Lesson Explainer: Displacement Science

In this explainer, we will learn how to distinguish the distance that an object moves between two points from the displacement of the object between those points.

We can begin by recalling that the distance an object travels is the length of the path traveled between the start point and endpoint of the motion of an object’s journey.

For example, let’s say that we have an object. Let’s label the starting point of the object A and label the endpoint of the object B. These two points are separated by a horizontal distance of 2 m. Let’s think about two different paths that the object could take from A to B, as shown in the following diagram.

The first path we will consider is shown in the diagram as a dashed red line. This path is simply the horizontal straight line from point A to point B. The length of path 1 is therefore the same as the distance between the start point and the endpoint, which is 2 m. This means that the length of path 1 is 2 m.

If an object travels from point A to point B on path 1, the distance that the object will travel is the length of path 1. Therefore, the object will have traveled a distance of 2 m.

Let’s now think about path 2. This is shown in the diagram above as a solid blue line from point A to point B.

Path 2 is a more complicated path that goes up, then across, and then back down. We can see that this path is much longer than path 1. This is because path 2 travels the same horizontal length as path 1, which is 2 m, but it also goes up and down. Since path 2 goes up 1 m before it travels horizontally 2 m and then travels down 2 m at the end, we need to add up these values to get the length of path 2.

So the length of path 2 is 1+2+1=4.mmmm

If an object travels from point A to point B on path 2, the distance that the object will travel is the length of path 2. This means that the object will travel a distance of 4 m.

We have therefore seen that an object traveling on path 1 will travel a shorter distance than an object traveling on path 2. Even though the start point and endpoint of the two paths are the same, the distances traveled on the two paths are different.

Distance is a “scalar” quantity, which means that it is just a number with units and does not have a direction associated with it.

Definition: Distance

The distance that an object travels is the length of the path that the object takes from the start point of its motion to the endpoint of its motion.

Example 1: Comparing Distances for Different Paths

A car is at the center of a circle. The arrows show paths that the car could travel to reach the circumference of the circle. Is the distance moved the same for both paths?

Answer

In this question, we are shown a car that is in the center of a circle. We are also shown two different paths that the car could travel to reach the circumference of the circle.

Let’s first consider the blue arrow, which denotes a straight path from the center of the circle to the circumference of the circle. This means that the length of the blue path must be equal to the radius of the circle.

Let’s assume that the radius of the circle is 10 m. This means that the length of the blue path is 10 m. If the car travels on the blue path, it will therefore travel a distance of 10 m.

Let’s now consider the red arrow. This path is also a straight line from the center of the circle to the circumference of the circle.

This means that the length of the red path is equal to the radius of the circle, which we have said is 10 m. If the car travels on the red path, it will travel a distance of 10 m.

This means that the distance the car would travel on the blue path and the red path is the same. This distance is equal to the radius of the circle, which is 10 m, for both paths.

We now want to think about the “displacement” of an object. This is a very similar word to “distance,” but it has a subtly different definition.

The magnitude of a displacement is the shortest distance between the start point and endpoint of the motion of an object. Displacement has a direction. The direction of a displacement is from the start point of the motion of an object to the endpoint of the motion of the object. This is because displacement is a “vector” quantity, which means that it is made up of a number and a direction. Examples of displacements are “2 m east” and “5 km south.”

The shortest distance between two points is the straight line between the two points. This means that the displacement between two points is always the straight-line distance between the two points plus the direction traveled.

The direction must be included when we give a displacement, because displacement is a vector quantity, and the direction is part of the displacement. For example, the displacement “2 m east” is different from the displacement “2 m west,” even though the number in the displacement is the same.

We say that the number part of a displacement is the “magnitude” of the displacement. For example, for the displacement “2 m east,” we say that the magnitude of the displacement is “2 m” and the direction of the displacement is “east.”

Definition: Displacement

The displacement of an object is a vector quantity. It has magnitude equal to the shortest distance between the start point and endpoint of the motion of an object, and the direction of the motion is specified too.

Example 2: Understanding the Definition of Displacement

Explain why 25 metres is not a displacement.

Answer

Displacement is a vector quantity, so it is made up of a magnitude and a direction.

Since 25 metres is just a magnitude and we are not given a direction, the 25 metres we are given is a scalar quantity instead of a vector quantity.

This means that 25 metres is not a displacement, because it is not stated what direction the 25 metres is in.

Instead, 25 metres is a distance, because this is a scalar quantity, and so a direction does not need to be stated.

Let’s think about the difference between distance and displacement. Let’s again consider two points, with A being the starting location of an object and B being the end location of the object. Point B is positioned 2 m to the right of point A. We will again consider two different paths between the two points, as shown again in the following diagram.

Let’s think about the distances along each of the paths. We have seen that the distance an objects travels if it follows path 1 is 2 m, and the distance an object travels if it follows path 2 is 4 m. These distance are different because path 2 has a longer length than path 1.

However, let’s see what happens if we consider displacement instead of distance. Recall that displacement is the shortest distance between the start point and the endpoint of the path.

For path 1, we have a start point of A and an endpoint of B. The magnitude of the displacement is then the shortest distance between these two points, which is the horizontal straight line between these points.

