The big idea: Watch a car pull away from the lights. That one journey can be drawn three different ways — how far it has gone, how fast it is going, and how hard it is accelerating.
The three graphs: s–t (displacement–time) — how far from the start.
v–t (velocity–time) — how fast, and which way.
a–t (acceleration–time) — how quickly the velocity is changing.
One journey — a car pulling away from the lights at 5 m s⁻² for 4 s — drawn three ways. Same motion, three completely different shapes.
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The one rule that links them: Gradient takes you DOWN the list (s → v → a).
Area takes you back UP (a → v → s).
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Every motion-graph question is really one of two moves: take a gradient, or take an area. Which one depends on the graph you're looking at.
| Graph | Its gradient gives | The area under it gives |
|---|---|---|
| s–t (displacement–time) | velocity | nothing useful |
| v–t (velocity–time) | acceleration | displacement |
| a–t (acceleration–time) | (not required) | change in velocity |
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Read the shape first: Before you calculate anything, look at the shape — it tells you the story of the motion.
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Now the same for a v–t graph: On a velocity–time graph the vertical axis is the velocity itself — so above the axis means forwards, below means backwards, and the area is the displacement.
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A runner's displacement–time graph is a straight line passing through (0 s, 0 m) and (5.0 s, 40 m). Determine her velocity, and state what a horizontal section of the graph would mean.
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How this is tested — graph questions are everywhere in physics. They come in two flavours:
Paper 1A
- 'Which graph shows the same motion?'
- Match an s–t to its v–t (or a–t).
Paper 2
- Calculate from a gradient or an area.
- Describe the motion in words.
The classic traps: Flat s–t means STOPPED — it's a flat v–t that means steady speed. Students mix these two up constantly.
The gradient of an s–t graph is the velocity, not the acceleration.
There is no useful area under an s–t graph.
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A cyclist's displacement–time graph is horizontal for the first 10 s, then a straight slope for the next 20 s, then a curve that gets steeper for the final 10 s. Describe the motion in each stage.
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A bouncing ball on a v–t graph: Between bounces only gravity acts, so every sloping part has the same gradient. At each bounce the velocity flips sign almost at once.
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