Integration of the Definitions in Special Cases 
General Time Integrals
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:
We are using η as the dummy variable of integration on the right because we are using t as the (arbitrary) endpoint of the integration, and so t is not available as the integration variable. You can just as easily use any other dummy variable.
and then use that velocity in the integral:
Two basic forms of the acceleration are widely useful and so we illustrate the procedure for these as special cases.
Case 1: Constant Acceleration
If the acceleration is constant then the velocity has the form:
which is substituted into the next integral to find:
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 of the One-Dimensional Motion with Constant Velocity model for the special case a = 0?
Case 2: Acceleration Varying Linearly with Position
The procedure illustrated above will work for any acceleration which given as an explicit and integrable function of time (though of course some integrable functions require more work to integrate than others). The procedure must be modified, however, if the acceleration is given as an implicit function of time. The simplest such specification that is commonly encountered in mechanics is:
where ω is a (real number) constant. Because the acceleration depends implicitly on time through the unknown time dependence of the position, we must solve for the position by construcing a second-order differential equation:
Solving a differential equation often requires specialized techniques, but in some simple cases (like this one) a good calculus student can guess the answer based upon the properties of common functions. In this case, the fact that the trig functions sine and cosine are proportional to the negative of their own second derivative is a big clue that the answer can be given in terms of these functions. We will explore the solutions to this differential equation in more detail when we learn the Simple Harmonic Motion model.
The character of the equation changes dramatically if the negative on the right hand side is removed, and sine or cosine is no longer a solution. The answer is still readily guessed by a student familiar with the basic functions of calculus, however. Can you think of a common function which would solve the case when the right hand side is positive?
Specializations and the Hierarchy of Models
The fact that we will study the two specializations discussed above in detail means that it makes sense to consider them separate models even though they are clearly special cases of the One-Dimensional Motion (General) model. Thus, if you examine the hierarchy of Models, you will see these two special cases listed below the General motion case as One-Dimensional Motion with Constant Acceleration and Simple Harmonic Motion. Of course, we could generate an infinite number of special cases, such as accelerations that are polynomial functions of time, accelerations that are proportional to velocity, and so on. In the hierarchy, we present only the specializations that are commonly taught and used in introductory mechanics. If you take a more advanced course in mechanics, you may want to expand the hierarchy to incorporate new specializations.
One-Dimensional Motion (General) - System: One point particle constrained to move in one dimension. — Interactions: Any that respect the one-dimensional motion.
- 1-D Motion (Constant Acceleration) — System: One point particle moving in one dimension either because it's constrained to move that way or because only one Cartesian component is considered. — Interactions: Constant force (in magnitude or in its component along the axis).
- 1-D Motion (Constant Velocity) — System: One point particle. — Interactions: No acceleration (zero net force).
- Simple Harmonic Motion — System: One point particle constrained to move in one dimension. — Interactions: The particle must experience a force (or torque) that attempts to restore it to equilibrium and is directly proportional to its displacement from that equilibrium.