Speed–time graphs and distance travelled

On a speed–time graph, height gives speed and the area between the graph and time axis gives distance travelled. The gradient gives rate of change of speed. For motion in one fixed direction this is the acceleration in that direction. A horizontal positive section means constant speed, not rest.

Example

A vehicle speeds up uniformly from \(0\) to \(10\) m/s in \(4\) s, continues at \(10\) m/s for \(6\) s, then slows uniformly to rest in \(5\) s.

The distance is \(\dfrac12(4)(10)+(6)(10)+\dfrac12(5)(10)=105\) m. Average speed is \(\dfrac{105}{15}=7\) m/s.

The initial acceleration is \(\dfrac{10}{4}=2.5\) m/s². During slowing, the gradient is \(\dfrac{0-10}{5}=-2\) m/s², corresponding to a deceleration of magnitude \(2\) m/s² for this fixed-direction motion.

Cumulative distances at \(4,10,15\) s are \(20,80,105\) m. The distance–time graph becomes progressively steeper, then straight, then progressively flatter. A speed graph alone does not show the direction of travel. Convert time units before calculating an area or gradient.