Example. How far does the earth travel in one year?
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.
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.
Distance (solid) and Displacement (dashed)
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. Medicine, it seems, would be without vocabulary were it not for this “dead” tongue — cardiac, referring to the heart; podiatry, the treatment of the feet; dentistry, the treatment of the teeth; etc. Examples are less common in physics, but they are there nonetheless. (There seem to be more Greek than Latin words in physics.)
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.
“Spatium transactum tempore longiori in eodem Motu aequabili maius esse spatio transacto tempore breuiori..”
For the same motion, with all other factors being equal, the distance traversed in a longer span of time is greater than the distance traversed in a shorter span of time.
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. 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 th of a second.
Multiples of the metre (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.
Speed
What’s the difference between two identical objects traveling at different speeds? Nearly everyone knows that the one moving faster (the one with the greater speed) will go farther than the one moving slower in the same amount of time. Either that or they’ll tell you that the one moving faster will get where it’s going before the slower one. Whatever speed is, it involves both distance and time. “Faster” means either “farther” (greater distance) or “sooner” (less time). Doubling one’s speed would mean doubling one’s distance traveled in a given amount of time. Doubling one’s speed would also mean halving the time required to travel a given distance. If you know a little about mathematics, these statements are meaningful and useful. (The symbol v is used for speed because of the association between speed and velocity, which will be discussed shortly).
Speed is directly proportional to distance when time is constant: v ~ s (t constant)
Speed is inversely proportional to time when distance is constant: v ~ 1/t (s constant)
Combining these two rules together gives the definition of speed in symbolic form.
Don’t like symbols? Well then, here’s another way to define speed.
Speed is the rate of change of distance with time.
In order to calculate the speed of an object we must know how far it’s gone and how long it took to get there. “Farther” and “sooner” correspond to “faster”. Let’s say you drove a car from New York to Boston. The distance by road is roughly 300 km (200 miles). If the trip takes four hours, what was your speed? Applying the formula above gives …
This is the answer the equation gives us, but how right is it? Was 75 kph the speed of the car? Yes, of course it was … Well, maybe, I guess … No, it couldn’t have been the speed. Unless you live in a world where cars have some kind of exceptional cruise control and traffic flows in some ideal manner your speed during this hypothetical journey must certainly vary. Thus, the number calculated above is not the speed of the car, it’s the average speed for the entire journey. In order to emphasize this point, the equation is sometimes modified as follows …
The line over the v indicates an average or a mean and the delta symbols indicate a difference or a change. This is the quantity we calculated for our hypothetical trip.
In contrast, a car’s speedometer shows its instantaneous speed, that is, the speed determined over a very small interval of time — an instant. Ideally this interval should be as close to zero as possible, but in reality we are limited by the sensitivity of our measuring devices. Mentally, however, it is possible imagine calculating average speed over ever smaller time intervals until we have effectively calculated instantaneous speed. This idea is written symbolically as …
or, in the language of calculus speed is the first derivative of distance with respect to time.
On a distance-time graph, speed corresponds to slope and thus the instantaneous speed of an object with non-constant speed can be found from the slope of a line tangent to its curve. We will deal with this in a later section of this chapter.