Video: Distance and Displacement

In this video, we will learn how to define distance as the length of a path between two positions and to define displacement as the straight line distance between two positions.

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Video Transcript

In this video, we will learn how to define distance and displacement, two similar words with subtly different definitions. So let’s get into it. Let’s start with the definition of distance.

Now, distance is defined as the length of a path between two positions. Now to understand what we mean by this, let’s first define our two positions. Let’s say our first position is position A and our second position is position B. Now let’s say that we got an object that’s at A initially and it’s trying to get to B. Now there are lots of different ways that the object could move.

The simplest way it could go is to just go directly from A to B, in which case the distance that the object travels — let’s call that distance 𝑑 — is simply the straight line distance between A and B. And more specifically, the distance that it travels is the length of the path taken by that object to get from A to B. So if A and B are five metres apart, then the distance travelled by the object when taking this path is five metres.

However there are lots of different ways for the object to get from A to B. For example, the object could start by going this way and then coming back this way. Well, in a situation like this, the distance travelled by the object is no longer five metres. It’s now going to be larger than five metres because this length and this length combined are going to be larger than five metres.

We can see this by adding a dotted line in the middle. This results in us forming two right-angled triangles. So one of the triangles is this triangle here and the other triangle is the identical one that’s been mirrored over here. Well in that case, we can see that the first half of the object’s trajectory was along the hypotenuse of the right-angled triangle. And the same is true for the second half.

Now, the hypotenuse is always the longest side in a right-angled triangle and it is the side opposite the right angle. So that means that this length, the length of the first half of the journey that the object took in this situation, is longer than this length which is half of the journey that the object took initially when we discussed it’s going straight from A to B. And the same is true for this length being longer than this half of the original journey.

Therefore, this length and this length added together is going to be longer than this length and this length added together. In other words, the journey that the object now takes means that the object travels a larger distance to get from A to B. And so, this distance is dependent on the path between the two positions.

And another important thing to know about distance is that it’s a scalar quantity. Now, a scalar quantity only has magnitude or size. And therefore, this means that the direction in which the object travels the distance doesn’t really matter. What’s relevant to distance is how long the path is between point A and point B in this case.

Now let’s compare that with displacement. Now, displacement is defined as the straight line distance between two positions. So if we consider point A and point B once again and then say that our object starts at A and goes straight to B, then the displacement of that object is simply defined as the straight line distance between the two positions.

So in this case, if we said that the distance between A and B was five metres, then the displacement of the object is also five metres. So how is that different to distance? Well, what we’ve done here is to discuss the scenario where distance and displacement coincide. Displacement really comes into its own when we discuss another path from A to B.

So now let’s consider that the object goes this way and then this way to get from A to B rather than going directly from A to B. Well, in that case, the displacement is still defined as the straight line distance between the two points. In other words, it’s the shortest distance, the straight line distance between the start point and the end point of the object. And so even though the object took this path, the displacement of the object is still five metres. And this is where the difference between distance and displacement comes into it.

But there is also one more thing that we haven’t mentioned yet. We said that distance was a scalar quantity. The reason we mentioned that in the first place is because displacement is a vector quantity. This means that it has magnitude or size and direction. In other words, we shouldn’t say that the object’s displacement is five metres. We should say it is five metres to the right because it starts at point A and finishes at point B. And point B is to the right of point A.

So a simple way to think about it is that the object going from A to B on whatever path it ends up taking is equivalent to it just having gone towards the right until it reached point B using the shortest distance, the straight line distance. So let’s clarify this slightly.

Let’s say that our object gets from A to B using this path. Well in that case, the distance travelled by the object is going to be something greater than five metres. Let’s say that distance ends up being eight metres. And that’s the total length of the path along which our objects go from point A to point B. However, this is not the displacement of the object.

The displacement is simply the straight line distance between the start point and the finish point which is five metres. But we also have to include the direction because displacement is a vector. So we say that the displacement is five metres to the right because the object went from left to right as it got from A to B. And so, this is the difference between distance and displacement.

Now let’s look at a couple of interesting scenarios. Let’s come back to the bee that we saw on the starting screen of this video. Now, let’s say that the bee is busy bumbling around taking whatever pass it wants to get from where it is now to where it needs to be. But then, let’s also imagine that it suddenly ends up coming back to its original starting position.

Well, in this situation, the distance travelled by the bee is simply the length of the path that the bee took to get from where it was which is the centre of the screen to where it is now which is also the centre of the screen. So this whole thing is the distance. And let’s call that distance 𝑑. It’s the length of the curvy, squiggly path. However, what is the displacement of the bee?

Well, the bee started here and the bee also ended up here in exactly the same position. Therefore, the straight line distance between the start point and the finish point of the bee is actually zero. So if were to label the displacement of the bee as 𝑠, we can say that that’s zero. But that’s quite a weird concept to think about.

The fact that the bee has travelled so far is actually travelled very large distances and yet its displacement is still zero. But this is why displacement is such a good measure of overall or net motion because it doesn’t care about how the bee got from where it was to where it is now. All it cares about is the position it was in initially and the position it’s in now and the difference between these two positions.

In other words, the bee travelling this huge long large curvy distance has made no difference to the overall position of the bee. It’s returned to the same position and it may as well have not moved at all. And the fact that the displacement of the bee is zero accurately reflects this.

