Remember that whether or not something is moving depends on the state of motion of the observer (Post 16.4). So, when we measure movement, we measure it compared to the movement of an observer. Providing the observer is not changing speed or direction, this does not present any problems, according to Newton’s laws of motion (Post 16.2). If this is the case, we say that the observer defines an inertial frame of reference. We can then restate the Galilean principle of relativity (Post 16.4) as “the laws of motion are the same in all inertial frames of reference”.
In this post we will assume that an object is moving in a straight line. This simplifies the mathematics and means the “speed” and “velocity” are the same (Post 16.2).
Suppose two points, A and B lie on a straight line, a distance Δx apart. If an object moves from A to B in a time Δt, then its average speed is Δx divided by Δt. What is the symbol Δ here for? It’s not essential, but it is conventional to use it when you want to show that you are writing about an interval in something – Δx is an interval in space (along a line) and Δt is an interval in time. I am following the convention here to make it easier if you want to pursue the subject further.
If the average speed does not change in any smaller interval within the time interval, Δt then the average speed will be equal to the true speed. We usually shorten the name “true speed” to speed.
However, the definition of average speed raises two questions:
- What is distance?
- What is time?
We all have an intuitive feeling about the meaning of these words. But can you define them? However hard you try, you will find that you can’t! If you think you have succeeded, you will find that you are simply using another word that you can’t define instead of using the words “distance” or “time”; in other words you will have created a circular argument.
There are three concepts involved in describing motion that we can’t really define:
- Mass (to be discussed in a later post)
- Length (which is the same as the distance between two points)
We overcome this problem by defining the units that we use to measure them. So we decide to measure length in metres and time in seconds.
A metre was originally defined, in the year 1793, to be the distance between two marks on a bar of platinum kept at constant temperature in Paris. However, there are all sorts of reasons why this is not very reliable. For example, over a very long period of time there may be changes in the bar of platinum. The metre is now defined to be 1 650 763.73 times the wavelength of the orange-red light emitted by atoms of krypton-86. For nearly all practical purposes, this gives the same result as the original definition. Its advantage is that it cannot change with time.
In about the year 1 000 the Persian (Iranian) scholar al-Biruni defined a second as 1/(24 × 60 × 60) times the length of a day. Since we define a day as 24 hours, an hour as 60 minutes and a minute as 60 seconds, this is still our usual definition of a second. However, it appears that days are very slowly getting longer. So a second is now defined to be 9 192 631 770 time the time it takes for a well-defined change to occur in an atom of caesium-133. This gives a definition which is very stable but, for all practical purposes, gives the same result as the original.
When we write a length, we need to make clear that it is measured in metres – to do this we use the standard abbreviation m; so that 1 m means “a length of one metre”. Similarly, when we measure a time, we need to make clear that it is measured in seconds – to do this we use the standard abbreviation s; so that 1 s means “one second”. The abbreviations for metre and second are standard, to make sure that everyone understands what you mean, so you shouldn’t invent your own – it’s incorrect to abbreviate 1 second to 1 sec.
However, one metre is very short for measuring the distance between cities. So we usually measure such distances in kilometres (abbreviated to km) where a kilometre is 1000 metres. The metre is very long for measuring distances in and between atoms. So we define a nanometre as 10-9 m (see Post 2016/7 if you can’t remember what this means). There are standard prefixes (like kilo- and nano-) that we can use to measure multiples of units. They are listed below with their standard abbreviations; there are some bigger and smaller ones but you are unlikely to ever meet them.
It is essential that to use exactly the correct abbreviation here. Sometimes you see advertisements for computers in which frequencies are said to be in mHz instead of MHz – they would be impossibly slow computers!
This system of prefixes is used for all scientific units. But for time we can also use the minute (min) and hour (h).
The units described here are called SI units. SI is an abbreviation for the French phrase Système Internationale (international system) and was introduced in 1960. It is now used by scientists throughout the world. But in the USA scientists sometimes use metric units that don’t strictly conform to the SI recommendations. Occasionally engineers in the USA will measure distances in inches. In principle, provided the units that are used are clearly stated, none of this should cause confusion.
Since speed is defined by dividing a distance (measured in m) by a time (measured in s), its units are metres per s, written as m/s, m.s-1 or m s-1. The dot or space in the last two abbreviations is very important because ms-1 is the abbreviation for a reciprocal millisecond (1/ms) which is a measure of frequency (the kilohertz) and not a speed. Don’t worry if you can’t understand everything about the example in the last sentence. The main point is that sometimes conventions are very important. Another important point is that 6 metres is abbreviated to 6 m; 6 ms would be 6 milliseconds.
Even though the idea of speed is simple, it gives us plenty to think about.