This is equal to 2 m, which is also the length of path 1. The displacement of an object traveling along path 1 is therefore “2 m to the right.” We have stated both the magnitude and the direction of travel. The displacement for path 1 is shown in the following diagram.

For path 2, the starting location is A and the end location is B. The displacement of an object is again the straight-line distance from A to B. This is again “2 m to the right.” The displacement for path 2 is shown below.

This means that the displacement of an object traveling along the two paths is the same, even though the paths are very different. Note that the distances along the two paths are different, despite the displacements being the same.

It is also possible to have a negative displacement. For example, a displacement of “2 m” east is a valid displacement. This is equivalent to a displacement of “2 m” west. However, distances are always positive.

Example 3: Understanding Distances in a Straight Line

If a distance is traveled in a straight line, which of the following is correct?

  1. The distance traveled is the magnitude of the displacement along the straight line.
  2. The distance becomes a vector quantity.

Answer

In this question, we are considering a distance that is traveled in a straight line. We are given two statements and asked to consider which of these statements is correct.

Let’s begin by considering the first statement, which says that the distance traveled in a straight line is the magnitude of the displacement along that straight line.

This looks like a correct statement. The straight-line distance cannot be a displacement as it is, because we are not told the direction in which this distance is traveled. Displacement is a vector quantity, and so it must have a direction as well as a magnitude.

A displacement has a magnitude equal to the shortest distance between two points, which is the straight-line distance between those points. Since we are told that the distance here is traveled in a straight line, we know that this is indeed equal to the magnitude of the displacement between these two points.

This means that the first statement here is correct.

Let’s now consider the second statement. This says that if a distance is traveled in a straight line, then the distance becomes a vector quantity.

A vector quantity has both a magnitude and a direction. The second statement in this question therefore cannot be correct, because we do not know the direction in which the distance is traveled. We do know that it is a straight-line distance, but that tells us nothing about which direction it is in. A straight line could point in any direction. The distance is a scalar quantity, not a vector quantity, and therefore it cannot be a displacement, even though it is a straight-line distance.

Therefore, only the first statement given in the question is correct, which says that the distance traveled is the magnitude of the displacement along the straight line.

Example 4: Comparing Displacements for Different Paths

A car is at the center of a circle. The arrows show paths that the car could travel to reach the circumference of the circle. Is the displacement of the car between its initial and final positions the same in both cases?

Answer

In this question, we are shown a car in the center of a circle. We are also shown two different paths that the car could take to reach the circumference of the circle. We need to consider the displacement of the car for each path it could take.

First of all, let’s consider the situation where the car travels along the blue path. This path is a straight line from the center of the circle to the circumference of the circle.

Recall that the displacement of an object is the shortest distance from the start point to the endpoint of the motion of the object, along with the direction of travel. Since the blue path is a straight path, it is the shortest distance from the start point to the endpoint. This means that the magnitude of the displacement in this case is the length of the blue path.

Since the blue path is a straight line from the center of the circle to the circumference of the circle, its length is equal to the radius of the circle. Let’s assume that the radius of this circle is 5 m. This means that if the car travels on the blue path, its displacement will be 5 m in the direction of the blue arrow. Note that we have given the direction, even though we do not explicitly know how to describe the direction in more detail than “in the direction of the blue arrow.”

Let’s now think about the displacement of the car if it travels along the red path shown above. In this case, the magnitude of the displacement is different from the length of the red path. This is because the magnitude of displacement is the straight-line distance between the start point and endpoint of the motion of the car, but the red path is curved.

The start point of the red path is the center of the circle, and the endpoint of the path is on the circumference of the circle. The shortest distance between these points is the straight line from the center of the circle to the circumference, which is the radius of the circle, 5 m.

Since the start point and endpoint of both of the paths here are the same, the direction of travel is the same too. The direction of both displacements is “in the direction of the blue arrow.” Since the magnitudes of both displacements are the same as well, we have found that both of the displacements in this question are exactly the same.

Example 5: Comparing Displacements for Different Paths

A car is at the center of a circle. The arrows show paths that the car could travel to reach the circumference of the circle. Is the displacement of the car between its initial and final positions the same in both cases?

Answer

In this question, we are shown a car in the center of a circle. We are also shown two different paths that the car could take to reach the circumference of the circle.

Let’s consider the displacement of the car for each of the paths.

Firstly, let’s consider the car traveling along the blue path. This path is a straight line from the center of the circle to the circumference of the circle.

Recall that the displacement of an object is the shortest distance from the start point of the motion of the object to the endpoint of the motion of the object, along with the direction of travel. Since the blue path is a straight path, it is the shortest distance from the start point to the endpoint. This means that the magnitude of the displacement in this case is the length of the blue path.

The blue path is a straight line from the center of the circle to the circumference of the circle; its length is the radius of the circle. Let’s say that the radius of the circle is 5 m. This means that if the car travels on the blue path, its displacement will be 5 m in the direction of the blue arrow. Note that we have given the direction, even though we do not explicitly know how to describe the direction in detail.

