Integration of the Definitions in Special Cases
The defnitions of the acceleration and velocity can be integrated, provided something is known about the functional form of the acceleration. The general procedure is to first find the velocity as a function of time by integrating the acceleration:
\begin
[ \int_{v_{i}}^
dv = \int_{t_{i}}^
a\:dt]\end
and then use that velocity in the integral:
\begin
[ \int_{x_{i}}^
dx = \int_{t_{i}}^
v\:dt ]\end
Two basic forms of the acceleration are widely useful and so we illustrate the procedure for these as special cases.
Constant Acceleration
If the acceleration is constant then the velocity has the form:
\begin
[ v - v_
= a(t-t_
) ]\end
which is substituted into the next integral to find:
\begin
[ x - x_
= \int_{t_{i}}^
\left(a(t-t_
)+ v_
\right)\:dt = v_
(t-t_
) + \frac
a(t-t_
)^
]\end
This equation is the basic Law of Change for the [One-Dimensional Motion with Constant Acceleration] model.
Does this agree with the Law of Change for One-Dimensional Motion with Constant Velocity for the special case a = 0?