A distance-time graph tells the story of a trip. Time runs along the bottom; the distance traveled from the start runs up the side. Its slope, the rise over the run, is meters per second: the speed.
Key idea
On a distance-time graph, slope = speed. A steeper line means faster. A flat line means stopped. A straight line means a steady speed.
Why it works
Over any piece of the graph, the rise is the distance covered and the run is the time it took. Rise ÷ run is distance ÷ time, which is the definition of speed. A flat line has zero rise: time passes but the distance does not change.
Worked example: Read the speed from the slope
A graph goes from (0 s, 0 m) to (4 s, 12 m). What is the speed?
- Rise: $12\,\mathrm{m} - 0\,\mathrm{m} = 12\,\mathrm{m}$.
- Run: $4\,\mathrm{s} - 0\,\mathrm{s} = 4\,\mathrm{s}$.
- Slope: $\dfrac{12\,\mathrm{m}}{4\,\mathrm{s}} = 3\,\mathrm{m/s}$.
Answer: $3\,\mathrm{m/s}$
Worked example: Compare the parts
In the graph above, which part is fastest?
- From 0 s to 4 s: $12 \div 4 = 3$ m/s.
- From 4 s to 7 s the line is flat: 0 m/s, stopped.
- From 7 s to 9 s: $(20 - 12) \div (9 - 7) = 4$ m/s. This part is the steepest.
Answer: The last part, at $4\,\mathrm{m/s}$
Common mistake
Reading the graph as a picture of the path. A line that slopes up does not mean the walker is going uphill, and a flat line does not mean a flat road. The graph only shows how the distance grows with time. A runner on a flat track makes a steep line; a hiker resting on a mountaintop makes a flat one.
Running and cycling apps record your distance every second and draw this graph for you. A curve that gets steeper means speeding up; one that gets flatter means slowing down.