The work done accelerating a particle during the infinitesimal time interval *dt* is given by the dot product of *force* and *displacement*:

where we have assumed the relationship **p** = *m* **v**. (However, also see the special relativistic derivation below.)

Applying the product rule we see that:

Therefore (assuming constant mass), the following can be seen:

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

This equation states that the kinetic energy (*E*_{k}) is equal to the integral of the dot product of the velocity (**v**) of a body and the infinitesimal change of the body's momentum (**p**). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

### [edit]Rotating bodies

If a rigid body is rotating about any line through the center of mass then it has *rotational kinetic energy* () which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

where:

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).