Analyzing motion can get complicated. Learning precise vocabulary will help.
What does position mean?
In physics, we love to precisely describe the motion of an object. Seriously, the first few chapters of basically every physics textbook are devoted to teaching people how to precisely describe motion since it is so important to everything else we do in physics.
But to describe an object's motion, we have to first be able to describe its position—where it is at any particular time. More precisely, we need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a professor’s position could be described in terms of where she is in relation to the nearby white board (Figure 1). In other cases, we use reference frames that are not stationary but rather 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 (Figure 2).
The variable
What does displacement mean?
If an object moves relative to a reference frame—for example, if a professor moves to the right relative to a whiteboard, or a passenger moves toward the rear of an airplane—then the object’s position changes. This change in position is known as displacement. The word displacement implies that an object has moved, or has been displaced.
Displacement is defined to be the change in position of an object. It can be defined mathematically with the following equation:
Displacement is a vector. This means it has a direction as well as a magnitude and is represented visually as an arrow that points from the initial position to the final position. For example, consider the professor that walks relative to the whiteboard in Figure 1.
Figure 1: A professor paces left and right while lecturing. The
The professor’s initial position is
Now consider the passenger that walks relative to the plane in Figure 2.
Figure 2: A passenger moves from his seat to the back of the plane. The
The airplane passenger’s initial position is
In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive—usually that will be to the right or up, but you are free to select positive as being any direction.
What do distance and distance traveled mean?
We must be careful when using the word distance since there are two ways in which the term distance is used in physics. We can talk about the distance between two points, or we can talk about the distance traveled by an object.
Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them.
Distance traveled is the total length of the path traveled between two positions. Distance traveled is not a vector. It has no direction and, thus, no negative sign. For example, the distance the professor walks is
It is important to note that the distance traveled does not have to equal the magnitude of the displacement (i.e., distance between the two points). Specifically, if an object changes direction in its journey, the total distance traveled will be greater than the magnitude of the displacement between those two points. See the solved examples below.
What's confusing about displacement?
People often forget that the distance traveled can be greater than the magnitude of the displacement. By magnitude, we mean the size of the displacement without regard to its direction (i.e., just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 meters during a lecture, yet still end up only two meters to the right of her starting point. In this case her displacement would be
People often forget to include a negative sign, if needed, in their answer for displacement. This sometimes occurs if they accidentally subtract the final position from the initial position rather than subtracting the initial position from the final position.
What do solved examples involving displacement look like?
Example 1: Displacement of four moving objects
Four objects move according to the paths shown in the diagram below. Assume the units of the horizontal scale are given in meters. (Image credit: altered from Openstax College Physics)
What was the displacement of each object?
Object A had an initial position of
Object B had an initial position of
Object C had an initial position of
Object D had an initial position of
Example 2: Distance traveled of four moving objects
Four objects move according to the paths shown in the diagram below. Assume the units of the horizontal scale are given in meters. (Image credit: altered from Openstax College Physics)
What was the total distance traveled by each object?
Object A travels a total distance of
Object B travels a total distance of
Object C travels a total distance of
Object D travels a total distance of