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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).

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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).

Info
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:

Explain the difference between how physicists use the term acceleration versus the everyday use of the terms "accelerate" and "decelerate".
Describe the features of a motion diagram representing an object moving with constant acceleration.
Summarize the givens needed to solve a problem involving motion with constant acceleration.
Construct a consistent sign convention for the initial velocity, the final velocity and the acceleration in the case of objects that are speeding up or slowing down.
Describe the features of a position versus time graph representing an object moving with constant acceleration.
Given a position versus time graph, determine whether the object represented is speeding up or slowing down.
Given a linear velocity versus time graph, determine the corresponding acceleration.
State the equation that relates position, initial velocity, acceleration and time for motion with constant acceleration.
State the equation that relates position, initial velocity, final velocity and acceleration for motion with constant acceleration.
Solve a quadratic equation for time.
Mathematically solve for the meeting time and location of two objects moving with constant acceleration by setting up and solving a system of equations.
Graphically locate the meeting point of two objects moving with constant acceleration.
Describe the trajectory of a projectile.
Describe the acceleration of a projectile throughout its trajectory.
State the conditions on the velocity and acceleration that describe the maximum height of a projectile.

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.

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

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

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

Note
In the above expressions, t_i is the initial time, the time as which the position and velocity equal x_i and v_i respectively. Often t_iis taken to equal 0, in which case these expressions simplify.

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

Note
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

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id oned
Examples Involving Purely One-Dimensional Motion

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id	oned

falsetruetrueAND501d_motion,constant_acceleration,example_problem

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id freefall
Examples Involving Freefall

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id	freefall

falsetruetrueAND50freefall,example_problem

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id catchup
Examples Involving Determining when Two Objects Meet

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id	catchup

falsetruetrueAND50catch-up,constant_acceleration,example_problem

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id all
All Examples Using this Model

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id	all

falsetruetrueAND50constant_acceleration,example_problem

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Photos courtesy US Navy by:
Cmdr. Jane Campbell
Mass Communication Specialist 1st Class Emmitt J. Hawks

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