From Words to Graphs: Describing Motion With Numbers

Picture a dog fetching a ball thrown straight down a long, flat park path. The ball lands 30 m from where the dog started. The dog runs the full 30 m to reach it, then trots back only 10 m before deciding to stop and chew on it for a while. How far has the dog actually travelled? Adding up every stretch it covered, 30 m out plus 10 m back, gives a total distance travelled of 40 m. But now ask a different question: how far is the dog now from where it originally started? That is only 20 m, since it went 30 m out and came 10 m back, and this second quantity, the straight-line change in position between where you started and where you ended up, is called displacement. Distance and displacement answer genuinely different questions, and in this example they even come out to different numbers, 40 m against 20 m, which shows they cannot simply be two names for the same thing. Distance only needs a number to describe it fully (its SI unit is the metre), but displacement needs a number and a direction, since '20 m back towards the ball' is a very different statement from '20 m further away from it'. Quantities like distance that need only a number are called scalars, and quantities like displacement that need both a number and a direction are called vectors.

You already know average speed, the total distance covered divided by the time taken, but speed on its own never tells you which way something is going, only how quickly ground is being covered. Consider a swimmer completing exactly one lap of a 25 m pool, swimming to the far end and back to the exact same starting point, in a total time of 40 seconds. The total distance covered is genuinely 50 m (25 m there, 25 m back), so the average speed works out to 50 m divided by 40 s, or 1.25 m/s. But the swimmer's displacement for that same lap is zero, since they end up exactly where they started, so a new quantity built from displacement instead of distance, average velocity, defined as displacement divided by time interval, comes out to 0 m/s for that same 40 seconds. Average speed and average velocity share the same SI unit, metres per second, but they are not interchangeable: speed only ever looks at ground covered, while velocity looks at net change in position, and for any motion that doubles back on itself even slightly, like this swimmer's full lap, the two numbers will not match.

Velocity itself can change over time, sometimes gradually, sometimes suddenly, and the quantity that captures how quickly it changes is called acceleration. Average acceleration over a time interval is defined as the change in velocity divided by that time interval: how much the velocity changed, divided by how long that change took. Picture an electric scooter that speeds up from 4 m/s to 10 m/s in exactly 3 seconds while pulling away from a traffic light: the change in velocity is 10 m/s minus 4 m/s, which is 6 m/s, and dividing this by the 3 second time interval gives an average acceleration of 2 m/s², read as 'two metres per second, per second', meaning the scooter's velocity is increasing by about 2 m/s with every second that passes. Acceleration, like displacement and velocity, needs a direction as well as a number to be fully described: if an object's speed is increasing, its acceleration points in the same direction as its velocity, but if it is slowing down instead, the acceleration points opposite to the velocity, working against the motion rather than adding to it. It is worth noticing something that feels a little surprising at first: an object can be moving very fast and still have zero acceleration, as long as its velocity simply is not changing, and equally, an object can have a large acceleration while barely moving at all, right at the very instant it starts from rest. Acceleration is about how quickly velocity is changing, not about how fast something happens to be going at any one moment.

Numbers in a table are useful, but a graph often reveals a motion's true character at a glance. Suppose you track a toy car's position every second as it rolls along a straight track, and plot each pair of readings, time along one axis, position along the other, then join the plotted points together: what you get is a position-time graph. If the toy car moves at a genuinely constant velocity, every point lines up perfectly to form a straight line, and the steeper that line is, the faster the car's velocity, since a steeper line means position is changing more rapidly for every second that passes. This steepness has a proper name: it is called the slope of the graph, and for a position-time graph, slope directly equals velocity, calculated exactly the same way you would calculate average velocity from any two points, by dividing the change in position between two points by the change in time between them. A perfectly horizontal line, one with zero slope, tells you the position is not changing with time at all, meaning the object is simply sitting still, at rest. When the car is speeding up instead of moving at a constant velocity, the graph is no longer a straight line, it curves, growing steeper and steeper as time goes on, since the car is covering more position-distance in each successive second than the one before it.

