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By Root 904 0

you have written down mathematical equations that contain them. Now suppose that someone, perhaps a colleague, tells you there is a very special vector, one that has the property that it never changes, no matter what happens to that part of the universe to which it corresponds. Your first reaction might be to express disinterest—if nothing changes then it is hardly likely to be capturing the essence of the matter at hand. Your interest would probably perk up if your colleague told you that the single, special vector is built up by adding together a whole bunch of other vectors, each associated with a different part of the thing you are trying to understand. The various parts of the thing can jiggle around and, as they do so, each of the individual vectors can change, but always in such a way that the sum total of all the vectors adds up to the same unchanging special vector. Incidentally, adding vectors together is easy, and we shall return to it in a moment.

To illustrate just how useful this idea of unchanging vectors can be, let’s think about a very simple task. We want to understand what happens when two billiard balls collide head-on. An example from billiards hardly sounds of earth-shattering significance but physicists quite often pick rather mundane examples like this, not because they can only study such simple phenomena or because they love billiards, but rather because concepts are often easiest to grasp first in simpler examples. Back to billiards: Your colleague explains that you should associate a vector with each ball. The vector should point in the direction of the ball’s motion. The claim is that by adding together the two vectors (one for each ball) we can obtain the special unchanging vector. That means that whatever happens in the collision, we can be sure that the two vectors associated with the balls after the collision will combine to make precisely the same vector as that obtained from the two balls before the collision. This is potentially a very valuable insight. The existence of the special vector severely limits the possible outcomes of the collision. We would be particularly impressed by our colleague’s claim that the “conservation of these vectors” works for every system of things in the whole universe, from colliding billiard balls to the explosion of a star. It will probably come as no surprise to know that physicists don’t go around referring to these as special vectors. Rather they speak of the momentum vector and the conservation of vectors is more commonly known as the conservation of momentum.

FIGURE 10

We have left a couple of points hanging: Just how long are the momentum arrows and exactly how are we to add them together? Adding them together is not hard; the rule is to place all of the arrows that we want to add together end-to-end. The net effect is to define an arrow that links the start of the first arrow in the chain to the tip of the last arrow. Figure 10 shows how it is done for three randomly chosen arrows. The big arrow is the sum of the little ones. The length of a momentum vector is something we can ascertain from experiments, and historically this is how it was arrived at. The concept itself dates back over a thousand years, simply because it is useful. In a crude sense, it expresses the difference between being hit by a tennis ball or an express train when both are traveling at 60 miles per hour. As we have discussed, it is closely related to the speed and, as the previous example illustrates all too vividly, it should also be related to mass. Pre-Einstein, a momentum vector has length that is simply the product of mass and speed. As we have already said, it points in the direction of motion. As an aside, the modern view of momentum as a quantity that is conserved relates to the work of Emmy Noether, as we discussed earlier. Then we learned of the deep connection between the law of conservation of momentum and translational invariance in space. In symbols, the size of the momentum of a particle of mass m moving with a speed υ can be expressed as p = mυ, where