Video: Position Distance and Displacement

In this video we learn about the position and displacement vectors, and also about the scalar quantity of distance. We learn how to calculate each one and how they are different from one another.


Video Transcript

In this video, we’re going to learn about position, distance, and displacement. What each term means, how they compare to one another, and how to work with them practically. As we start out, imagine for a moment that you are a world-class swimmer and are just finishing up a long career involving world championships and Olympic medals. Over the course of your career, you swam tens, if not hundreds of thousands, of laps in a pool. Your talent and ability has led you to the top of the winners podium on more than one occasion. And tonight, at the end of your long career, you’re being celebrated and inducted into the international swimmers hall of fame.

During the evening’s events, the master of ceremonies unveils a special banner honoring your lifetime achievement. The MC explains that a careful accounting has been performed of all of your swimming both training and competition over the years. On this special evening, your total career swimming displacement is being recognized. And it is zero meters! Wait, that can’t be right. The audience is surprised. Not least of all, you are. The master of ceremonies is flustered. But it turns out, with a better understanding of distance displacement and position, we can find that this result does make sense. To get to the bottom of this, let’s consider what the definitions of position, distance, and displacement are.

Say that we have a three-dimensional coordinate system, 𝑥, 𝑦, and 𝑧. And imagine further that we have an insect, such as a fly, moving around in this coordinate space. At any given moment, we can draw a vector starting at the origin and going towards the fly’s position. That vector, which we can call 𝑝, is known as the position of the object, in this case our fly, relative to the origin of a coordinate axis we’ve defined. So that’s what the position vector of an object is. It’s the object’s location relative to some defined origin. And notice that position is a vector quantity. Going back to our fly, we know that flies often move around within a space. And this one is no different. It follows a path seemingly randomly throughout the open space. If we take a snapshot of the fly’s position at a second moment in time, we can draw in its position vector calling it 𝑝 sub two. And with these two vectors defined, we now have a start point and an end point for a journey that the fly makes.

If we drew a vector from the fly’s initial position to the fly’s final position, we can call that vector 𝑑. And it would represent the fly’s displacement over this journey. That’s the second term whose meaning we wanted to come to understand, displacement. And we see that displacement also is a vector quantity. What about distance? What is the distance that the fly travelled? To find that out, we need to follow the fly’s path through every twist and turn that it takes. It’s that total path length which is the fly’s distance travelled. We can call that capital 𝐷. Unlike position and displacement, distance is a scalar quantity. It doesn’t have a direction associated with it. And we can go further in showing the difference between this scalar quantity distance and the vector quantity’s displacement and position.

Notice that when it comes to distance, the location of our origin, or our coordinate system in general, doesn’t matter. The distance the fly travels is independent of our coordinate system. On the other hand, the position of the fly and its displacement are very much dependent on the location of our origin and our coordinate system in general. One last thing about the difference between distance and displacement. Imagine that our fly continued on its way lazily buzzing about the room and finally ending up at the same location at which we started. And we ask the question, what is the fly’s distance travelled over this journey? To find that out, we would calculate the total length of all of its travels.

But now, what if we ask what is its displacement over the loop? When we recall that displacement is equal to the vector difference between initial and final position, if initial and final position of an object are the same, then that displacement vector is zero. So over this path, even though the fly may have travelled a great distance, its displacement is zero. This is the answer to our question of how a hall of fame swimmer could have a lifetime swimming displacement of zero meters. Every time a swimmer swim a lap, they start and end in essentially the same position. So their displacement in total is zero. Now that we have an idea of what these terms mean, let’s get some practice using them with a few example questions.

On a journey, a cyclist rides 3.0 kilometers west and then turns around and rides 2.0 kilometers east. Assume that east corresponds to positive displacement. What is the cyclist’s displacement from the starting point of their journey to its end point? What distance does the cyclist travel during their journey? What is the magnitude of the cyclist’s displacement from the starting point of their journey to its end point?

In this exercise, we want to solve first for the cyclist’s displacement. Then we want to solve for the distance that they’ve travelled on their journey. And finally, we wanna solve for the displacement magnitude of the cyclist’s trip. Let’s start by drawing a sketch of their journey. If the cyclist begins on their journey at a location we call the origin, we’re told that they ride to the west 3.0 kilometers and then turn around and return to the east a distance of 2.0 kilometers. We want to solve for their displacement. We can call it lower case 𝑑. The distance they travel on their journey, we can call it capital 𝐷. And finally, we want to know the magnitude of their displacement.

In order to solve for the displacement the cyclist experiences over their journey, we mark out their starting and ending locations along our axis. And we draw a vector starting at the start point and ending at their ending location. That vector, joining their starting and ending point, is their displacement. And looking at our distance scale, we see that that displacement is negative 1.0 kilometers. So we record that value for lowercase 𝑑. Notice that this answer shows the vector nature of displacement. We’re told that movement towards the east is movement in the positive direction. So our minus sign shows that indeed displacement is a vector.

Next, we want to solve for the distance the cyclist travelled on their journey. To figure this out, we move alongside the cyclist all throughout their journey, recording each bit of distance we cover. We see that on the first leg of their journey, the cyclist moves 3.0 kilometers. Then on the second leg, they move 2.0 kilometers. So the distance travelled is the sum of these parts. So the cyclist’s total distance travelled is 5.0 kilometers. And finally, we want to solve for the magnitude of the displacement of the cyclist. We know that the magnitude of any vector — in this case, the vector 𝑑 for displacement — is a positive value. The magnitude of the cyclist’s displacement is equal to the absolute value of negative 1.0 kilometers, or 1.0 kilometers. This is a measure of the distance between the cyclist’s start point and end point.

Let’s now look at one more example helping us understand the difference between distance, displacement, and position.

Which of the following statements comparing position, distance, and displacement is correct? An object may record a nonzero distance while recording a displacement of zero. An object may record a distance of zero while recording a nonzero displacement. An object may record a nonzero distance while maintaining a position of zero. An object may record a nonzero displacement while maintaining a position of zero.

In this exercise, we want to evaluate each of these four statements and find which one of the four is correct. Let’s begin with the first statement. An object may record a nonzero distance while recording a displacement of zero. That’s accurate. We can imagine an object moving in such a way that it starts and ends at the same location. And yet, it’s travelled a total distance in order to complete that journey. In that case, its distance travelled would indeed be nonzero. But its displacement, that is the separation from its start and end point, would be zero. We highlight this first statement as correct. Even though we know this is the right one, let’s continue on and look at the remaining statements. The second statement says that an object may record a distance of zero while recording a nonzero displacement. We know this statement can’t be accurate because distance is always equal to or greater than displacement.

The third statement says that an object may record nonzero distance while maintaining a position of zero. If an object maintains a position of zero, that means that it doesn’t move at all. In which case, it couldn’t record a nonzero distance. So we know this third statement is not accurate either. And finally, the fourth statement says an object may record a nonzero displacement while maintaining a position of zero. Once again, if an object maintains a position of zero, then its displacement as well as its distance cannot be anything other than zero. This means it’s unable to record a nonzero displacement. So this fourth statement is not correct either.

Let’s summarize what we’ve learned about position, distance, and displacement. Position, distance, and displacement are all terms that describe object location as well as the past motion of that object. Also, position and displacement are vectors, while distance is a scalar quantity. This means that we can define distance without reference to a coordinate frame or an origin. But position and displacement require that framework. And finally, distance and displacement do not equal one another, if the moving object’s direction changes. When an object moves while changing direction, then the distance it travels will be greater than the magnitude of the displacement it experiences.

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