Long summer days are perfect to go hiking. Let’s say you decide to pay a visit to a friend who lives 3 km away. You both meet and decide to have an ice cream in a store nearby, just 500 m away. After having a good time, you walk back to your friend’s house, and then go back home.
You sure did some exercise that day! Nevertheless, according to physics, your total displacement was… well, zero. This is because the magnitude of your displacement is defined as the difference between your starting and ending points, which in this case were the same: your house. On the other hand, the distance you travelled was indeed 7 km.
Don’t worry. It might sound absurd, but this definition of displacement is quite useful when considering physical phenomena such as body motion, potential energies, etc. Let’s discover why!
How to calculate displacement
- Strategically establish a coordinate system, for example, with its origin on the starting point of the trajectory.
- Determine the coordinates of the starting and ending points of the trajectory: (xi, yi) and (xf, yf), respectively.
- Calculate the displacement along the x- and y-axes by subtracting each corresponding coordinate pair: dx=xf-xi and dy=yf-yi, respectively.
- Calculate the magnitude of the displacement using the Pythagorean theorem:
| D=dx2+dy2 | (1) |
- Express the direction of the displacement vector through the angle it forms with the x-axis using this equation:
| =tan-1(dy/dx) | (2) |
What is displacement
If you could attach a color smoke bomb to a bird’s tail and let it fly free, you would be able to observe the path it draws in the sky, like the following image shows. Physicists call the pathway followed by moving bodies their trajectory.
Trajectories can have infinite shapes, and can be 1-, 2- or 3-dimensional. An example of a 1-dimensional trajectory is that of an object in free fall. All objects on the Earth tend to fall to its center due to the force of gravity. This force acts in a straight line connecting the object’s center of mass and the center of gravity of the Earth. Since the movement occurs along a straight line, we can say it is 1-dimensional.
An example of a 2D trajectory is that of an ant walking around to find food on your kitchen floor. The ant’s position changes in two directions parallel to the floor’s surface. If we use a coordinate system, we can call these directions the x- and the y-axes. Every direction the ant moves can be described with an x- and a y-component. Since our kitchen ant can’t fly, its trajectory is bound to the ground, so it only changes in two possible directions and is therefore 2-dimensional.
Lastly, the bird from our previous example draws a 3-dimensional trajectory in the sky. In this case its position changes in all possible directions. If you think about it in terms of its location on the map, the bird could fly to the north, east, south or west, but also change its altitude. If we use a coordinate system, the x- and y-axes could be used to describe motion along the four cardinal directions. Nevertheless, we would need a third axis, namely the z-axis, to describe the bird’s flying altitude.
In physics, when an object moves in space, we say it covers a certain distance. This is the length of the trajectory followed by the object. To measure the distance our kitchen ant travels, we would have to divide its entire trajectory in very small sections, measure the length of each section, as the following image shows, and then add all of the results together. The more sections we divide the pathway in, the more accurate the calculated distance travelled by the ant will be.
Knowing the distance travelled by a body in motion is very useful to calculate, for example, the instantaneous velocity at any point of the trajectory. Nevertheless, sometimes we only want to know what is the last position of the body after following a certain trajectory. In these cases we consider its displacement, rather than the distance it travelled.
Displacement is defined as a vector, whose magnitude is the distance between the final and initial positions of the moving body along a straight line connecting both points. The resulting vector, usually denoted as s, simply points from the start to the end of the trajectory. In the case of our kitchen ant, the displacement vector would look like this:
How to calculate displacement
As mentioned in the previous section, the magnitude of the displacement vector is the distance between the starting and ending points in a straight line. To measure it, in our kitchen ant example, we can locate a coordinate system with its origin on the starting point, as the following image shows. This way, that point will have coordinates (0, 0). We can now measure the x- and y-components of the ant’s final position, namely xf and yf.
