Displacement–time graphs
A displacement–time graph is a record of position, not a picture of the road. Its vertical coordinate tells you where the particle is; its gradient tells you how quickly and in which direction that position is changing.
Read velocity from the gradient
Gradient means vertical change divided by horizontal change. The symbol means “change in”, calculated as final value minus initial value.
Symbols
- = velocity over a straight graph segment (m s⁻¹)
- = signed change in displacement (m)
- = change in time (s)
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- Positive gradient: motion in the positive direction.
- Zero gradient at one instant means instantaneous rest. A horizontal interval means the particle stays at rest throughout that interval.
- Negative gradient: motion in the negative direction.
- A steeper graph means a larger speed.
Use a tangent for an instantaneous velocity
A secant joins two points and gives average velocity over an interval. A tangent touches the curve at one instant and gives the instantaneous velocity there.
- Choose two clear tangent points. Use points far apart on the tangent line, not necessarily points on the curve.
- Keep the sign. Calculate signed rise divided by positive run.
- Attach units. Displacement divided by time gives .
Common mistake
Check whether the story is physically possible
- Two position graphs intersect when the particles occupy the same position at the same time.
- A particle can pass through the origin: the graph crosses . It need not stop there.
- Position cannot jump from one value to another in zero time. An impact may change the gradient suddenly, producing a corner, but the position graph remains connected.
A displacement–time graph consists of straight segments joining , , and , with time in seconds and displacement in metres. Find the velocity on each stage and state when the particle is at rest.
Show worked answer
From 0 to 4 s:
From 4 to 7 s:
From 7 to 10 s: