A bird flies 2.5 kilometres north, then four kilometres west, then one kilometre south, then three kilometres east, and finally two kilometres north. What is the bird’s net displacement north from its starting position? What is the bird’s net displacement west from its starting position?
Okay, so, in order to answer these two questions, let’s first start by drawing a diagram to show the bird’s entire journey. So, let’s start by first defining this direction upwards on the screen to be north. This automatically means that to the right is east, this way is west, and this way is south.
So, now that we’ve defined which direction is which, let’s choose a starting position for the bird. Let’s say it starts here. Now we’ve been told that from here the bird first flies 2.5 kilometres north, so we can draw an arrow pointing in the northward direction to represent this motion. And we can say that the length of this arrow is representing 2.5 kilometres. Then we’ve been told that the bird flies four kilometres west. So, we draw an arrow in the westerly direction and label it four kilometres. After which the bird flies one kilometre south, one more arrow, southernly pointing this time labelled one kilometre.
And then it moves three kilometres east. So, there’s the three-kilometre-east arrow. And then the bird moves two kilometres north. Now if we’re about to draw this final arrow, then we’re gonna crash into this four-kilometer label. So, let’s make the label smaller and shift it to the left slightly. And now we can draw our two-kilometre-north arrow. And there it is, labelled two kilometres. Now this just goes to show that when we’re drawing diagrams, it’s probably a good idea to draw them in pencil first. This way we can modify things as we go and the diagram is as neat as possible.
But anyway, so we said that the bird started here. And its final point after its entire journey was here. What we’ve been asked to do is to find the bird’s net displacement north from its starting position and its net displacement west from its starting position. So, first of all, what do we even mean by the word displacement?
Well, we can recall that displacement is defined as the straight-line distance between two positions. So, if we say that some object is trying to get from this point, which we’ll call point A, to this point, which is point B, and they take whatever path they want between A and B, the distance that they travel is whatever the length of this pink path is. But the displacement is the shortest distance, or the straight-line distance, between A and B.
It’s also worth noting that displacement is a vector quantity. In other words, not only does it have a magnitude or size, but it also has a direction. So, for the object trying to get from A to B, the displacement is whatever the straight-line distance between A and B is. But we also need to include a piece of information telling us that the object went from A to B or, in other words, it traveled to the right. Or if we use this convention, it traveled towards the east. And that’s what it means for displacement to be a vector quantity.
But anyway, so we’ve been asked to find the bird’s net displacement north and net displacement west from its starting position. Now when we’re talking about net displacement, the word net is used to mean that we’re talking about its entire journey. In other words, the overall displacement of the bird from its starting position once it’s completed its entire journey. And so, we’re trying to find the net displacement north and the net displacement west from here, the starting position, to here, the finish position, of the entire journey.
Now the net displacement by itself of the bird over its entire journey is going to be the straight-line distance between the start and the finish points. In other words, it’s going to be this distance here. And it’s going to be in this direction because we started here and finished here. However, we’re not simply trying to find the net displacement. We’re trying to find the net displacement north and the net displacement west.
And to do this, we can recall that displacement, because it’s a vector quantity, can be broken up into components. Specifically, we can break up the bird’s entire journey’s displacement into a northward component and a westward component. And in fact, these two components are exactly what we’re trying to find in order to answer our two questions. So, let’s start by trying to find the northward component.
Now, interestingly, the bird is only ever moving either north, or east, or south, or west. It’s never moving, for example, north east, or south west, or anything like that. It only ever moves in one of the four compass directions. And this actually makes life easier for us because this way the only parts of the bird’s journey affecting the northward components are whenever the bird travels north or whenever it travels south.
This is because if the bird is travelling directly east or directly west, this does not change how far north or south the bird has moved. And hence, for finding the northward component, we can actually ignore the eastward and westward parts of the bird’s journey.
So, the northward component of the bird’s journey then, well, we know the bird starts its journey by moving two and a half kilometres north. As well as this, at a later point in its journey, it moves one kilometre south. And then, to finish off its journey, it moves two kilometres north. So, the bird’s net displacement north, which we’ll call 𝑑 subscript 𝑛, is equal to the two and a half kilometres it travels at the beginning of its journey. So, that’s this distance here, minus the one kilometre south. Because the one kilometre is in the opposite direction to north, and then, plus the two kilometres north it travels once again to finish its journey.
At which point, we can evaluate 2.5 kilometres minus one kilometer plus two kilometers as being 3.5 kilometres. And this is the bird’s net displacement north from its starting position. Now before we give our final answer, we need to realise that we’ve just given a component of a displacement. And specifically, we’ve given the northward displacement of the bird over its entire journey. But then we said earlier that displacement is a vector quantity. Does that mean that we need to give its direction as well?
Well, the answer to that is no, in this case, because we’ve been told that this is the northward component in the question. So, we don’t actually need to state that the northward component of the displacement that we found is 3.5 kilometres to the north. And hence, we can simply do with giving 3.5 kilometres as the answer to the first part of our question. At which point, we can now move on to finding the bird’s net displacement west from its starting position.
Using the same logic as earlier, the only thing affecting the bird’s net displacement west will be any part of the bird’s journey where it travels west or where it travels east. Because any part of the journey where the bird goes either north or south is not going to result in a change in how far west the bird has travelled.
Hence, the only parts of the journey that we need to consider are the four kilometres west that the bird travels and the three kilometres east that it travels at a later point in the journey. So, that’s four kilometres west and three kilometres east. Hence, we can say that the net displacement to the west of the bird over its entire journey is equal to the four kilometres it travels west minus the three kilometers it travels. Because the three kilometres are to the east and travelling east reduces how far west the bird is.
So, evaluating the right-hand side of this equation, we find that this net displacement, this time to the west, is one kilometre. And hence, we found our final answer to this part of the question. The bird’s net displacement west from its starting position is one kilometre.