Velocity

Chapter 3 ended on a cliff. An address, x, is a single frozen frame, and the story only begins when you ask how the frame changes. This chapter asks it – and notices, first, that everyday life already has a word waiting. The word is speed, and it is the number on a car’s speedometer. The trouble is that the speedometer reads 50 whether you are driving toward home or away from it. It answers “how fast?” and politely refuses to hear “which way?”

In physics, direction almost always matters – whether the ball is rising or falling, whether the planet is approaching or receding – so speed has to graduate. The graduated version is called velocity: how fast, and which way, folded into one signed number. The tuition fee is a single minus sign, borrowed straight from chapter 3, and the reward is the primer’s biggest idea, waiting in the figure below: a motion and its graph, becoming two views of the same thing.

Velocity is a rate, and a rate is a kind of sentence: “this much change, for every unit of waiting.” A velocity of 3 m/s says that every second you let pass, the address x grows by three metres. Read the unit exactly as it is written – metres, per second – and it stops being notation and becomes the sentence saying itself. Wait two seconds and you expect six metres of change; wait half a second, one and a half. The rate is the exchange rate between seconds spent and metres moved.

To measure one, run chapter 2’s recipe from its other side. There, v = d ÷ t was one of the three rearrangements of d = v × t; here it becomes the definition: velocity is the displacement divided by the time it took. And notice which quantity sits on top – the displacement, chapter 3’s signed arrow, not the odometer’s count. Walk +6 m of displacement in two seconds and your velocity was +3 m/s, whatever wandering route the odometer billed you for on the way.

Because displacement carries a sign, velocity inherits one, and it means exactly what it meant in chapter 3: a direction, nothing more. A car at +3 m/s and a car at −3 m/s are doing the same work – same engine note, same tyre wear, same fuel bill – one heading up the street’s positive side, the other down its negative side. The minus sign is still a pointing finger; it has simply climbed aboard something that moves.

Strip the sign off a velocity and what remains is speed. Sometimes that is precisely what you want – a speed camera does not care which way you were going too fast – but more often the sign was the valuable part, and dropping it throws information away. The courses ahead keep the two words distinct on purpose: velocity when the direction is part of the answer, speed when it honestly is not. Get that habit here and a hundred later sentences parse themselves.

Now for the picture this chapter exists to give you. In the figure below, a dot drives along a track – that is the motion itself, the thing your eyes would see. Underneath, a pen writes the same story onto graph paper: time marching across the page, position read off upward. Press drive and watch both at once. A steady velocity writes a perfectly straight line. Set the slider faster and the line comes out steeper; set it negative and the line runs downhill, because the address is shrinking while time goes on growing.

Then do the thing the figure is really for: drag the slider mid-run. The line bends at exactly that instant, and the dashed ghost ahead of the pen shows the line your new setting is about to write. This is the crucial reading lesson: the graph is not a picture of the road. The dot never leaves its straight track. The graph is a diary of the trip – when you were where – and learning to flick at will between the two views, motion and diary, is the single most useful skill this primer teaches. Every course ahead assumes you have it.

Everything the figure showed compresses into one sentence, and it is the chapter’s theorem: however much the line climbs in one second of graph paper, that is the velocity. Always. A line that gains two metres each second is a motion at +2 m/s; a line that drops four metres each second is −4 m/s; a flat line gains nothing and is standing still. Steepness even wears the right name – the slope of the line – so the theorem fits in three words: slope is velocity.

One quiet promise before moving on. You found the slope here by comparing two moments a full second apart. Mathematics has a way of asking the same question of a single instant – how steep is the diary right now, even mid-bend? – and its name is calculus. Nothing in this catalogue will spring it on you unprepared; when it matters, it will be built the way this figure was, slowly and on screen. For now it is enough to know that the seed in your hand grows into that tree. Next: what happens when the slope itself starts changing.