5 Constant Acceleration

Stable Interactions Required    

According to Newton's Laws of Motion, a system experiencing constant acceleration is necessarily experiencing a constant net interaction. Thus, this model can only be applied when the interactions are fairly steady. It is important to remember that a constant acceleration must have a constant direction as well as a constant size. Thus, if the forces on a system are clearly changing direction during the motion being analyzed, it is not reasonable to apply the One-Dimensional Motion with Constant Acceleration model.

An Abundance of Laws of Change    

Integrating the definitions of acceleration and velocity for the special case that acceleration is constant leads to four expressions that are commonly encountered in descriptions of motion with constant acceleration:

Four Useful Equations

It is clear from these equations that there are seven possible unknowns in a given problem involving motion between two points with constant acceleration:

1. initial time ( t_i )
2. final time ( t_f )
3. initial position ( x_i )
4. final position ( x_f )
5. initial velocity ( v_i )
6. final velocity ( v_f )
7. acceleration (constant) a

Looking at the four equations, you can see that each is specialized to deal with problems involving specific combinations of these unknowns.

An Exercise in Derivation

Because the initial position and initial time can generally be arbitrarily chosen, it is often useful to rewrite all these equations in terms of only five variables by defining:

If you replace the initial and final positions and times with these "deltas", then each of the equations given above involves exactly four unknowns. Interestingly, the four equations represent all but one of the unique combinations of four variables chosen from five possible unknowns. Which unique combination is missing? Can you derive the appropriate "fifth equation"?

Graphical Representations    

One way to graphically represent constant acceleration is to draw a velocity versus time graph. By the definition:

it is clear that the acceleration is equal to the slope of such a graph, so that if the acceleration is constant, the velocity versus time graph will necessarily be linear.

Another way to graphically represent acceleration is to note that the equation:

implies that for the case of constant acceleration, the position versus time graph describing the motion will be a parabola.

Freefall    

The importance of the One-Dimensional Motion with Constant Acceleration model in the mechanics curriculum owes much to the fact that one of the most prevalent interactions in our environment, gravity gives rise to a constant force on every object near the earth's surface. Thus, any object moving purely under the influence of gravity near the earth's surface will experience a constant acceleration. This type of motion is common enough in everyday experience that it receives the special name freefall.

Freefall: An object that is subject only to the force of gravity is in freefall.

More surprisingly, the acceleration is also almost exactly the same on every object in freefall near the earth! The acceleration of gravity on any object in freefall near the earth's surface is well approximated by the value of 9.8 m/s² directed toward the ground. The magnitude of the gravitational acceleration is so widely used in mechanics that we assign it a special constant, g.

Decomposition of Complicated Motions    

A wide variety of realistic one-dimensional motions can be closely approximated by decomposing the motion into segments of constant acceleration. By matching the initial conditions of each segment (time, position and velocity) to the final conditions of the previous segment, the entire motion can be described using only the simple Law of Change for the One-Dimensional Motion with Constant Acceleration model.

Decomposition of "Meeting" Problems into Multiple Systems    

One common type of problem that often involves the One-Dimensional Motion with Constant Acceleration model has the goal of determining where two objects meet. Because this model can only describe the motion of a single particle, solving for the point at which two objects meet requires that the problem be approached by considering the two objects to be separate systems governed by separate models (or separate realizations of the same model). Finding the meeting point requires that appropriate constraints be applied to each system in order to ensure that their relative positions be accurate at the initial time and that they be equal at the meeting time.