Describing linear motion with graphs
Linear motion is movement along a straight line. Two graphs help us describe it: the displacement–time (s–t) graph and the velocity–time (v–t) graph. Reading the shape of each line tells us how an object is moving.
Displacement–time graph
Time is on the horizontal axis and displacement on the vertical axis. The gradient (slope) equals the velocity.
- A horizontal line means the displacement is not changing – the object is at rest.
- A straight sloping line means uniform (constant) velocity.
- A curved line means the velocity is changing – the object is accelerating.
Key idea
On an s–t graph: gradient = velocity = displacement ÷ time. On a v–t graph: gradient = acceleration, and the area under the line = displacement.
Velocity–time graph
Time is on the horizontal axis and velocity on the vertical axis.
- The gradient equals the acceleration.
- A horizontal line (above zero) means constant velocity and zero acceleration.
- A line sloping upward shows acceleration; sloping downward shows deceleration.
- The area under the line equals the displacement.
Curved lines carry extra meaning. On a velocity–time graph, a line that curves upward more and more steeply shows an acceleration that is increasing, while a curve that flattens out shows an acceleration that is falling. To find the displacement for a straight-line v–t graph, we usually split the area into simple rectangles and triangles and add them together.
Example
A car on the North–South Expressway moves 20 m in 4 s at steady speed. On the s–t graph the gradient is 20 ÷ 4 = 5 m s⁻¹. If instead its velocity rises from 0 to 20 m s⁻¹ in 5 s, the acceleration is 20 ÷ 5 = 4 m s⁻².
Remember
Do not mix up the two graphs. A flat line on an s–t graph means at rest, but a flat line on a v–t graph means moving at constant velocity.