Read a distance-time graph and calculate speed at different intervals

Question

A distance-time graph shows the journey of a cyclist. From the graph:

0 to 2 hours: distance increases from 0 to 30 km (straight line)
2 to 3 hours: horizontal line at 30 km (rest stop)
3 to 5 hours: distance increases from 30 km to 70 km (straight line)

Calculate the speed during each interval and the average speed for the entire journey.

Solution — Step by Step

On a distance-time graph:

The y-axis shows distance (km)
The x-axis shows time (hours)
Speed = slope of the graph = rise/run = change in distance / change in time

A steep slope = fast speed. A gentle slope = slow speed. A horizontal line = at rest (speed = 0).

From the graph:

Distance at $t = 0$ : $0$ km
Distance at $t = 2$ hours: $30$ km
Change in distance: $30 - 0 = 30$ km
Change in time: $2 - 0 = 2$ hours

\text{Speed}_1 = \frac{\Delta d}{\Delta t} = \frac{30}{2} = 15 \text{ km/h}

From the graph:

The line is horizontal — distance stays at 30 km
Change in distance: $30 - 30 = 0$ km
Change in time: $3 - 2 = 1$ hour

\text{Speed}_2 = \frac{0}{1} = 0 \text{ km/h}

The cyclist is at rest (resting or stopped). The horizontal line always represents zero speed.

From the graph:

Distance at $t = 3$ hours: $30$ km
Distance at $t = 5$ hours: $70$ km
Change in distance: $70 - 30 = 40$ km
Change in time: $5 - 3 = 2$ hours

\text{Speed}_3 = \frac{40}{2} = 20 \text{ km/h}

The cyclist is faster in this interval than in the first interval (20 km/h vs 15 km/h) — shown by the steeper line on the graph.

\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}

Total distance = 70 km (from 0 to 70 km).

Total time = 5 hours (from $t = 0$ to $t = 5$ ).

\text{Average speed} = \frac{70}{5} = 14 \text{ km/h}

Note: Average speed is NOT the average of the three speeds (0, 15, 20). It must be calculated from total distance and total time.

Why This Works

The slope of a distance-time graph gives speed because speed is defined as distance per unit time. The slope formula is $\text{rise}/\text{run}$ = $\Delta d / \Delta t$ , which is exactly the speed formula.

This is a fundamental graph-reading skill: recognise that steepness encodes information (here, speed). A flat line encodes zero of whatever the slope represents.

Alternative Method

For a uniform speed (straight line segment), you can also read off any two points on that segment and use the formula:

\text{Speed} = \frac{d_2 - d_1}{t_2 - t_1}

Any two points on the segment work — you don’t have to use the endpoints. All points on a straight-line segment give the same slope (same speed), which is why the line is straight for uniform motion.

Common Mistake

Students sometimes calculate average speed as the average of the individual speeds: $(15 + 0 + 20)/3 = 11.7$ km/h. This is wrong. Average speed = total distance / total time = 70/5 = 14 km/h. The “average of speeds” formula only works if each speed is maintained for the same duration — here they’re not (1, 2, and 2 hours respectively).

Practice reading values from graph axes carefully. In exams, graphs often have irregularly spaced axis markings. Before calculating, read off the exact coordinate values for both $d$ and $t$ at the start and end of each interval. Write down the coordinates as $(t, d)$ pairs before applying the formula — this avoids confusion.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next