Let’s now think about the displacement of the car if it travels along the red path. In this case, because the red path is curved, the magnitude of the displacement will be different from the length of the path. The magnitude of the displacement is equal to the straight-line distance from the center of the circle to the endpoint of the red path, which is on the circumference of the circle. This will have a length equal to the radius of the circle, 5 m. This is shown in the following diagram.

The direction of the displacement associated with the red path is in the direction of the green arrow we have drawn onto the diagram here.

Note that the magnitudes of both of the displacements here are the same and equal to the radius of the circle, 5 m. However, the directions of the two displacements are different, since the blue arrow and the green arrow we have drawn are pointing in different directions. This means that the two displacements are not equal.

Let’s now imagine two points, A and B, and again these points are separated by a horizontal distance of 2 m. First of all, imagine a car traveling from point A to point B along a straight line, as shown in the diagram below.

In this diagram, the car started at point A and traveled horizontally to point B. Now, the distance that the car traveled is equal to the length of the path that it followed. This means that the car traveled a distance of 2 m.

The displacement of the car is equal to the straight-line distance from A to B, which is again 2 m, along with the direction that the car has traveled in. This means that the displacement of the car is 2 m to the right. Note that here the magnitude of the displacement is equal to the distance that the car has traveled.

Let’s now imagine a second stage to this journey. After the car has traveled from A to B along the red path shown above, the car then reverses back to point A, also along a horizontal path, as shown in the diagram below.

The second part of the journey is shown by the blue arrow. This means that the car ends its journey at point A. Let’s think about the distance that the car has traveled in total.

The whole journey that the car undergoes is starting at point A, traveling horizontally to point B, and reversing horizontally back to point A. To find the total distance traveled, we need to add up the lengths of the paths it traveled along.

The car traveled 2 m to the right and then 2 m to the left to return to point A. The total distance traveled is therefore 2+2=4mmm.

Let’s now calculate the displacement of the car for the whole journey. The starting location of the car is A, and the car travels to point B and then back to point A, which is the endpoint of the car. So the start point and endpoint of the motion of the car are the same. This means that the distance from the start point to the endpoint of the motion of the car is zero metres. Note that we do not need to give a direction when the displacement is zero. A displacement must be nonzero to have a direction.

For a journey that ends at the same point that it started at, the displacement will always be zero, but the distance traveled will be nonzero and equal to the length of the path traveled.

Let’s think about how this idea works on a circular path. Imagine a car that starts at the leftmost point of a circular path, marked with a blue circle below. The circular path has a circumference of 5 m and a diameter of 𝑑 m. This is shown in the following diagram.

Let’s first imagine that the car drives halfway around the circle, to the point marked with a pink circle. First of all, let’s calculate the distance the car has traveled from the blue point to the pink point. This distance is shown by the blue arrow in the following diagram.

The distance traveled is equal to the length of the path around the circle from the start point to the pink point. This is equal to half of the circumference of the circle, which is half of 5 m. Half of 5 m is equal to 2.5 m, so this is the distance traveled by the car.

Let’s now compare this to the displacement of the car. The magnitude of the displacement is the straight-line distance from the start point to the pink point, and the direction is the to the right. This is shown by the purple arrow in the diagram above.

The length of the purple arrows, which is a straight line from one side of the circumference to the other side of the circumference, is equal to the diameter of the circle. We know this because it is a straight line that passes through the center of the circle.

This line therefore has length equal to the diameter of the circle, which is 𝑑 m. The displacement of the car is therefore 𝑑 m to the right.

Note that if the circumference of a circle is 𝑐 and the diameter of the circle is 𝑑, then the circumference of the circle is given by the formula 𝑐=𝜋×𝑑.

This means that 𝑐 and 𝑑 cannot be the same, so for our car, the magnitude of the displacement is not equal to the distance.

Let’s now see what happens if the car carries on on the circular path and returns to its start point, having driven around the entire circle. This entire journey is shown by the blue arrow in the following diagram.

If the car travels the full journey around the circle, returning to the same point it started from, then the distance the car has traveled is equal to the length of the path it followed. This length is equal to the entire circumference of the circle, which is equal to 5 m. This is the distance that the car traveled around the entire circle.

We have seen that the displacement of an object that ends at the same point it started from is equal to zero. Thus, the displacement of our car is equal to zero, because the endpoint of its motion is the same as the start point of its motion.

As the car goes on the path around the circle, the distance it has traveled is always increasing. On the other hand, the displacement of the car increases for a while and then decreases, until it reaches zero again when the car is at its endpoint. This is a key difference between distance and displacement for an object that is not traveling in a straight line.

Let’s now summarize the key points we have seen in this explainer.

Key Points

  • The distance that an object travels is equal to the length of the path that object has moved along. If an object travels along the same part of a path multiple times, then the lengths of the repeated parts of the path need to be added on to the total length in order to find the distance traveled.
  • Distance is a scalar quantity, so it does not specify the direction that the object has traveled in.
  • The displacement of an object is a vector quantity, with size equal to the straight-line distance between the start point and endpoint of the object.
  • Since displacement is a vector quantity, the direction traveled must be specified as well.
  • A displacement is always a straight line, while a distance can be a curved path.

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