Discussion

definitions

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	possible answer	standard answer
earth to sun	1 astronomical unit	1.496 × 1011 m
66th to 86th Street in NYC	1 mile	1.609 × 103 m
heel to toe on a man's foot	1 foot	3.048 × 10−1 m

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

how far is it	possible answer	standard answer
International Space Station	ninety minutes per orbit	40,000,000 m
Chicago to Milwaukee	ninety minutes by car	150,000 m
Central Park to Battery Park	ninety minutes on foot	10,000 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.

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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.

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 down river 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.

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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.

symbols

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.

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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 location, r for the location any time after that, and Δr [delta r] for the change in location — 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 location on a path, s for the location on the path any time after that, and Δs [delta s] for the space traversed going from one location 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 — Galileo's Discourses on Two New Sciences in 1640.

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.

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.

Enough with the languages.

One important thing to notice in the diagram above is that the location of the observer does not really matter. You may think that the observer must be located at the origin, but this is not the case. It is merely convenient for the sake of illustration. If the observer were not at the origin, we could always move the origin to the observer. In addition, the x-axis need not be horizontal nor must the y-axis be vertical. No matter how you twist the coordinate system, the essence of the diagram remains unchanged. Distance and displacement are said to be isotropic, that is, they remain unchanged even if the coordinate system undergoes translation or rotation. All properly formulated physical laws must be isotropic.

units

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 thousandth the distance from the equator to the north pole (as measured through Paris); 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.

• A nautical mile is now 1852 m (6080 feet), but was originally defined as one minute of arc of a great circle, or 1/60 of 1/360 of the earth's circumference. Every sixty nautical miles is then one degree of latitude anywhere on earth or one degree of longitude on the equator. This was considered a reasonable unit for use in navigation, which is why this mile is called the nautical mile. The ordinary mile is more precisely known as the statute mile; that is, the mile as defined by statute or law. Use of the nautical mile persists today in shipping, aviation, and at NASA.
• Distances in near outer space are sometimes compared to the radius of the earth: 6.4 × 10⁶ m. Some examples: the planet Mars has ½ the radius of the earth, the size of a geosynchronous orbit is 6.5 earth radii, and the earth-moon separation is about 60 earth radii.
• The mean distance from the earth to the sun is called an astronomical unit: approximately 1.5 × 10¹¹ m. The distance from the sun to Mars is 1.5 AU; from the sun to Jupiter, 5.2 AU; and from the sun to Pluto, 40 AU. The star nearest the sun, Proxima Centauri, is about 270,000 AU away.
• For really large distances, the light year is the unit of choice. A light year is the distance light would travel in a vacuum after one year. It is equal to some nine quadrillion meters (six trillion miles). This unit is described in more detail in the next section.