Position

Ask a small child where something is and you get a beautifully useful answer: “by the lamppost”, “next to the red door”, “over there”. It is genuinely enough to go and find the thing. But physics can do nothing with “over there” – it cannot add it, or graph it, or predict from it what happens next. So physics answers the same question a fussier way. Not “by the lamppost” but x = +3 m: three metres, in the positive direction, from a point we have agreed to call zero. Same information; one version is fit for equations.

That is the whole of this short chapter. Where a thing is turns out to be an address measured from a zero that somebody chose, and once you see that the choosing is yours, the two things that scare newcomers – the minus sign, and the gap between how far you walked and where you ended up – turn out to be the friendliest ideas in the catalogue.

Every address needs a reference point to be measured from, and physics gives it a name: the origin, marked x = 0. The liberating part, the part worth stopping on, is that you get to put the origin wherever it suits you. Timing a race? Zero is the start line. Describing a lift? Zero is the ground floor. Studying the planet’s pull? Zero might be the centre of the Earth. No law of physics prefers one choice, and no experiment can tell which you picked.

That last sentence is quietly one of the deepest in the book. Slide your origin three metres to the left and every address in town changes, yet nothing physical has moved – the walker is exactly where she was, the shop exactly as far away. When something can be relabelled freely without changing what actually happens, physics has learned to pay close attention. This is the first, gentlest taste of an idea that Course III – special relativity – eventually grows into a full theory: that where you measure from is a convenience, and the real physics is whatever survives the choice.

Draw a single straight line through the origin and you have a street with two ends. Physics calls everything on one side positive and everything on the other side negative, and – this is the part worth saying out loud – which side is which is pure convention, settled for convenience and meaning nothing deeper. Right is usually called positive because most of us read left to right, and for no grander reason than that.

So x = −2 m is not a smaller, sadder, or more broken position than x = +2 m. It is the exact same two metres from the lamppost, taken the other way. The minus sign is not a wound; it is a pointing finger. Read a handful of addresses aloud – “plus five, five metres to the right”; “minus three, three metres to the left” – and the sign stops being arithmetic and starts being a direction, which is all it was ever trying to be.

Now the figure earns its keep, because it keeps two honest books about the same walk and they routinely disagree. The first book is the odometer, the number ticking up in a car: it counts every single metre you cover and never asks which way you were facing. The second is the displacement – the arrow drawn straight from where you started to where you finished, sign and all. Walk five metres right and then five metres back, and the odometer proudly reads ten while the displacement reads zero: you are home, having gone nowhere on net.

Both books tell the truth; they just answer different questions. “How much shoe leather did you spend?” is the odometer’s question. “Where are you now, compared with where you began?” is displacement’s, and it is the one physics almost always wants, because the courses ahead care where things end up. Notice that displacement carries a direction baked in – a size and a way, five metres and to the left – while the odometer carries only a size. That distinction, a quantity that includes its direction versus one that does not, is the seed of the vector, an idea the classical course will lean on hard. You met it here first, on a pavement, for the price of a short walk.

For all its usefulness, an address is a strangely forgetful thing. x = +3 m tells you where the walker is and then refuses to say another word: not where she was a moment ago, not which way she is heading, not how quickly. It is a single frame lifted out of a film, frozen and silent. Everything about the motion – the direction, the speed, the whole plot – has been left on the cutting-room floor.

Which raises the obvious next question, and it is the one the rest of the primer is built to answer: how does x change? Line up two frames instead of one, ask how much the address moved between them, and you have taken the first step toward the primer’s biggest idea. That step is the next chapter.