Velocity-time graphs work the same way, but now plot an object's velocity against time instead of its position, and they reveal a different kind of information. A horizontal line on a velocity-time graph means the velocity is not changing, which means zero acceleration. A straight, sloped line means the velocity is changing at a constant rate, that is, constant acceleration, and the slope of this line, exactly like before, is the rate of change, this time giving you the acceleration directly rather than the velocity. There is a second, less obvious piece of information hiding inside a velocity-time graph: the area enclosed between the plotted line and the time axis, for any chosen time interval, turns out to equal the displacement covered during that same interval. This works because, for constant velocity, that enclosed area is literally a rectangle, velocity multiplied by time, which is exactly the formula for displacement; and for constant acceleration, the enclosed shape becomes a rectangle plus a triangle, whose combined area still correctly works out to the displacement, even though the velocity was not constant throughout. A velocity-time graph, in other words, quietly encodes both acceleration (in its slope) and displacement (in its area) at the same time, using nothing more than a single drawn line.

For the special, common case of motion with constant acceleration in a straight line, the relationships between an object's initial velocity (u), final velocity (v), acceleration (a), time (t), and displacement (s) can all be captured in three compact equations: v = u + at, which finds a final velocity; s = ut + ½at², which finds displacement directly from time; and v² = u² + 2as, which finds a final velocity without needing time at all, useful whenever time was not measured or is not the quantity you need. These are not three independent facts to memorise separately, they can all be derived from one single starting idea, the definition of average acceleration itself, so understanding where the first equation comes from is enough to eventually work out the rest. Picture a delivery drone descending straight down, decelerating steadily from 12 m/s to a gentle landing speed of 2 m/s over a measured 5 seconds: using v = u + at, the acceleration works out to (2 m/s minus 12 m/s) divided by 5 s, which is -2 m/s², the negative sign confirming the drone is slowing down rather than speeding up. Using this same acceleration in s = ut + ½at², the vertical distance it covers during that 5 second descent comes out to (12 x 5) + (½ x -2 x 5²), which is 60 - 25, or 35 m.

Every example so far has involved motion confined to a single straight line, but most real motion is not nearly so restrained: a footballer curving a free kick, a cricket ball's arc through the air, a car overtaking another on a bending road, all of these move across a flat surface in two directions at once, called motion in a plane. The core ideas, displacement, velocity, acceleration, still apply perfectly well, but now direction is not just a plus or minus sign along one line, it is a genuine compass direction across a surface, and this is exactly where the distinction between distance and displacement becomes even more dramatic than it was in a straight line. Picture a park path that loops in a perfect circle: walk all the way around it once, back to your exact starting bench, and the distance you have walked is the full length of the loop, a substantial number, but your displacement for that same walk is exactly zero, since you are back precisely where you began, having gone nowhere at all in the net sense that displacement measures.

Take that circular walking path and add one more condition: walk it at a perfectly steady pace the entire way round, never speeding up or slowing down. Motion along a circular path at constant speed like this has its own name, uniform circular motion, and it holds a small surprise: even though the speed never changes, this motion is still accelerated motion. The reason is that velocity is not just speed, it also includes direction, and on a circular path, direction is constantly changing, every single instant you are heading a slightly different way around the curve, even while your speed stays exactly the same. Since acceleration is defined as any change in velocity, and a changing direction absolutely counts as a change in velocity, uniform circular motion always involves acceleration, continuously, throughout the entire circle, even at a dead-constant speed. For an object taking time T to complete one full lap of a circular path of radius R, the distance covered in that single lap is the circle's full circumference, 2 x pi x R, so the average speed works out to 2πR divided by T; but the displacement for that same complete lap is zero, exactly like the looping park path, since the object returns precisely to its starting point.