Video: Distance and Displacement

A rover on Mars is initially facing north. The rover travels 300 m in the direction it is facing, turns to its left and travels 200 m in this new direction, turns left again and travels 500 m in the direction it is now facing, and turns left once more and travels 100 m in the direction that it is now facing. What is the rover’s net displacement east of its initial position? What is the rover’s net displacement north of its initial position?

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Video Transcript

A rover on Mars is initially facing north. The rover travels 300 metres in the direction it is facing, turns to its left, and travels 200 metres in this new direction, turns left again, and travels 500 metres in the direction it is now facing, and turns left once more, and travels 100 metres in the direction that it is now facing. Now the first part of the question says what is the rover’s net displacement east of its initial position?

But before we go about answering this question, let’s first draw a diagram showing the full journey that the rover took. So, let’s say that this is where the rover started. And we’ve been told that the rover is initially facing north. So, let’s say that this direction is north. Now we’ve been told that the rover travels 300 metres in the direction that it’s facing, which is north. So, we can draw that like this, 300 metres north.

Then, we’ve been told that it turns left and travels 200 metres in this new direction. So, the rover here turns left and travels 200 metres, now in the westward direction. Then, we’ve been told it turns left again and travels 500 metres. So, the rover goes from travelling westward and turns left to go southward.

In this direction, it travels for 500 metres. And then, once it’s done that, we’ve been told that it turns left once more and travels 100 metres in this new direction. So instead, of going southward any further, it turns left and travels for 100 metres. At which point, we’ve reached the end of the rover’s journey.

Now what we’ve been asked to do is to find the rover’s net displacement east of its initial position. So, first of all, what do we even mean by displacement? Well, we can recall that displacement is defined as the straight-line distance between two positions. And it is a vector quantity. In other words, it’s got a magnitude and direction.

Now in this question, we’ve been asked to find the rover’s net displacement. Now the net displacement basically means the overall displacement, or in other words, the displacement of the rover after completing its entire journey. In other words, when we’re trying to find the net displacement of the rover, the two positions that are mentioned in the displacement definition, in this case, will be the start position and the finish position. Because this way we’ve got the net displacement of the rover considering its entire journey.

So, in this case, the net displacement is going to be from the start point all the way to the finish point. And it’s the straight-line distance between the two. And, of course, because it’s a vector quantity, which, in other words, means it’s got magnitude and direction, we not only need to worry about the straight-line distance between the start and the finish points. But we also need to worry about the direction, which is going from the start point to the finish point. In other words, then, the displacement vector that we’ve drawn is an overall representation of the rover’s entire journey. The rover has been displaced from here to here.

However, this is not the answer to our question either. We’ve been asked to find the rover’s net displacement east of its initial position. In other words, how far east is the finish position of the rover compared to its start position? To answer this question, we can actually split up the net displacement vector in two components. We can split it up into a north-south component and an east-west component. In other words, going from the start to the finish directly is equivalent to going this distance south and this distance west.

Now notice, we’ve been asked to find the net displacement east not west. However, this is not something we need to worry about. Because being displaced a certain distance to the west is the same thing as being displaced the negative of that distance to the east. In other words, the object’s net displacement to the west, that’s this distance here, is the same thing as been displaced negative that distance to the east.

So, for now, let’s see that the net displacement that we’re trying to calculate to the west is called 𝑑 subscript 𝑤, for displacement to the west. So, what is that distance? Well, to calculate this we actually only need to consider the east-west motion of the rover because regardless of how far north or south the rover goes, its east-west position is not affected by north-south motion.

So, the only parts of the rover’s motion that go east or west are the 200 metres it travels to the west and the 100 metres that travels to the east. And so, the distance that we’re trying to calculate, 𝑑 subscript 𝑤, is equal to the 200 metres west minus the 100 metres. Because the 100 metres are travelled to the east, in the opposite direction to west.

In other words, the net, or overall, westerly displacement of the rover is 100 metres. Over its entire journey the rover travels 100 metres to the west. Or if we want to find the easterly displacement of the rover, that’s negative 100 metres. Because it travels negative 100 metres to the east, which is just a fancy way of saying 100 metres to the west. And so, we have our answer to this part of the question. The rover’s net displacement east of its initial position is negative 100 metres. Now let’s look at the second part of this question.

What is the rover’s net displacement north of its initial position?

Okay, so, in this question we’ve been asked to find how far north the rover travels on its entire journey. Well, we can see that if the start point is here and the finish point is here, the rover doesn’t actually travel north. It travels southward overall. However, this is the same as the previous part of the question. We can simply work out the southerly displacement and say that this is equal to the negative of the northerly displacement.

So, what even is the southerly displacement of the rover? Well, to answer this, we only need to consider the north-south components of the rover’s motion because the east-west component are going to have no effect on how far north or how far south the rover is going. So, what are the north-south components? Well, firstly, the rover travels 300 metres to the north, and then, secondly, it travels 500 metres to the south.

So, this time let’s just directly work out the northerly displacement of the rover. Let’s say that the rover travels a distance 𝑑 subscript 𝑁 to the north. And of course, this is the net displacement rather than just the overall distance travelled by the rover. And this is equal to the 300 metres north that the rover initially travels minus 500 metres because the 500 metres are travelled in the southerly direction. In other words, in the opposite direction.

And so, evaluating the right-hand side, we find that the northerly displacement of the rover is equal to negative 200 metres. Hence, we have our final answer. The rover’s net displacement north of its initial position is negative 200 metres because it actually travels 200 metres south of its initial position over its entire journey.

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