle.

It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is

The line integral that defines work along curve C takes a special form if the force F is related to a scalar field U′(x) so that

Given a force field F(x), evaluation of the work integral using the gradient theorem can be used to find the scalar function associated with potential energy.

The power applied to a body by a force field is obtained from the gradient of the work, or potential, in the direction of the velocity v of the point of application, that is

Examples of work that can be computed from potential functions are gravity and spring forces.

In classical physics, gravity exerts a constant downward force F = (0, 0, Fz) on the center of mass of a body moving near the surface of the Earth.

The work of gravity on a body moving along a trajectory r(t) = (x(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, v = (vx, vy, vz), to obtain

The work of this spring on a body moving along the space curve s(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (vx, vy, vz), to obtain

For convenience, consider contact with the spring occurs at t = 0, then the integral of the product of the distance x and the x-velocity, xvx, is x2/2.

The work W required to move q from A to any point B in the electrostatic force field is given by the potential function

When the book hits the floor this kinetic energy is converted into heat, deformation, and sound by the impact.

"Height" in the common sense of the term cannot be used for gravitational potential energy calculations when gravity is not assumed to be a constant.

The amount of gravitational potential energy held by an elevated object is equal to the work done against gravity in lifting it.

The upward force required while moving at a constant velocity is equal to the weight, mg, of an object, so the work done in lifting it through a height h is the product mgh.

However, over large variations in distance, the approximation that g is constant is no longer valid, and we have to use calculus and the general mathematical definition of work to determine gravitational potential energy.

in the formula for gravitational potential energy means that the only other apparently reasonable alternative choice of convention, with

at infinity is by far the more preferable choice, even if the idea of negative energy in a gravity well appears to be peculiar at first.

[11] Gravitational potential energy has a number of practical uses, notably the generation of pumped-storage hydroelectricity.

The process is not completely efficient and some of the original energy from the surplus electricity is in fact lost to friction.

Gravitational potential energy is also used to power clocks in which falling weights operate the mechanism.

Another practical use is utilizing gravitational potential energy to descend (perhaps coast) downhill in transportation such as the descent of an automobile, truck, railroad train, bicycle, airplane, or fluid in a pipeline.

As an example, when a fuel is burned the chemical energy is converted to heat, same is the case with digestion of food metabolized in a biological organism.

An object can have potential energy by virtue of its electric charge and several forces related to their presence.

It is defined as the work that must be done to move it from an infinite distance away to its present location, adjusted for non-electrical forces on the object.

Conversely, like poles will have the highest potential energy when forced together, and the lowest when they spring apart.

In the Sun, the process of hydrogen fusion converts about 4 million tonnes of solar matter per second into electromagnetic energy, which is radiated into space.

If the work done by a force on a body that moves from A to B does not depend on the path between these points, then the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field.

This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.