Newton's second law of motion

where the momentum, , is the product of the object's inertial mass, , and its velocity, . If is not a function of time then Equation (2.13) reduces to the familiar equation

Note that this equation is only valid in a inertial frame. Clearly, the inertial mass of an object measures its reluctance to deviate from its preferred state of uniform motion in a straight line (in an inertial frame). Of course, the preceding equation of motion can only be solved if we have an independent expression for the force, (i.e., a law of force). Let us suppose that this is the case.

An important corollary of Newton's second law is that force is a vector
quantity. This must be the case, because the law equates force to the
product of a scalar (mass) and a vector (acceleration).^{2.2}
Note that acceleration is obviously a vector because it is directly related to displacement, which is the prototype of all vectors. One consequence of force being a vector is
that two forces,
and
, both acting at a given
point, have the same effect as a single force,
,
acting at the same point, where the summation is performed according to the
laws of vector addition. Likewise, a single force,
, acting at
a given point, has the same effect as two forces,
and
,
acting at the same point, provided that
. This
method of combining and splitting forces is known as the *resolution of
forces*; it lies at the heart of many calculations in Newtonian mechanics.

Taking the scalar product of Equation (2.14) with the velocity, , we obtain

(2.15) |

This can be written

where

(2.17) |

The right-hand side of Equation (2.16) represents the rate at which the force does work on the object; that is, the rate at which the force transfers energy to the object. The quantity represents the energy that the object possesses by virtue of its motion. This type of energy is generally known as

Suppose that, under the action of the force, , our object moves from point at time to point at time . The net change in the object's kinetic energy is obtained by integrating Equation (2.16):

because . Here, is an element of the object's path between points and , and the integral in represents the net

As is well known, there are basically two kinds
of forces in nature: first, those for which line integrals of the type
depend on the end points, but not
on the path taken between these points; second, those for which
line integrals of the type
depend
both on the end points, and the path taken between these points.
The first kind of force is termed *conservative*, whereas the
second kind is termed *non-conservative*. It can be
demonstrated that if the line integral
is path-independent, for all
choices of
and
, then the force
can
be written as the gradient of a scalar field. (See Section A.5.) In other words, all
conservative forces satisfy

for some scalar field . [Incidentally, mathematicians, as opposed to physicists and astronomers, usually write .] Note that

(2.20) |

irrespective of the path taken between and . Hence, it follows from Equation (2.18) that

(2.21) |

for conservative forces. Another way of writing this is

Of course, we recognize Equation (2.22) as an