We now have the coordinates of the ant’s starting and ending points: (0, 0) and (xf, yf), respectively. If you examine the previous picture, and draw a straight line connecting the start and the end of the ant’s trajectory, you will notice two triangles are formed with the red lines. These triangles have a horizontal length of dx=xf-0, and a vertical height of dy=yf-0. They are also right triangles, which means one of its angles is 90°, for example, the one formed between the vertical red line and the x-axis. Finally, their hypotenuse is the magnitude of our displacement vector.
To calculate it, we may use equation 1, which is simply the definition of a right triangle’s hypotenuse or the Pythagorean theorem. This yields distance D=dx2+dy2. Now, remember that the difference between D and s is that the former is the magnitude of the displacement vector, and is therefore a scalar (it does not have a particular direction). To express the displacement vector’s direction, we can define angle in our coordinate system as the one formed between the displacement and the x-axis.
Again, considering the triangle formed by the vector and the coordinate system’s axes, we can calculate its tangent as: tan =dy/dx. Look at the next image for help. In order to solve for the angle , we need to take its inverse tangent or arctan, using equation 2. This way, we have calculated the displacement vector’s magnitude, D, and its direction, using angle .
Exercise 1: Your friend Lucy decides to go and have a cup of coffee in a store near her house. She goes out and walks around the park, as the following image shows.
Calculate the displacement vector (magnitude and direction) of Lucy’s motion around the town. Use the provided grid as help to calculate positions. All numbers are in meters. Keep in mind, Lucy’s trajectory does not start at the origin of the selected coordinate system, so you will have to calculate the horizontal and vertical distances before using equations 1 and 2.
- What is the magnitude of Lucy’s displacement?
- What is the angle the displacement vector forms with respect to a horizontal line?
- What distance did Lucy’s travel?
Answers:
- 15,23 m
- 23,2°
- 20 m
What is displacement useful for
In linear motion, the displacement vector is the root that allows us to formally calculate velocity as a vector. In this case, velocity is defined as the first time derivative of the displacement:
| v=dsdt | (3) |
Objects certainly do not move along straight lines, but in free trajectories. So, in order to apply this definition of velocity, the actual trajectory is divided into very small straight lines, called differentials. That is why we write ds in equation 3 instead of just s. Consequently, since the distance travelled by the body is divided into differentials and we want to calculate its velocity, we have to divide the time needed to cover said distances into very small periods of time, called time differentials. These are denoted as dt in the previous equation.
In the examination of more advanced physical phenomena, the total change in an object’s position, which is measured through its displacement, is more important than the actual trajectory it follows. This happens, for example, when measuring the magnetic potential energy of a magnet inside a magnetic field.
A magnet produces its own magnetic field around it. Therefore, when it is introduced in an external magnetic field, both interact and forces are generated. This is similar to you putting two magnets close to each other: each magnet’s field interacts with the other’s, and you will probably feel a very strong force acting on both of them. In very general terms, our magnet will tend to move inside an external magnetic field due to said forces.
This is somewhat similar to you skydiving from an airplane. Before you jump, your body will tend to fall to the ground, but it does do so thanks to the aircraft carrying you. Since you have the potential to fall to the ground, which you will eventually use once you jump, physicists describe this as your gravitational potential energy.
In the case of our magnet, we can also associate the potential of moving inside an external magnetic field to a magnetic potential energy. Now, if you want to calculate the potential energy difference from one point to another and see if it increases or decreases as you move the magnet around, you will need to do so considering its displacement. This is because the magnetic potential depends solely on the magnet’s current position —as your gravitational potential depends only on your current altitude—, and not the trajectory it followed to get there in the first place.
A similar situation arises when calculating a charged particle’s electric potential difference when moved inside an electric field. The particle’s potential difference is a function of its displacement, and not of the distance it travels.
Other helpful sources
If you want to learn more about the difference between distance and displacement, go ahead and check out this cK-12 simulator, where you can explore these concepts further.