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One-Dimensional Motion with Constant Acceleration

DescriptionandAssumptions"> Description and Assumptions

This model is applicable to a single point particle moving in one dimension either because it's 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 positive (e.g. a rocket) or negative (e.g. gravity).   Note: Multi-dimensional motion can often be broken into components, as for the case of projectile motion. where there constant acceleration along one axis. The constnt acceleration model can be used describe the system's motion in any situation where the net force on the system is constant (a point particle subject only to near-earth [gravitation] is a common example). It is a subclass of the One-Dimensional Motion (General) model defined by the constraint da/dt = 0 (i.e. a(t)=constant).

ProblemCues"> Problem Cues

PriorModels"> Prior Models

Vocabulary"> Vocabulary

Model

Compatible Systems"> Compatible Systems

A single point particle, or a system such as a rigid body or many bodies that is treated as a point particle with position specified by the center of mass. (The c of m involves the MOMENTUM MODEL.)

Relevant Interactions"> Relevant Interactions

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

Laws of Change"> Laws of Change

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

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\begin

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$v(t) =  v_

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)$\end


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\begin

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$x(t) = x_

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+\frac

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(v_

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+v_

)(t - t_

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)$\end



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\begin

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$ x(t) = x_

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(t-t_

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)+ \frac

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a(t-t_

)^

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$\end

In these expressions, ti~ is the initial time, the time as which the position and velocity equal xi~ and v{~}i~ respectively.

Here's an expression that relates the velocity at initial and final times - it follows algebraically from the two expressions above. 

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\begin

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$v^

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= v_

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^

+ 2 a (x - x_

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)$\end

This is an important expression, because the velocity can be regarded as a function of initial and final position, hence time is eliminated from the expression.  This realization is the gateway to deriving the relationship between [work] and [kinetic energy].

Diagrammatic Representations"> Diagrammatic Representations

Relevant Examples

ExamplesInvolvingPurelyOne-DimensionalMotion"> Examples Involving Purely One-Dimensional Motion

ExamplesInvolvingFreefall"> Examples Involving Freefall

ExamplesInvolvingDeterminingwhenTwoObjectsMeet"> Examples Involving Determining when Two Objects Meet

AllExamplesUsingthisModel"> All Examples Using this Model

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



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