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One-Dimensional Motion with Constant Acceleration

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 (universal) 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).

Problem Cues

The problem will often explicitly state that the acceleration is constant, or else will indicate this by giving some quantitative information that implies constant acceleration (e.g. a linear plot of velocity versus time). The model is also sometimes useful (in conjunction with Point Particle Dynamics) in dynamics problems when it is clear that the net force is constant in magnitude - in fact if one axis lies along the net force, the perpendicular axes will have no acceleration and hence will exhibit motion with constant velocity.

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.

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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In these 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.

Relations between velocity, position, and acceleration when acceleration is constant

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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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

Position as a Function of Time
From the formulas given in the Laws of Change, it is clear that a plot of position vs. time will give a parabola.

If the acceleration is positive the parabola will open upwards. The position at t = t_i will be x_i , as shown in the graph below (time at the origin is t_i ):

In this case the position is positive. The fact that the plot of position vs. time is increased means that the initial velocity, v_i , is also positive.

If the acceleration and the initial position x_i were the same, but the initial velocity was negative , then the graph of position vs. time would look like this:

The parabola has a minimum value at the time t_min

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This information is intended to familiarize the reader with the shape of the curve and how it behaves. Obviously, if the object starts out at time t = t_i its real motion will not be described by the portion of the curve for t < t_i, and so an object moving with positive initial velocity and positive acceleration will not have such a "minimum" position - it will move in the same direction, with increasing speed, for all t > t_i .

A plot of velocity vs. time for this case looks like this. The position of the intercept with the time axis is t_min :

On the other hand, an object with negative initial velocity v_i and positive acceleration will encounter a "minimum" position, at which it will have zero velocity. After slowing to zero it will reverse direction and gather speed with increasing time. (Note that the slope of the graph is the same as in the above case, which is what you expect, since it is determined only by the value of the acceleration.) The intercept of zero velocity occurs, again, at t_min = t_i- v_i/a. Since v_i < 0, the intercept is to the right of t_i.

The plots for the case of negative acceleration (a < 0) are similar, only the parabola opens downward in that case.

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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Child pages

1-D Motion (Constant Acceleration)

One-Dimensional Motion with Constant Acceleration

Description and Assumptions

Problem Cues

Learning Objectives

Model

Compatible Systems"> Compatible Systems

Relevant Interactions"> Relevant Interactions

Laws of Change"> Laws of Change

Relations between velocity, position, and acceleration when acceleration is constant

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