Position and velocity

Before we can explain why anything moves, we have to describe motion cleanly: where a thing is, how fast it is going, and which way. Get those three straight and half of the confusion in physics disappears.

You track motion all the time. Walking to a shop, you sense both how far you have come and how quickly. A car's speedometer reads one number while the odometer counts another. A little dot slides across a map app, and you read its progress at a glance. Each of these is quietly separating ideas that beginners tend to smear together. This lesson pulls them apart. We are only describing motion here, not yet asking what causes it.

Play first: watch position and its slope

Here is a car on a track with two live graphs: its position over time, and its velocity over time. Drag the speed slider and watch the position line tilt. Then scrub the clock and see the car crawl along. Notice one thing before we name it: a faster speed makes the position line steeper.

Position, distance, speed, velocity

Start with the simplest picture: a thing moving along a straight line, like a train on a track. Its position is just where it is, a distance from some agreed starting mark. Watch the position change and two different questions appear. How far did it travel in total? That is distance. How quickly is the position changing? That is speed, distance divided by time.

Speed alone still hides something. Walk three steps east, then three steps west, and your speed was never zero, yet you ended exactly where you began. Direction clearly matters. Speed with a direction attached is velocity. Two cars at the same speed heading opposite ways have opposite velocities, and that difference is the whole reason they drift apart.

Velocity is the slope of the position graph

Plot position on the vertical axis and time on the horizontal, exactly the top graph in the widget. A thing sitting still traces a flat line. A thing moving steadily traces a straight, tilted line, and here is the key link: how steeply that line rises is its velocity. Cover more ground in the same time and the line climbs faster, so a steeper position graph means a faster velocity.

The Greek letter $\Delta$ just means "the change in." So $\Delta x/\Delta t$ reads "change in position divided by change in time," which is rise over run, which is slope. If you want the fully sharpened version, where the time interval shrinks to an instant, that is the derivative, and the derivatives lesson builds it carefully. It is optional deepening, not a prerequisite: slope of position is all you need right now.

Position and velocity, the way code sees them

A moving thing, to a computer, is not a smooth line. It is an array of position samples, one reading every so often, like a GPS trace or the frames of a video. Velocity is then the discrete derivative: the difference between each sample and the next. Steady steps in position mean a steady velocity.

xs = [0.0, 1.5, 3.0, 4.5, 6.0]      # position each second, in metres
vs = [xs[i + 1] - xs[i] for i in range(len(xs) - 1)]
# vs == [1.5, 1.5, 1.5, 1.5]  ->  a steady 1.5 m/s

# The reverse trip - rebuilding position by adding velocities back up - is
# integration, the subject of the simulating-motion lesson.

That reverse trip, adding up velocities to recover position, is integration, and it is exactly how a game or a physics engine moves a character forward one frame at a time. We build it in simulating motion.

Check yourself