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# What is displacement?

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 x is often used to represent the horizontal position. The variable y is often used to represent the vertical position.

## 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:
start text, D, i, s, p, l, a, c, e, m, e, n, t, end text, equals, delta, x, equals, x, start subscript, f, end subscript, minus, x, start subscript, 0, end subscript
x, start subscript, f, end subscript refers to the value of the final position.
x, start subscript, 0, end subscript refers to the value of the initial position.
delta, x is the symbol used to represent displacement.
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 plus, 2, point, 0, start text, space, m, end text displacement of the professor relative to the whiteboard is represented by an arrow pointing to the right. (Image credit: Openstax College Physics)
The professor’s initial position is x, start subscript, 0, end subscript, equals, 1, point, 5, start text, space, m, end text and her final position is x, start subscript, f, end subscript, equals, 3, point, 5, start text, space, m, end text. Thus, her displacement can be found as follows, delta, x, equals, x, start subscript, f, end subscript, −, x, start subscript, 0, end subscript, equals, 3, point, 5, start text, space, m, end text, −, 1, point, 5, start text, space, m, end text, equals, plus, 2, point, 0, start text, space, m, end text. In this coordinate system, motion to the right is positive, whereas motion to the left is negative.
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 −, 4, point, 0, start text, space, m, end text displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. (Image credit: Openstax College Physics)
The airplane passenger’s initial position is x, start subscript, 0, end subscript, equals, 6, point, 0, start text, space, m, end text and his final position is x, start subscript, f, end subscript, equals, 2, point, 0, start text, space, m, end text, so his displacement can be found as follows, delta, x, equals, x, start subscript, f, end subscript, −, x, start subscript, 0, end subscript, equals, 2, point, 0, start text, space, m, end text, −, 6, point, 0, start text, space, m, end text, equals, minus, 4, point, 0, start text, space, m, end text. His displacement is negative because his motion is toward the rear of the plane, or in the negative x direction in our coordinate system.
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 2, point, 0, start text, space, m, end text. The distance the airplane passenger walks is 4, point, 0, start text, space, m, end text.
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.

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 plus, 2, start text, space, m, end text, the magnitude of her displacement would be 2, start text, space, m, end text, but the distance she traveled would be 150, start text, space, m, end text. In kinematics we nearly always deal with displacement and magnitude of displacement and almost never with distance traveled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The displacement is simply the difference in the position of the two marks and is independent of the path taken when traveling between the two marks. The distance traveled, however, is the total length of the path taken between the two marks.
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 0, start text, space, m, end text and a final position of 7, start text, space, m, end text. The displacement of object A can be shown with this equation:
delta, x, start subscript, A, end subscript, equals, 7, start text, space, m, end text, minus, 0, start text, space, m, end text, equals, plus, 7, start text, space, m, end text
Object B had an initial position of 12, start text, space, m, end text and a final position of 7, start text, space, m, end text. The displacement of object B can be shown with this equation:
delta, x, start subscript, B, end subscript, equals, 7, start text, space, m, end text, minus, 12, start text, space, m, end text, equals, minus, 5, start text, space, m, end text
Object C had an initial position of 2, start text, space, m, end text and a final position of 10, start text, space, m, end text. The displacement of object C can be shown with this equation:
delta, x, start subscript, C, end subscript, equals, 10, start text, space, m, end text, minus, 2, start text, space, m, end text, equals, plus, 8, start text, space, m, end text
Object D had an initial position of 9, start text, space, m, end text and a final position of 5, start text, space, m, end text. The displacement of object D can be shown with this equation:
delta, x, start subscript, D, end subscript, equals, 5, start text, space, m, end text, minus, 9, start text, space, m, end text, equals, minus, 4, start text, space, m, end text

### 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 7, start text, space, m, end text.
Object B travels a total distance of 5, start text, space, m, end text.
Object C travels a total distance of 8, start text, space, m, end text, plus, 2, start text, space, m, end text, plus, 2, start text, space, m, end text, equals, 12, start text, space, m, end text.
Object D travels a total distance of 6, start text, space, m, end text, plus, 2, start text, space, m, end text, equals, 8, start text, space, m, end text.

## Want to join the conversation?

• I didn't understand why they picked the values c = 8m + 2m +2m= 12m, at least not by reading the diagram. What are these numbers?
• Its because the object first travels 8 m straight and take a 180 degree turn. After taking the turn it travels 2 m and then again it takes a 180 degree turn and travels more 2 m.
Thus making the distance 8 m+2 m+2 m=12 m.
• Regarding the person who is walking to the back of the plane, does it matter that the plane is actually moving forward when we are considering his displacement?
• What is the difference between "distance traveled" and "Displacement"??
• Displacement has to be the shortest path between the two points.
If you go around in a circle back to where you started, distance is the circumference of the circle. Displacement is zero.
• what is the difference between displacement and magnitude of displacement? Thank you!
• Bare minimum: Displacement gives where it was and where it is, but magnitude of displacement gives where it was and infinitely many places where it could be.

Slightly more detail: Displacement will give you both the current and previous locations of the object in question, but magnitude of displacement will give you the previous location of the object, and a radius from the original location. This radius can be used to describe a circle to represent all of the possible current locations of the object in question.
• What is the difference in saying "distance" and "distance travelled"? Isn't "distance" a scalar quantity?
• Distance can refer to the magnitude of displacement, or it can refer to the length of the path taken from one point to another. To clarify, people sometimes use the term distance traveled when they mean path length.
• What is the difference between circular motion and projectile motion?
• In most cases with projectile motion you have a constant acceleration in the negative Y direction where as in circular motion the acceleration is a constant magnitude but is constantly changing direction. If you deal with a more generic case of projectile motion where the distance traveled along the earth's surface is far enough so that you have to start to factor in the curvature of the earth the projectile motion get closer and closer to being circular motion.
• Hi, I did not exactly understand the difference between distance and distance traveled. Thanks!
• Conciser walking around a 1 km race track, if you make 1 full lap around the track you will have traveled 1 km but you are at a distance of 0 from where you started.
• i am a little bit confused about the difference of distance and displacement.