A graph turns a journey into a picture you can read at a glance. On a distance-time graph the gradient is the speed; on a speed-time graph the gradient is the acceleration and the area underneath is the distance. Confusing those two graphs is where marks easily slip away.
On a distance-time graph, the gradient (slope) is the speed. On a speed-time graph, the gradient is the acceleration and the area under the line is the distance travelled.
For a speed that changes, you cannot just multiply the final speed by the total time. You must find the actual area under the line, splitting it into triangles and rectangles. Adjust the asymmetrical journey below and compare the correct area with that tempting shortcut.
| On the graph | Distance-time graph | Speed-time graph |
|---|---|---|
| Gradient | the speed | the acceleration |
| Horizontal line | stationary (not moving) | constant speed (no acceleration) |
| Area under line | has no physical meaning | the distance travelled |
| Curve getting steeper | speeding up | increasing acceleration |
When the speed is changing, the distance is not the final speed multiplied by the final time. That treats the whole graph as one massive rectangle. The distance is the area under the line, found by splitting it into triangles and rectangles and adding them. Use area = ½ × base × height for each triangle and length × width for each rectangle.
A car accelerates from rest to 20 m/s in 8.0 s, holds 20 m/s for 12 s, then brakes to rest in 5.0 s. Find the total distance travelled.