A tourist wandering around a city walks 965 meters east, then 244 meters south, then 139 meters west, then 760 meters north, then 826 meters west, and then finally 44 meters south before stopping for a rest. How far is the tourist displaced from the point they started walking from?
Okay, so in this question, we have a tourist wandering around a city, and we’re given information about the various parts of the journey that they take. Let’s begin by using this information from the question to draw a sketch of the path that they follow. The tourist starts at some position and the first thing that they do is walk 965 meters east. If we label the compass directions like this, then we can add this part of the journey to our sketch as follows. Our tourist has walked a distance of 965 meters in the direction we’ve labeled east. The next part of the tourist’s journey is to walk 244 meters south. Then they walk 139 meters west, followed by 760 meters north, then 826 meters west, and finally 44 meters south. This sketch now shows the whole journey taken by the tourist.
The question is asking us how far the tourist is displaced from the point they started walking from. So, we need to work out the displacement of their end position relative to their start position. We should recall that displacement is the straight-line distance between two positions. And since displacement is a vector quantity, it has a direction as well as a magnitude.
Our tourist is walking along the compass directions north, south, east, and west. In general, someone walking along these four directions will end up displaced in both the north–south direction and the east–west direction. We can see from our set of compass axes that south is negative north and west is negative east. So, we can say that the displacement to the north, which we’ve labeled 𝑑 subscript N, is equal to the sum of all of the distances walked in the north direction minus the sum of all of the distances walked in the south direction and similarly for the displacement to the east, which we’ve labeled 𝑑 subscript E. This is the sum of all of the distances walked in the east direction minus the sum of all of the distances walked in the west direction.
In the general case in which someone is displaced in both the east direction and the north direction, then their overall displacement will be given by this orange arrow. The magnitude of this overall displacement is given by Pythagoras’s rule, which says that the length of the hypotenuse of a right-angled triangle is given by the square root of the sum of the squares of the other two sides. In order to fully specify this overall displacement, we would also need to work out this angle here so that we could give the direction. Getting back to the question itself, we can see that we’re going to need to calculate the displacement of the tourist in the north–south direction and also in the east–west direction.
So, let’s work out the values of 𝑑 subscript N, the displacement to the north, and 𝑑 subscript E, the displacement to the east, for our tourist’s journey. For the displacement to the north, we need to start by finding all of the distances our tourist walks in the north direction. So, that’s all of the arrows pointing in the north direction on our sketch. In this case, that’s just the one arrow with a distance of 760 meters. Then, we need to subtract all of the distances walked in the south direction. So, that’s all of the arrows pointing south in our sketch.
We have this arrow with a length of 244 meters and this one with a length of 44 meters. Then, we just need to evaluate this expression. When we do this, we get a result of 472 meters. Since this value is positive, this means that the displacement along the north–south direction is indeed to the north. Remember that since south is negative north, then if our value was negative, this would indicate the displacement was actually to the south.
So, now let’s calculate 𝑑 subscript E. This is the displacement in the east direction. We need to start by finding all of the distances walked to the east. So, that’s all of the arrows pointing east on our sketch. We have just the one arrow pointing east, and it has a length of 965 meters. Then we need to subtract from this all of the distances walked to the west. So, that’s all of the arrows pointing west on our sketch. In this case, that’s this arrow with a length of 139 meters and this one with a length of 826 meters.
And then, we just need to work out this expression here. When we do this, we get a result of zero meters. What this means is that the tourist does not end up displaced at all along the east–west direction relative to where they started. In other words, the net result is that the distances they walked to the east exactly cancel out with the distances they walked to the west. This means that our situation turns out to be quite a lot more straightforward in this general case we described here. Since our value of 𝑑 subscript E is zero, then the overall displacement of our tourist is just their displacement to the north since this is the only direction in which they are displaced.
Since we know that displacement is a vector, then in order to fully specify it, we need to give its magnitude and its direction. So, we could say that the overall displacement of the tourist, which we’ve labeled 𝑑, is given by 472 meters to the north. But if we look back at the question, we see that we’re not actually asked to find the displacement of the tourist, but rather how far they are displaced. Since the question only asks us how far, this means that we need to give the magnitude of the displacement for our answer rather than the displacement itself. So, that’s just the magnitude of this straight-line distance between their start position and their end position, which we have calculated as 472 meters.
And so our final answer to the question how far is the tourist displaced from the point they started walking from is 472 meters.