The kinetic energy

Suppose that in the case of a material point movement from position 1 to position 2 by a force it’s velocity changes from to . The motion of material point is described by the Newton's second law:

Both parts of the equation (2.47) multiply by the scalar vector of motion d :

As then (2.48) takes the form:

Next consider that . This is due to the obvious equality . Actually Differentiating the last ratio, we get 2

So,

It is important to emphasize that this relation is confirmed not only for the velocity vector , but also for any other vectors, e.g. d = r dr.

Taking it into account, formula (2.49) has form:

Integrating (2.51) along the particle's trajectory from position 1 to position 2, which receives velocity values of and , we get:

or

Thus, the work force, in case of moving of material point, equal to the change of some magnitude. So, quantity can characterize body's ability to do the work. Physical quantity that describes the ability of body or system of bodies to carry out the work called energy. Energy, which body has while moving with velocity v, is called the kinetic and denote ()

If the kinetic energy of a point in the final state is denoted as and in initial the ratio (2.52) can rewritten as:

So, the work force when moving material point is equal to the change of the kinetic energy of this point. If the work is positive then ie, kinetic energy increases. If the work is negative (effect of force is directed against the direction of movement and inhibits it), the kinetic energy is reduced:

It is important to emphasize that we are talking about the resultant of all the forces, applied to the point. This follows directly from the fact that the ratio (2.54) obtained from the equation of Newton's second law (2.47), which under the force is the resultant of all forces. Relation (2.54) is called Theorem of kinetic energy.

The result (2.54) can be generalized to an arbitrary system of n material points.

Kinetic energy of the system is the sum of the kinetic energies of material points that make up the system: