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). 










Introduction to the Model



Description and Assumptions



This model is applicable to a single point particle moving in one dimension either because it is physically constrained to move that way or because only one Cartesian component is considered.  The force, or component of force along this direction, must be constant in time. The force can be in the same direction of motion or in the opposite direction of motion.  Equivalently, the model applies to objects moving in one-dimension which have a position versus time graph that is parabolic and a velocity versus time graph that is linear.  It is a subclass of the One-Dimensional Motion (General) model defined by the constraint da/dt = 0 (i.e. a(t)=constant).




Multi-dimensional motion can often be broken into components, as in the case of projectile motion.  In this manner, the 1-D motion with constant acceleration model can be employed to describe the system's motion in any situation where the net force on the system is constant, even if the motion is multi-dimensional. 



Learning Objectives



Students will be assumed to understand this model who can:







S.I.M. Structure of the Model




Compatible Systems



A single point particle, or a system such as a single rigid body or a grouping of many bodies that is treated as a point particle with position specified by the system's center of mass. 



Relevant Interactions



Some constant net external force must be present to cause motion with a constant acceleration.



Laws of Change



Mathematical Representations



This model has several mathematical realizations that involve different combinations of the variables for position, velocity, and acceleration.



\begin{large}\[v(t) =v_{i}+ a (t - t_{i})\]\end{large}





\begin{large}\[x(t) = x_{i}+\frac{1}{2}(v_{f}+v_{i})(t - t_{i})\]\end{large}





\begin{large}\[ x(t) = x_{i}+v_{i}(t-t_{i})+ \frac{1}{2}a(t-t_{i})^{2}\]\end{large}


In the above expressions, ti is the initial time, the time as which the position and velocity equal xi and vi respectively. Often tiis taken to equal 0, in which case these expressions simplify.


\begin{large}\[v^{2}(x)= v_{i}^{2}+ 2 a (x - x_{i})\]\end{large}


This is an important expression, because time is eliminated.



Diagrammatic Representations












Click here for a Mathematica Player application illustrating these representations.














Click here to download the (free) Mathematica Player from Wolfram Research








Relevant Examples




Examples Involving Purely One-Dimensional Motion








Examples Involving Freefall








Examples Involving Determining when Two Objects Meet








All Examples Using this Model












 

Photos courtesy US Navy by:

Cmdr. Jane Campbell

Mass Communication Specialist 1st Class Emmitt J. Hawks



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