3.1 Position, Displacement, and Average Velocity

Position

To describe the motion of an object, you must first be able to describe its position (x): where it is at any particular time. More precisely, we need to specify its position relative to a convenient frame of reference. A frame of reference is an arbitrary set of axes from which the position and motion of an object are described. Earth is often used as a frame of reference, and we often describe the position of an object as it relates to stationary objects on Earth. For example, a rocket launch could be described in terms of the position of the rocket with respect to Earth as a whole, whereas a cyclist’s position could be described in terms of where she is in relation to the buildings she passes Figure 3.2. In other cases, we use reference frames that are not stationary but are in motion relative to Earth. To describe the position of a person in an airplane, for example, we use the airplane, not Earth, as the reference frame. To describe the position of an object undergoing one-dimensional motion, we often use the variable x. Later in the chapter, during the discussion of free fall, we use the variable y.

Displacement

If an object moves relative to a frame of reference—for example, if a professor moves to the right relative to a whiteboard Figure 3.3—then the object’s position changes. This change in position is called displacement. The word displacement implies that an object has moved, or has been displaced. Although position is the numerical value of x along a straight line where an object might be located, displacement gives the change in position along this line. Since displacement indicates direction, it is a vector and can be either positive or negative, depending on the choice of positive direction. Also, an analysis of motion can have many displacements embedded in it. If right is positive and an object moves 2 m to the right, then 4 m to the left, the individual displacements are 2 m and $\mathrm{-4}$ m, respectively.

We use the uppercase Greek letter delta (Δ) to mean “change in” whatever quantity follows it; thus, $\text{Δ}x$ means change in position (final position less initial position). We always solve for displacement by subtracting initial position $x_0$ from final position $x_{\text{f}}$. Note that the SI unit for displacement is the meter, but sometimes we use kilometers or other units of length. Keep in mind that when units other than meters are used in a problem, you may need to convert them to meters to complete the calculation (see Appendix B).

Objects in motion can also have a series of displacements. In the previous example of the pacing professor, the individual displacements are 2 m and $\mathrm{-4}$ m, giving a total displacement of −2 m. We define total displacement $\text{Δ}{x}_{\text{Total}}$, as the sum of the individual displacements, and express this mathematically with the equation

$$\text{Δ}{x}_{\text{Total}}={\displaystyle \sum \text{Δ}{x}_{\text{i}}},$$

where $\text{Δ}{x}_i$ are the individual displacements. In the earlier example,

$$\text{Δ}{x}_1=x_1-x_0=2-0=2\text{m.}$$

Similarly,

$$\text{Δ}{x}_2=x_2-x_1=\mathrm{-2}-(2)=\mathrm{-4}\text{m.}$$

Thus,

$$\text{Δ}{x}_{\text{Total}}=\text{Δ}{x}_1+\text{Δ}{x}_2=2-4=\mathrm{-2}\text{m}\text{.}$$

The total displacement is 2 − 4 = −2 m along the x-axis. It is also useful to calculate the magnitude of the displacement, or its size. The magnitude of the displacement is always positive. This is the absolute value of the displacement, because displacement is a vector and cannot have a negative value of magnitude. In our example, the magnitude of the total displacement is 2 m, whereas the magnitudes of the individual displacements are 2 m and 4 m.

The magnitude of the total displacement should not be confused with the distance traveled. Distance traveled $x_{\text{Total}}$, is the total length of the path traveled between two positions. In the previous problem, the distance traveled is the sum of the magnitudes of the individual displacements:

$$x_{\text{Total}}=\left|\text{Δ}{x}_1\right|+\left|\text{Δ}{x}_2\right|=2+4=6\text{m}\text{.}$$

Average Velocity

To calculate the other physical quantities in kinematics we must introduce the time variable. The time variable allows us not only to state where the object is (its position) during its motion, but also how fast it is moving. How fast an object is moving is given by the rate at which the position changes with time.

For each position $x_{\text{i}}$, we assign a particular time $t_{\text{i}}$. If the details of the motion at each instant are not important, the rate is usually expressed as the average velocity $\stackrel{\text{–}}{v}$. This vector quantity is simply the total displacement between two points divided by the time taken to travel between them. The time taken to travel between two points is called the elapsed time $\text{Δ}t$.

It is important to note that the average velocity is a vector and can be negative, depending on positions $x_1$ and $x_2$.

Summary

• Kinematics is the description of motion without considering its causes. In this chapter, it is limited to motion along a straight line, called one-dimensional motion.
• Displacement is the change in position of an object. The SI unit for displacement is the meter. Displacement has direction as well as magnitude.
• Distance traveled is the total length of the path traveled between two positions.
• Time is measured in terms of change. The time between two position points $x_1$ and $x_2$ is $\text{Δ}t=t_2-t_1$. Elapsed time for an event is $\text{Δ}t=t_{\text{f}}-t_0$, where $t_{\text{f}}$ is the final time and $t_0$ is the initial time. The initial time is often taken to be zero.
• Average velocity $\stackrel{\text{–}}{v}$ is defined as displacement divided by elapsed time. If $x_1,t_1$ and $x_2,t_2$ are two position time points, the average velocity between these points is

  $$\stackrel{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t}=\frac{x_2-x_1}{t_2-t_1}.$$

Conceptual Questions

Problems