Motion — Speed, Velocity & Acceleration

To describe how something moves, we need to answer several questions: how far has it gone, how fast is it moving, in which direction, and is its speed changing? Physics gives each of these a precise measure, and two graph types that let us see the whole story at a glance.

Speed is a scalar that tells you only how fast something moves:

speed = distance divided by time, written v = d divided by t.

Velocity is a vector that tells you both the speed and the direction, found from displacement divided by time. The distinction is real, not just wording. A car driving at 80 kilometres per hour around a circular track has a constant speed, but its velocity is constantly changing, because its direction keeps turning. Whenever direction changes, velocity changes, even if the speedometer reading stays fixed.

Acceleration is the rate at which velocity changes:

a = (v minus u) divided by t,

where u is the initial velocity, v is the final velocity, and t is the time taken. Acceleration is measured in metres per second squared, which means the velocity changes by so many metres per second during every second. A positive acceleration means speeding up, a negative one, often called deceleration, means slowing down. Changing direction at constant speed also counts as acceleration, because velocity is changing.

A distance-time graph plots position up the side against time along the bottom. A flat horizontal line means the object is stationary. A straight rising line means constant speed, and a steeper line means a faster speed. A curve that steepens shows acceleration, while a curve that flattens shows deceleration. The crucial rule is that the gradient, or slope, of a distance-time graph equals the speed.

A velocity-time graph plots velocity up the side against time along the bottom, and it reveals two things at once. The gradient of a velocity-time graph equals the acceleration, so a rising line means speeding up and a falling line means slowing down. Even more usefully, the area under a velocity-time graph equals the distance travelled. For a rectangular section the area is simply base times height. For a triangular section it is one half times base times height. Splitting a journey into rectangles and triangles lets you total up the distance exactly.

Motion always depends on your point of view. A passenger on a train moving at 100 kilometres per hour sees the platform rush past at 100 kilometres per hour, while a person on the platform sees the train rush past. Neither is wrong. Each description is correct within its own reference frame, and physics is happy with both.

Reading these graphs is a real engineering skill. Formula 1 teams analyse velocity-time data every lap, crash-test engineers track a car panel frame by frame, and air traffic controllers watch each plane's velocity to keep them safely apart.

Students often think a horizontal line on a velocity-time graph means the object is not moving. It actually means constant velocity. Only a horizontal line sitting at zero velocity means the object is truly stationary. A horizontal line at 5 metres per second means steady motion, not rest.