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h1. 1-D Motion - General

We consider now the motion of a pint particle in one dimension, which can occur either because the particle is constrained to move along a track, or because we restrict attention of one Cartesian component of a particle's motion.  In general, the force applied to this particle can vary arbitrarily with time.  Hence the particle will have time-varying acceleration in proportion, a(t).  The resulting motion may then be found using calculus: the velocity v(t) is the integral of the acceleration a(t) plus the velocity at the start of the interval of integration.  Mathematically we'd say that the arbitrary constant of integration is constrained by the initial condition on the velocity.  To get x(t) we integrate v(t) and add the initial position. 

If we start knowing the position vs. time x(t), the velocity, v(t) is the derivative of its position, and the derivative of this velocity is the particle's acceleration, a(t). The force is the particle's mass times a(t).

In fact, the velocity and acceleration are defined as derivatives of the position, a fact acknowledged by the phrase "the calculus of motion".  Newton had to invent calculus of one variable to deal with motion\!

The quantities v(t) and a(t) are important in physics because these mathematically defined quantities appear in experimentally discovered physical laws.  The acceleration is related to the force applied via F=ma, and the velocity determines things like the Doppler Shift and the pressure measured in a Pitot tube - those little bent tubes that stick out of fuselage of an airplane and are bent to face into the airflow so that a pressure-measuring instrument can determine the plane's airspeed.
In our hierarchy of models there are two mutually exclusive special cases: Simple Harmonic Motion caused by a restoring force that varies linearly with the particle's displacement from some center of force, and Motion with Constant Acceleration, caused by a constant force.  Motion with Constant Velocity is a special case of Motion with Constant Acceleration specified by the constraint a(t) = 0.
 
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h3.

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