Now, the other thing is that we said earlier displacement is a vector quantity. So why haven’t we given a direction? Well, this is one of the special cases that you don’t have to give a direction. Because if the displacement is zero, then the bee hasn’t moved in any direction. In other words, zero displacement to the left is the same thing as zero displacement to the right is the same thing as zero displacement up or down or in and out. The bee hasn’t moved at all. So we don’t need to give a direction.

So the interesting thing to take away from this example is that an object may travel very large distances, but it still may have a zero displacement value. But that only happens if it returns to its starting position at the end of its journey. Now as well as this, let’s also think about an object trying to get from point A to point B. Let’s now not worry about how it got there. Let’s just worry about its displacement.

The displacement is the shortest or straight line distance between A and B. However because displacement is a vector quantity, in other words it’s a distance — let’s say 10 metres — and it’s in this direction because to get from A to B you have to go in this direction, what we can actually do is to break up this displacement vector into components. In other words, we can say for example that A and B are the start point and end point of a very short walk I took the other day. Well, it doesn’t have to be very short. I could have taken a very long path to get from A to B, but that’s not really relevant right now.

The fact of the matter is that I was displaced 10 metres to the northeast let’s say. If we say that this direction is north which automatically means this way is east, this way is south, and this way is west. Well in that case, I could break up my walks displacement vector into components. I could break it up into an easterly component and a northerly component as well.

And notice that we’ve got a right-angled triangle here. So basically, on my walk, I was displaced 10 metres to the northeast-ish. But I could say that that was equivalent to breaking up into this distance, which would be my easterly displacement. And it would be less than 10 metres because 10 metres is the hypotenuse of my right-angled triangle. And this distance would be my northerly displacement.

Now this comes in very handy when considering for example bearings or directions. And so, it’s important to know that we can break up vectors into their components. So now that we’ve understood the difference between distance and displacement and looked at a couple of interesting cases for displacement, let’s look at an example question.

A leaf is blown by the wind. The leaf moves five metres forward and then three metres backward. What is the distance moved by the leaf? What is the leaf’s net forward displacement?

Okay, so in this question, we’ve got a leaf. And that leave is blown by the wind so that it moves five metres forward and then it moves three metres backward. So the leaf starts out in this position here and it ends up in this position here. What we’ve been asked to do is to firstly find the distance moved by the leaf.

Well, we can recall that distance is defined as the length of a path between two positions. In this case, the two positions are the start position and the end position of the leaf. And so, the path taken by the leaf is to firstly go five metres forward and then come back three metres. And hence, the total distance that the leaf has travelled — let’s call that distance 𝑑 — is equal to the five metres forward plus the three metres it travelled back.

Because even though the leaf came back and travelled in the opposite direction to its initial motion, that doesn’t matter when we try to calculate distance. All that matters is the total path length. And this total path length is five metres plus three metres which ends up being eight metres. Therefore, we can say that the distance moved by the leaf is eight metres.

Now the second part of the question asks us to find the leaf’s net forward displacement. Now, net simply means overall or resultant. And so, we’re just trying to find the overall forward displacement of the leaf. Now, displacement is defined as the straight line distance or in other words the shortest distance between two positions, in this case between the starting position and the ending position once again.

Well, in this situation, the straight line distance between those two points is this distance here. But then, that distance is equal to this whole five metres minus this three-metre distance here. And so, we can say that the displacement of the leaf which we will call 𝑠 is equal to five metres — that’s the whole distance it moves forward — minus the three metres that it moves back. And five metres minus three metres is equal to two metres.

Now there’s a couple of things we haven’t considered here. Firstly, we need to recall that displacement is a vector quantity. So this means that it has magnitude or size and direction. So why have we not stated direction here when we’ve worked out the displacement of the leaf? Well, it’s because we don’t need to; the question has done it for us. We’ve been asked to find the leaf’s net forward displacement.

And in fact, overall, the leaf does move forward because the net effect the overall effect of moving five metres forward and then three metres back is equivalent to the leaf just having moved two metres forward. And so, what we’ve done is calculated the forward displacement of the leaf. And it would be unnecessary to write two metres forward when we’ve been asked to find the forward displacement.

So in a situation like this, we don’t need to write the direction. But in general, we should. It’s a good idea to write the direction. But anyway, so we found our final answer. Now the leaf’s net forward displacement is two metres.

Okay, so now that we’ve had a look at this example, let’s summarize what we’ve learnt in this lesson.

Firstly, we saw that distance is the length of a path between two positions. It’s also a scalar quantity. So it only has a magnitude. Secondly, we saw that displacement is the straight line distance between two positions. It’s also a vector quantity. So it has a magnitude and a direction.

Thirdly, we saw that the magnitude or size of the displacement of an object can be zero, even for a nonzero distance. In other words, an object can start at a certain point, take any path, and then return back to its original starting position. And it will have travelled some distance, where that distance will be nonzero, but its displacement is zero because it’s starting and finishing at the same point. And hence, the straight line distance between the start and the end points is zero.

And finally, we saw that displacement can be split into components. In other words, if this, for example, is our object’s displacement, then we can choose to break this up into a left right component and an up down component. In this specific case, the components are pointing to the left and upward. And so, this is how we would deal with distances and displacements.

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