Distance–Time Graphs

A distance–time graph tells the story of a journey. Time runs across the bottom, and distance from a starting point runs up the side. A rising line means you're moving away. A flat line means you've stopped — time keeps passing but your distance doesn't change. A falling line means you're coming back towards the start. The steeper the line, the faster you're going. The one thing to be careful about is the vertical axis: it's distance FROM home, not total distance travelled. That's exactly why the journey home makes the line go down instead of continuing up.

The gradient of the line is the speed. Speed is distance divided by time, and on a graph that's exactly rise over run — so a steeper line means more distance in the same time, which means faster. To find a speed, pick a straight section, read how far it climbed and how long that took, and divide. If a section rises 12 km across 30 minutes, that's 12 divided by half an hour, giving 24 km per hour. The step people lose marks on is units: 30 minutes is 0.5 hours, and if you divide by 30 you'll get an answer that's sixty times too small.

Exam questions often ask you to describe a journey in words. Work section by section and give each one a sentence that says three things: which direction, how fast, and for how long. So section A becomes 'travels away from home at a steady speed for 20 minutes'; B becomes 'stops for 15 minutes'; C becomes 'travels away again, faster than before'; and D becomes 'returns home, arriving after 70 minutes'. Comparing steepness between sections is worth doing even when you aren't asked for numbers — saying a section is faster than the one before it earns marks that 'then it goes up again' does not.

Going the other way — from words to a graph — is easier than it looks if you plot the key points first and join them afterwards. Take 'cycles 6 km in 20 minutes, rests 10 minutes, rides home in 30 minutes'. The points are (0, 0) at the start, (20, 6) at the end of the first leg, (30, 6) at the end of the rest, and (60, 0) back home. Join them with straight lines, because a steady speed always gives a straight line. The mistake to guard against is treating each duration as a fresh start: the times accumulate along the bottom axis, so the rest ends at 30 minutes, not at 10.

Average speed for a whole journey is total distance travelled divided by total time taken — and the stops count as time. You cannot get it by averaging the two speeds, because that ignores how long each leg lasted. For the cycling trip: 6 km out and 6 km back is 12 km travelled, over a total of one hour including the rest, so 12 km/h. Notice that 12 km travelled leaves you 0 km from home, and questions ask for both. 'How far from home' is read straight off the vertical axis; 'distance travelled' means adding up every leg regardless of direction.