# Module 4 -- Graphing Motion and Acceleration vs. Deceleration

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#### Learning Goals

After working through this module, you should be able to:

• Recognize or construct a velocity versus time graph illustrating 1-D motion with constant acceleration.
• Recognize or construct a position versus time graph illustrating 1-D motion with constant acceleration.
• Given a velocity versus time graph illustrating 1-D motion with constant acceleration, determine the acceleration.
• Given a position versus time graph illustrating 1-D motion with constant acceleration, determine the sign of the acceleration.
• Define "deceleration".
• Describe the conditions on velocity and acceleration that give rise to deceleration.
• Given a position versus time graph illustrating 1-D motion with constant acceleration, find any time intervals over which the object is decelerating.

## Graphical Representation of Acceleration

One way to represent a system described by the One-Dimensional Motion with Constant Acceleration Model graphically is to draw a velocity versus time graph for that system. According to the definition

$a_{x} = \frac{dv_{x}}{dt}$

it is clear that the acceleration is equal to the slope of the velocity versus time graph. Thus, if the acceleration is constant, the velocity versus time graph will necessarily be linear (the only type of graph with a constant slope).

Another way to graphically represent the Model is to note that the equation

$x(t) = x_{i} + v_{i,x}(t-t_{i}) + \frac{1}{2} a_{x}(t-t_{i})^{2}$

implies that a system moving with constant acceleration will be described by a parabolic position versus time graph (the position is a quadratic function of the time).

Simulation courtesy PhET Interactive Simulations
at the University of Colorado
http://phet.colorado.edu

## Position vs. Time Graphs and Acceleration

The concavity (or equivalently, the second derivative) of a position versus time graph can be used to determine the sign of the acceleration. A concave up position versus time graph has positive acceleration. The reason can be seen by considering the case of a system with constant positive acceleration. The position versus time graph for such a system will be an upward-opening parabola like that shown below.

The vertex of this parabola is a point where the slope of the graph goes to zero. A point of zero slope in a position vs. time graph implies that the velocity goes to zero at that time. Thus, the system is momentarily at rest at the time corresponding to the vertex of the parabola. Everywhere to the right of the vertex in the graph, the slope of the parabola is positive and increasing. Thus, the velocity is increasing in the positive direction, implying positive acceleration. Everywhere to the left of the vertex, the velocity is negative and approaching zero (becoming smaller in magnitude). This lessening of a negative velocity also corresponds to positive acceleration.

The case of a concave down position versus time graph is analogous. The position versus time for a system experiencing constant negative acceleration is shown below.

Again, the vertex is a point with zero velocity. This time, however, points to the right of the vertex have negative slope that is growing steeper as time goes on, and points to the left of the vertex have positive slope that is lessening. Each of these cases correspond to negative acceleration.

## Acceleration vs. Deceleration

It is important to discuss one problem with the specialized vocabulary of physics. So far, we have introduced three different aspects of motion. Each one can be discussed in terms of a vector concept (magnitude and direction) or in terms of a scalar concept (magnitude only). For instance, we discussed displacement, a vector, and distance, a scalar. For motion in one direction, distance is the magnitude of displacement. We discussed velocity, a vector, and speed, a scalar. If we are considering instantaneous velocity, then speed is the magnitude of velocity. Our last quantity, acceleration, can also be discussed in terms of a vector acceleration or simply the magnitude, but for acceleration we have no special term for the magnitude. The vector is called "the acceleration" and the magnitude is "the magnitude of the acceleration". This can result in confusion.

This problem is exacerbated by the fact that in everyday language, we often use the terms distance, speed and acceleration. The everyday definitions of distance and speed are basically equivalent to their physics definitions, since we rarely consider direction of travel in everyday speech and these quantities are scalars in physics (no direction). Unfortunately, in physics, we usually use the term "acceleration" to refer to a vector, while in everyday speech it denotes a magnitude.

The difficulties do not end there. Everyday usage does make one concession to the vector nature of motion. When we talk about acceleration in everyday speech, we usually specify whether the object is "accelerating" (speeding up) or "decelerating" (slowing down). Both terms imply a change in velocity, and so in physics we can call either case "accelerating". In physics, the difference between accelerating and decelerating is determined by the relative directions of the velocity and the acceleration.

Everyday Term Physics Equivalent
acceleration acceleration and velocity point in the same direction
deceleration acceleration and velocity point in opposite directions

The difference between acceleration and deceleration can also be illustrated graphically.

## Check Your Understanding

By looking at the position versus time graph shown above, determine the following at each of the eight numbered instants of time.

A.) Is the system's position positive or negative?

B.) Is the system's velocity positive or negative?

C.) Is the system's acceleration positive or negative?

D.) Is the object speeding up ("accelerating") or slowing down ("decelerating")?