The gradient of a velocity–time graph tells us whether the object’s velocity has been changing at a high rate or a low rate, or not at all (Figure 2.4). We can deduce the value of the acceleration from the gradient of the graph:

acceleration = gradient of velocity–time graph

The graph (Figure 2.5) shows how the velocity of a cyclist changed during the start of a sprint race. We can find his acceleration during the first section of the graph (where the line is straight) using the triangle as shown.
The change in velocity $\Delta v$ is given by the vertical side of the triangle. The time taken $\Delta t$ is given by the horizontal side.

$\begin{array}{l}
acceleration = \frac{{change\,in\,velocity}}{{time\,taken}}\\
 = \frac{{25 - 0}}{5}\\
 = 4.0\,m\,{s^{ - 2}}
\end{array}$

A more complex example where the velocity–time graph is curved is shown in Figure 2.18.

Figure 2.4: The gradient of a velocity–time graph is equal to acceleration.