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.
 

Introduction to the Model Description and Assumptions This model is applicable to a single point particle subject to an acceleration that is constrained to one dimension and which is either parallel to or anti-parallel to the particle's initial velocity. Learning Objectives Students will be assumed to understand this model who can: Choose the one graph possible velocity or acceleration vs. time graphs which corresponds to a model position versus time graph. Differentiate position given as a polynomial function of time to find the corresponding velocity and acceleration. Integrate the velocity or acceleration when given as a polynomial function of time along with appropriate initial conditions to find the functional form of the position. S.I.M. Structure of the Model Compatible Systems A single point particle (or a system treated as a point particle with position specified by the center of mass). Relevant Interactions Some time-varying external influence that is confined to one dimension. Laws of Change Mathematical Representation Differential Forms Integral Forms Diagrammatic Representations position versus time graph velocity versus time graph acceleration versus time graph motion diagram Click here to run a simulation demonstrating position, velocity and acceleration graphs for general 1-D motion Simulation provided by: PhET Interative Simulations University of Colorado http://phet.colorado.edu Relevant Examples All Examples Relevant to the Model Page: When 7000 hp Just Isn't Enough — Explore the kinematics of constant power (as opposed to constant force). non-conservative_work example_problem 1d_motion nonuniform_acceleration