Distance & Displacement



I've got to assume that everybody reading this has an idea of what distance is. It's one of those innate concepts that doesn't seem to require explanation. Nevertheless I've come up with a preliminary definition that I think is rather good. Distance is a measure of the interval between two locations. (This is not the final definition.) The "distance" is the answer to the question, "How far is it from this to that or between this and that?"

How far is it?
how far is it possible answer standard answer
Earth to sun 1 astronomical unit 1.5 × 1011 m
66th to 86th Street in NYC 1 mile 1.6 × 103   m
heel to toe on a man's foot 1 foot 3.0 × 10−1 m

You get the idea. The odd thing is that sometimes we state distances as times.

How far is it?
how far is it possible answer standard answer
International Space Station 90 minutes per orbit 40,000,000 m
Chicago to Milwaukee 90 minutes by train 00,150,000 m
Central Park to Battery Park 90 minutes on foot 00,010,000 m

They're all ninety minutes, but nobody but a fool would say they were they were all the same distance. What's being described in these examples is not distance, but time. In casual conversation, it's often all right to state distances this way, but in most of physics this is unacceptable.

That being said, let me deconstruct the definition of distance I just gave you. Every year in class, I do the same moronic demonstration where I start at one side of the lecture table and walk to the other side and then ask "How far have I gone?" Look at the diagram below and then answer the question.

There are two ways to answer this question. On the one hand, there's the sum of the smaller motions that I made: two meters east, two meters south, two meters west; resulting in a total walk of six meters. On the other hand, the end point of my walk is two meters to the south of my starting point. So which answer is correct? Well, both. The question is ambiguous and depends on whether the questioner meant to ask for the distance or displacement.

Let's clarify by defining each of these words more precisely. Distance is a scalar measure of the interval between two locations measured along the actual path connecting them. Displacement is a vector measure of the interval between two locations measured along the shortest path connecting them.

How far does the Earth travel in one year? In terms of distance, quite far (the circumference of the Earth's orbit is nearly one trillion meters), but in terms of displacement, not far at all (zero, actually). At the end of a year's time the Earth is right back where it started from. It hasn't gone anywhere.

A trip to New Jersey

Your humble author occasionally rides his bicycle from Manhattan to New Jersey in search of discount そば (soba) and さけ (sake) at a large Japanese grocery store on the other side of the Hudson River. Getting there is a three step process.

  1. Follow the Hudson River 8.2 km upriver.
  2. Cross using the George Washington Bridge (1.8 km between anchorages).
  3. Reverse direction and head downriver for 4.5 km.

The distance traveled is a reasonable 14 km, but the resultant displacement is a mere 2.7 km north. The end of this journey is actually visible from the start. Maybe I should buy a canoe.

Distance and displacement are different quantities, but they are related. If you take the first example of the walk around the desk, it should be apparent that sometimes the distance is the same as the magnitude of the displacement. This is the case for any of the one meter segments but is not always the case for groups of segments. As I trace my steps completely around the desk the distance and displacement of my journey soon begin to diverge. The distance traveled increases uniformly, but the displacement fluctuates a bit and then returns to zero.

This artificial example shows that distance and displacement have the same size only when we consider small intervals. Since the displacement is measured along the shortest path between two points, its magnitude is always less than or equal to the distance.

How small is small? The answer to this question is, "It depends". There is no hard and fast rule that can be used to distinguish large from small. DNA is a large molecule, but you still can't see it without the aid of a microscope. Compact cars are small, but you couldn't fit one in your pocket. What is small in one context may be large in another. Calculus has developed a more formal way of dealing with the notion of smallness and that is through the use of limits. In the language of calculus the magnitude of displacement approaches distance as distance approaches zero.

Last, but not least, is the subject of symbols. How shall we distinguish between distance and displacement in writing. Well, some people do and some people don't and when they do, they don't all do it the same way. Although there is some degree of standardization in physics, when it comes to distance and displacement, it seems like nobody agrees.


What would be a good symbol for distance? Hmm, I don't know. How about d? Well, that's a fine symbol for us Anglophones, but what about the rest of the planet? (Actually, distance in French is spelled the same as it is in English, but pronounced differently, so there may be a reason to choose d after all.) In the current era, English is the dominant language of science, which means that many of our symbols are based on the English words used to describe the associated concept. Distance does not fall into this category. Still, if you want to use d to represent distance, how could I stop you?

All right then, how about x? Distance is a simple concept and x is a simple variable. Why not pair them up? Many textbooks do this, but this one will not. The variable x should be reserved for one-dimensional motion along a defined x-axis (or the x-component of a more complex motion). Still, if you want to use x to represent distance, how could I stop you?

