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h1. MIT 8.01 Lesson 4:  1-D Motion - General

h4. Lesson Summary

{excerpt}We consider the motion of a [point 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.{excerpt}  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]. 

h4. Learning Objectives

By the end of this Lesson, you should be able to:

* Read and understand the [One-Dimensional Motion (General)|One-Dimensional Motion (General)] [model] summary.
* Accomplish the Learning Objectives listed in that summary.
* Summarize the structure of the [Hierarchy|ModelHierarchy of HierarchyModels] of [models|model] that fall under the [One-Dimensional Motion (General)] [model], and state the specializations required for each sub-[model] to apply.

h3. Introduction

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.
 
{include:Motion -- 1-D General (Definitions)}
{include:Motion -- 1-D General (Cases)}

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