1.6.1 Determining the Signs of Position, Velocity, and Acceleration

Position, velocity, and acceleration are measured with respect to a coordinate system, a grid or axis that you impose on a problem to analyze the motion. We will find it convenient to use an x-axis to describe both horizontal motion and motion along an inclined plane. A y-axis will be used for vertical motion. A coordinate axis has two essential features:

An origin to define zero; and
An x or y label (with units) at the positive end of the axis.

NOTE In this textbook, we will follow the convention that the positive end of an x-axis is to the right and the positive end of a y-axis is up. The signs of position, velocity, and acceleration are based on this convention.

TACTICS BOX 1.3 Determining the sign of the position, velocity, and acceleration

The figure illustrates the sign of the position, velocity, and acceleration of an object.

Full Alternative Text

The sign of position $(x or y)$ tells us where an object is.
The sign of velocity $(v_{x} or v_{y})$ tells us which direction the object is moving.
The sign of acceleration $(a_{x} or a_{y})$ tells us which way the acceleration vector points, not whether the object is speeding up or slowing down.

The figure illustrates the sign of the position of an object. The y axis with the marking 0 at its center is shown. A dot is above zero. y greater than 0. — Position above origin.

The figure illustrates the sign of the position of an object. The y axis with the marking 0 at its center is shown. A dot is below zero. y less than 0. — Position above origin.

Workbook Exercises 30–31

Acceleration is where things get a bit tricky. A natural tendency is to think that a positive value of $a_{x}$ or $a_{y}$ describes an object that is speeding up while a negative value describes an object that is slowing down (decelerating). However, this interpretation does not work.

Acceleration is defined as $\vec{a}_{avg} = Δ \vec{v}^{} / Δ t$ . The direction of $\vec{a}$ can be determined by using a motion diagram to find the direction of $Δ \vec{v}$ . The one-dimensional acceleration $a_{x}$ (or $a_{y})$ is then positive if the vector $\vec{a}$ points to the right (or up), negative if $\vec{a}$ points to the left (or down).

FIGURE 1.16 shows that this method for determining the sign of a does not conform to the simple idea of speeding up and slowing down. The object in Figure 1.16a has a positive acceleration $(a_{x} > 0)$ not because it is speeding up but because the vector $\vec{a}$ points in the positive direction. Compare this with the motion diagram of Figure 1.16b. Here the object is slowing down, but it still has a positive acceleration $(a_{x} > 0)$ because $\vec{a}$ points to the right.

The figure consists of two parts labeled ("a") and (b) that illustrate the motion diagram of a car speeding toward the right and a car slowing down toward the left.

FIGURE 1.16 Full Alternative Text

In the previous section, we found that an object is speeding up if $\vec{v}$ and $\vec{a}$ point in the same direction, slowing down if they point in opposite directions. For one-dimensional motion this rule becomes:

An object is speeding up if and only if $v_{x}$ and $a_{x}$ have the same sign.
An object is slowing down if and only if $v_{x}$ and $a_{x}$ have opposite signs.
An object’s velocity is constant if and only if $a_{x} = 0$ .

Notice how the first two of these rules are at work in Figure 1.16.