Motion: Displacement, Velocity and Acceleration

$\text{Distance}$ is the total length of the path travelled — a scalar, always positive, that keeps accumulating regardless of direction. $\text{Displacement}$ is the straight-line distance from the starting point to the finishing point, in a specified direction — a vector, which can be zero even after a long journey if you end up back where you started.

The same distinction carries over to $\text{speed}$ (distance ÷ time, a scalar) and $\text{velocity}$ (displacement ÷ time, a vector). A runner completing one lap of a circular track has covered a large distance at a high average speed, but their displacement — and hence their average velocity — is exactly zero.

$\text{Acceleration}$ is the rate of change of velocity — how quickly velocity itself is changing, in metres per second, per second (m s⁻²). Because velocity is a vector, acceleration is too: an object can accelerate by speeding up, slowing down, or simply changing direction at constant speed.

On a $\text{displacement–time graph}$, the gradient at any point gives the instantaneous velocity — a steeper slope means a faster velocity, a horizontal section means the object is stationary, and a negative gradient means it is moving back the way it came.

On a $\text{velocity–time graph}$, the gradient gives acceleration, and — crucially — the area enclosed between the line and the time axis gives displacement. This area interpretation works for any shape of graph, not just constant acceleration, by finding the area of the appropriate geometric shape (or summing several shapes together).