English is currently the dominant language of science, but this has not always been the case nor is there any reason to believe that it will stay this way forever. Latin was preeminent for a very long time, but it is little used today. Still, there are thousands of technical and not so technical words of Latin origin in use in the English language.

Imagine some object traveling along an arbitrary path in front of an observer. Let the observer be located at the origin. The vector from the origin to the object points away from the observer much like the spokes of a wheel point away from its center. The Latin word for spoke is radius. For this reason, we will use r0 (r nought) for the initial position, r for the position any time after that, and Δr (delta r) for the change in position — the displacement. Unlike the spokes of a wheel, however, this radius is allowed to change.

Much more directly, but less poetically, the Latin word for distance is spatium. For this simple reason, we will use s0(s nought) for the initial position on a path, s for the position on the path any time after that, and Δs (delta s) for the space traversed going from one position to the other — the distance.

If you think Latin deserves its reputation as a "dead tongue" then I can't force you to use these symbols, but I should warn you that their use is quite common. Old habits die hard. Use of spatium goes back to the first book on kinematics as we know it — Dialogues Concerning Two New Sciences (1638) by Galileo Galilei.

In uno stesso moto equabile, lo spazio percorso in un tempo più lungo è maggiore dello spazio percorso in un tempo più breve.   In the case of one and the same uniform motion, the distance traversed during a longer interval of time is greater than the distance traversed during a shorter interval of time.
Galileo Galilei, 1638   Galileo Galilei, 1638

OK, that was actually Italian. Galileo wrote to the people of the Mediterranean boot in his regional dialect, but the rest of Europe would have most likely read a Latin translation.

Spatium transactum tempore longiori in eodem Motu aequabili maius esse spatio transacto tempore breuiori.   In the case of one and the same uniform motion, the distance traversed during a longer interval of time is greater than the distance traversed during a shorter interval of time.
Galilaeus Galilaei, 1638   Galileo Galilei, 1638

Enough with the languages — dead or living. Let's finish with an important observation about the diagram above.

  1. First, the location of the observer does not matter. Place it wherever it's convenient. Distance and displacement will not be affected by a translation of the origin. There is no special place when it comes to measuring distance and displacement. All locations are equivalent. Space is homogeneous.
  2. Second, the orientation of the axes is irrelevant. Point them in any direction that's convenient. (The x-axis must remain perpendicular to the y-axis, however. This you cannot change.) Distance and displacement will not be affected by a rotation of the axes. There is no special direction when it comes to measuring distance or displacement. All directions are equivalent. Space is isotropic.
  3. Third, and most difficult to state in words, the chirality or handedness of the coordinate system is also irrelevant. Traditionally, the x-axis points to the right (of a page, blackboard, whiteboard, computer monitor, etc.) and the y-axis points toward the top. Hold your hand in front of you with your thumb pointing along the x-axis, your fingers pointing along the y-axis, and your palm facing you. The only way this is possible is if you use your right hand, which is why the traditional coordinate system is said to be right handed. (This is a two-dimensional example. There is a similar rule for three-dimensional axes that will be discussed later in this book.) Reverse any one of the axes and try this experiment again. The only way you can get your hand to line up the way I described with this new coordinate system is to use your left hand. This change of handedness — technically called chirality after the Greek word for hand, χερι (kheri) — is what one experiences when looking at the reflection of a hand in a mirror. It also happens when two people use a clearboard. A right handed coordinate system drawn on one side becomes a left handed coordinate system when viewed from the other side (when viewed through the looking glass, to use a literary reference). Distance and displacement will not be affected by the reflection of an axis. There is no special side of the looking glass for measuring distance and displacement. Space is orthomorphic. (I believe that is the correct term.)

No matter how you translate, rotate, or reflect the coordinate system, the values of Δs and Δr are unaffected. The essence of the diagram remains the same. All properly formulated physical laws work this way.


The SI unit of distance and displacement is the meter [m]. A meter is a little bit longer than the distance between the tip of the nose to the end of the farthest finger on the outstretched hand of a typical adult male. Originally defined as one ten millionth of the distance from the equator to the north pole as measured through Paris (so that the Earth's circumference would be 40 million meters); then the length of a precisely cut metal bar kept in a vault outside of Paris; then a certain number of wavelengths of a particular type of light. The meter is now defined in terms of the speed of light. One meter is the distance light (or any other electromagnetic radiation of any wavelength) travels through a vacuum after 1/299,792,458 of a second.

Multiples (like km for road distances) and divisions (like cm for paper sizes) are also commonly used in science.

There are also several natural units that are used in astronomy and space science.