Distance & Displacement

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 1.5 × 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 possible answer standard answer
International Space Station 90 minutes per orbit 40,000,000 m
Chicago to Milwaukee 90 minutes by car 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.

[slide]

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.

[slide]

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.