The distance travelled is the area between the speed-time line and the time axis. Split the shape into triangles, rectangles and trapeziums, find each area, and add them.
This lesson follows reading speed-time graphs in the SPM Mathematics motion graphs chapter.
How do you split the area?
Mark each point where the line changes direction and drop a vertical line to the time axis. The pieces are simple shapes.
- Triangle: ½ × base × height
- Rectangle: width × height
- Trapezium: ½ × (sum of parallel sides) × distance between them
The parallel sides can be vertical (two speeds) or horizontal (two time lengths), depending on how the shape lies.
Worked example: a school bus
A school bus has points (0, 0), (10, 15), (30, 15) and (40, 0) on a graph of speed in m/s against time in seconds.
| Part | Shape | Working | Area |
|---|---|---|---|
| 0 to 10 s | Triangle | ½ × 10 × 15 | 75 |
| 10 to 30 s | Rectangle | 20 × 15 | 300 |
| 30 to 40 s | Triangle | ½ × 10 × 15 | 75 |
The total distance is 75 + 300 + 75 = 450 m.
Trapezium check. The whole shape is a trapezium lying on its side. Its parallel sides are the top edge (20 s long) and the bottom edge (40 s long), and its height is 15.
Area = ½ × (20 + 40) × 15 = ½ × 60 × 15 = 450 m. Both methods agree.
The mistake that costs marks
The common slip is to multiply the top speed by the total time: 15 × 40 = 600 m. That treats the bus as moving at 15 m/s the whole time, although it started at rest and ended at rest.
| Wrong | Right | |
|---|---|---|
| Method | Top speed × total time | Add the areas |
| Answer | 600 m | 450 m |
| Why | Ignores speeding up and slowing | Uses the real shape |
The wrong answer is always too large here, because it fills a rectangle bigger than the graph.
Check yourself
A cyclist has a graph with points (0, 4), (5, 4), (9, 12) and (12, 0), with speed in m/s and time in seconds. Find the total distance.
Answer
From 0 to 5 s, the speed is constant at 4. Area = 5 × 4 = 20 m.
From 5 to 9 s, the speed rises from 4 to 12. This is a trapezium with parallel sides 4 and 12 and width 4. Area = ½ × (4 + 12) × 4 = 32 m.
From 9 to 12 s, the speed falls from 12 to 0. Area = ½ × 3 × 12 = 18 m.
Total distance = 20 + 32 + 18 = 70 m.
What to study next
Keep the gradient and the area separate in your mind. The next lesson, distinguishing gradient from area in motion graphs, trains that choice. Try changing a graph in the motion-graph explorer, and record slips in the mistake log and paper-error review.
If you want a teacher to check how you split the shape, see online one-to-one Mathematics tuition.