A-level Mathematics/Advanced/Basic Mechanics/Space and Time

From Wikibooks, open books for an open world
Jump to: navigation, search

I will ignore special relativity for the time being and consider Newtonian space and time.

Space[edit]

For our purposes Space is the background in which mechanics takes place. Before we can say more than this we require a better understanding on the concepts of co-ordinates.

Co-ordinates[edit]

Co-ordinates are simply a collection of labels. As an example consider telephone numbers in an office building. I can label all of the rooms in the building (assuming they all have phones) by the phone number of the room. This would then be a set of co-ordinates. I could also label the rooms by a room number, this is another set of co-ordinates. Co-ordinates are useful in that they let me specify something by a label (like room number) but they are in no way fundamental, at least from the perspective of modern mechanics. Notice that with the case of rooms in an office there are many ways one could form co-ordinates of the office. If one had a directory of room numbers and phone numbers then one could convert from one set of co-ordinates to another if one should so choose. This is useful because in mechanics sometimes it is better to operate in one set of co-ordinates rather than another. In the case of the office if you are looking for Dave in Room 112 then one wouldn't use the telephone number of 112 to go looking for it. Rooms generally have room numbers on their doors, not their phone numbers. But if you were going to phone Dave you would find his room number rather useless unless you converted to the phone number (via the phone directory). In mechanics we specify position via co-ordinates. That is we assign to each location of interest a label, almost always a number. NEED SOME EXAMPLES HERE. Just as when looking for someone in a building, in mechanics one must pick sensible co-ordinates to get something done. When is one co-ordinate system better than another? This is something you have to learn or intuit via problems, which is why just looking at a text book is a waste of time in mathematics, you will need experience if you are to solve problems in mechanics. Experience will usually allow you to pick, based on the 'I've seen one like this before' principle. We could have more than one number to specify a location via co-ordinates. For example consider a building with many floors. The rooms might be numbered 1, 2, 3 etc. on each floor. If I say to you to go to room 43, you wont be able to unless I tell you what floor it is on. If I say go to room 43 on floor 4 you will be able to find the room in question. We now have two co-ordinates, the room number and the floor number. This is the way most people first meet co-ordinates - on a graph, where the two values indicate the location.

Dimensions[edit]

Dimensionality is difficult to pin down. One simple way to think of dimensionality is to define it as the number of numbers needed to give a label to every point in some small region in a space. In our previous example we could if we wanted just use phone numbers to classify all the rooms, so the space could be thought of as one dimensional (the space in this case is the list of rooms, not the spatial distributions of the rooms, which is something different). Ordinary space is three dimensional, I can classify every position in space using three numbers. An example of a set of three numbers I could use would be my distance from three (distinct, non-colinear) points. I could also classify my position using cartesian co-ordinates, or by my distance from some point and the angle the line from said point to my position to two fidicule directions. In this sense ordinary space is 3 dimensional. Sometimes we examine problems in less dimensions because additional dimensions are superfluous or we can generalise from special cases. This is another clever trick one must learn to become truly proficient at mechanics, know when you can reduce the number of dimensions in a problem, you will correspondingly reduce the complexity of the problem and make a solution much easier to find.

Time[edit]

If one wanted, an entire book on what time is could be written. But we want a simple definition. In short time is defined to make motion look simple, an in a desperate effort to correspond to psychological time. Another way to think about time is to define it empirically. Time is something that is measured by clocks. For the purposes of mechanics we define time to be a parameter which always increases, and which a set of devices known as clocks will agree upon at some given configuration of our system (an instant). Put another more simplistic way, time is something that all good clocks will agree upon at any one instant.