Speed–Time Graphs: Calculating Acceleration

A speed–time graph shows how fast something is moving at different times. The slope (gradient) of the graph tells you the acceleration. Acceleration is how quickly speed changes.

Key idea

Acceleration is change in speed per second. Its unit is metres per second squared, written as m/s².

$$a = \frac{\Delta v}{\Delta t} \quad \text{and on a speed–time graph, } a = \text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in speed}}{\text{change in time}}$$

How to calculate acceleration from a straight line

• Choose two clear points on the straight section of the graph (far apart for accuracy).
• Read their speeds and times carefully from the axes.
• Find change in speed: $\Delta v = v_2 - v_1$.
• Find change in time: $\Delta t = t_2 - t_1$.
• Compute $a = \frac{\Delta v}{\Delta t}$. Include the sign and units (m/s²).

What the sign means

• Positive gradient: speeding up (accelerating).
• Negative gradient: slowing down (decelerating). Deceleration is a negative acceleration.
• Zero gradient (horizontal line): constant speed, so acceleration is 0.

Curved graphs (changing acceleration)

If the line is curved, the acceleration is not constant. To find the acceleration at a particular time, draw a tangent touching the curve at that point and find its gradient. This gives the instantaneous acceleration.

Common mistakes

• Using a distance–time graph by accident (its gradient gives speed, not acceleration).
• Forgetting to subtract times and speeds (use changes: $\Delta v$ and $\Delta t$).
• Dropping the negative sign when the object slows down.
• Missing units or writing m/s instead of m/s².