A ship sails 7.5 kilometers west and then 10 kilometers east. What is the distance moved by the ship? And what is the ship’s net displacement to the west from its starting position?
This is a question inviting us to explore the difference between distance and displacement. And this question comes in two parts. The first part is asking us to find the distance moved by the ship. And the second part of the question is asking us to find the ship’s net displacement to the west from where it starts. Let’s begin by drawing out the ship’s journey.
Let’s imagine that the ship begins at this position marked 𝑥. The question tells us that the ship begins by traveling 7.5 kilometers to the west. We’re told that then the ship travels 10 kilometers to the east. And so, we have our diagram showing the ship’s motion from the start of its journey to the end of it. Looking back at the question, we see that the first part we need to answer is to find the distance moved by the ship. We should recall that distance is defined as the length of the path traveled between two positions.
So, let’s look at the path traveled by our ship. This path consists of two segments. First, the ship travels 7.5 kilometers to the west, and then it travels 10 kilometers to the east. So, the total length of this path or, in other words, the distance moved by the ship is given by 7.5 kilometers plus 10 kilometers. Doing the sum, we find that the distance moved by the ship is 17.5 kilometers. And this is our answer to the first part of the question.
The second part of the question is asking us about the ship’s net displacement to the west. Recall that displacement is defined as the straight-line distance between the object’s start position and its end position, and that the displacement includes the direction that this distance is in. Looking back at our diagram, we can see that we’ve already identified the start position and the end position of the ship. And so, we see that we can add the displacement onto this diagram as follows.
This blue arrow we have drawn extends from the start position to the end position. Notice that this blue arrow appears to be pointing to the east. But if we look back at the question, we see that we’re asked for the net displacement to the west. But this doesn’t present a problem. We can still find the ship’s displacement to the west. Since east and west are opposite directions, we can think of them like an axis, where our zero is the position in which the ship starts and west points one way and east the other. In this sense, we can think of east as being equivalent to negative west.
To actually calculate the ship’s net displacement to the west, we need to take the distance that it travels to the west and subtract from that the distance that it travels towards the east. Remember that we’ve said that east is essentially negative west. The distance the ship travels towards the west is 7.5 kilometers. And then we subtract from this the distance that it travels towards the east, which is 10 kilometers. If we do this subtraction, we get a result of negative 2.5 kilometers. So, we’re saying that the ship has a net displacement of negative 2.5 kilometers to the west, which is equivalent to saying it has a displacement of positive 2.5 kilometers to the east.
We should note that when we defined displacement, we said that it also included a direction. But we don’t need to explicitly give a direction along with our answer because the direction is already implicitly included in the question asking us for the displacement to the west. And so, our answer to the second part of the question is that the ship’s net displacement to the west from its starting position is negative 2.5 kilometers.