Velocity-Time Graphs — An Examiner's Guide

Velocity-time graphs sit at the boundary between Maths and Physics — they appear on both syllabi, and the GCSE Maths Higher version is usually worth six to nine marks. The two skills you need are simple to state but easy to mix up: gradient gives acceleration; area under the graph gives distance. This guide walks through both, plus the worded scenarios that come up most often.

The two key facts

That's the whole topic, distilled. Everything else is geometry.

Reading acceleration from the gradient

Pick two points on a straight section of the graph. Acceleration = change in velocity / change in time.

If the line slopes downwards, the gradient is negative — that's deceleration (or "negative acceleration"). If the line is horizontal, the gradient is 0 — the object is moving at constant velocity (no acceleration).

Reading distance from the area

The distance travelled in any time interval equals the area under the graph in that interval. Break the area into rectangles, triangles and trapeziums.

The trapezium shortcut

Many velocity-time questions involve a trapezium-shaped region (object accelerates from one speed to another over some time). Use the trapezium area formula:

Where a and b are the parallel sides (the two velocities) and h is the time.

Average speed across the whole journey

For the cyclist above: 84 / 15 = 5.6 m/s.

Curved velocity-time graphs (Higher tier)

If the graph is curved instead of made of straight lines:

• Acceleration at a point = gradient of the tangent at that point.
• Distance = area under the curve, estimated using the trapezium rule (split the area into thin trapeziums and add).

For more detail on these techniques, see gradients of curves and area under curves.

The mistakes that cost the most marks