Graphs of Motion · Form 4
Why is the area under a speed–time graph the distance?
Because distance = speed × time, and on a speed–time graph that product is exactly the area between the line and the time axis, so measuring the area gives the distance travelled.
Distance = speed × time
For a steady speed, distance is speed multiplied by time. On a speed–time graph with a horizontal line at 20 m/s for 5 s, that speed and that time are the two sides of a rectangle, and its area (20 × 5 = 100) is the distance in metres.
Multiplying the two axes together is the same thing as finding the area.
Why it still works when speed changes
When the line slopes because the speed is changing, the region becomes a triangle or a trapezium instead of a rectangle, but the same 'speed times time' idea holds and the area still equals the distance. This is why the region under a multi-stage graph can be seen as triangles and rectangles whose areas add up to the total distance.
As long as the vertical axis is speed and the horizontal axis is time, the area carries the units of distance.
Where students go wrong
Many try to find the area under a distance–time graph, which is meaningless, area only becomes distance when the axes are speed and time. The area also gives the total distance travelled, not the final position; a speed–time graph does not by itself tell you whether the object turned back.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Why does the area under a speed–time graph give the distance travelled?
Because distance = speed × time, and on a speed–time graph, speed is the height and time is the base of each strip. Multiplying height by base is exactly finding area, so adding up the areas of the shapes under the line gives total distance.
This works even when speed changes.
How do I find the total distance when the speed–time graph has several straight sections?
Split the graph into simple shapes, triangles, rectangles, trapeziums, for each section, find each shape's area using the correct formula, then add all the areas together. The total area under the whole graph equals the total distance travelled for the journey.
Why can't I just use the gradient formula to find distance from a speed–time graph?
Gradient on a speed–time graph gives acceleration, not distance, that's a common exam mix-up. Distance comes from the area under the graph, while gradient (change in speed ÷ change in time) belongs to distance–time graphs where it gives speed.
Mixing up the two graphs loses easy marks.