
The simplest physical example of a situation with such a
force is the motion of an object on the surface of the earth
subject to the pull ot the earth’s gravity.
When the acceleration is constant and is given by gk,
we can deduce that the velocity, v(t), of the object
whose derivative this acceleration must be will obey:
v(t) = v(t_{0})  g(t
 t_{0})k,
and this velocity is the derivative of a position vector
r(t) which must take the form:
We have not discussed at this point how we deduce
these things, but we can verify them by differentiating r
and v, and checking their correctness at time t_{0}.
(In general, if the acceleration were given to us as a polynomial
function of t, we could produce forms like these containing
polynomials of degree one and two greater than that of the
acceleration to describe velocity and position vectors, and
adjust their coefficients to yield correct formulae for them.
However, physical phenomena are rarely described by forces
and hence accelerations that depend explicitly on time. Rather
the most important and fundamental situation is one in which
the object experiences a force that depends linearly on its
position.)
Though these equations provide a complete solution to the
problem of motion in three dimensions subject to a constant
acceleration, they do not provide convenient answers to all
the questions we might raise about the situation. Thus, we
can use them to determine r and v at any time
t, but they are not particularly convenient for deducing what
r is as a function of v, or vice versa.
To help answer such questions, we define the Energy function,
E by
The first of these terms is called the kinetic
energy, T, of the object, and the second its potential energy,
V. The force experienced by the object can be written here
as V;
the derivative of E with respect to time is then given by
This fact, called the conservation of energy,
holds whenever F can be written as a gradient. Here if we
know the value of E at any time, we can use this fact to compute
z as a function of v or vice versa from the expression above
for E.

