Travel Graphs: Distance-Time and Speed-Time Graphs
Travel graphs show a journey as a picture, and reading them is a core IGCSE Maths skill. This page covers both distance-time graphs, where the gradient of the line is the speed, and speed-time graphs, where the gradient is the acceleration and the area underneath is the distance travelled. Clear worked examples show how to find a speed, handle stops, calculate average speed, and work out distance from the area under a graph, followed by auto-marked practice rooms with fresh questions on every load.
A travel graph is a picture of a journey. The shape of the line tells you how fast something is moving, when it stops, and how far it goes. There are two kinds you need for IGCSE: the distance-time graph and the speed-time graph (sometimes called a velocity-time graph). The skill is always the same idea wearing two hats: a gradient and an area. Read the gradient to find a rate, and read the area to find a total.
The two graphs at a glance
The trick is to keep straight what each axis shows, because the words look similar but the graphs mean very different things.
Distance from the start is on the vertical axis, time on the horizontal axis. The gradient of the line is the speed. A flat line means the object is not moving.
\(\text{speed}=\dfrac{\text{distance}}{\text{time}}\)
Speed is on the vertical axis, time on the horizontal axis. The gradient is the acceleration. The area underneath the line is the distance travelled.
\(\text{acceleration}=\dfrac{\text{change in speed}}{\text{time}}\)
Distance-time: reading the line
On a distance-time graph the line tells a story from left to right:
- A sloping line going up: the object is moving away from the start. The steeper the slope, the faster it goes.
- A horizontal (flat) line: the object is stationary. Time passes but the distance does not change.
- A sloping line coming back down to zero: the object is returning to the start.
To find a speed, pick any straight section and divide the distance it covers by the time it takes. For a whole trip, the average speed uses the total distance over the total time:
\(\text{average speed}=\dfrac{\text{total distance travelled}}{\text{total time taken}}\)
Read the question carefully: some ask you to include a stop in the total time, and some ask you to leave the stops out. The words decide which time you use.
Speed-time: gradient and area
On a speed-time graph a sloping line means the speed is changing. A line sloping up is acceleration; a line sloping down is deceleration (also called retardation). A flat line means a steady speed, so the acceleration is zero.
The area between the line and the time axis gives the distance travelled. Break the area into rectangles and triangles, work out each piece, then add them up:
\(\text{distance}=\text{area under the line}\)
A common shape is the trapezium: speeding up, holding a steady speed, then slowing down. Its area is one triangle, plus a rectangle, plus another triangle.
Gradient gives the rate: speed on a distance-time graph, acceleration on a speed-time graph.
Area under a speed-time graph gives the total distance travelled.
Worked examples
💡 Example 1: speed from a distance-time graph
A ferry leaves a harbour and sails in a straight line, covering \(36\) km in the first \(1.5\) hours at a steady speed. What is its speed?
\(\text{speed}=\dfrac{\text{distance}}{\text{time}}\)
\(\text{speed}=\dfrac{36}{1.5}\)
\(\text{speed}=24\text{ km/h}\)
What is happening?
The line is straight, so the speed is constant. Divide the distance by the time. The gradient of the line and the speed are the same thing.
💡 Example 2: average speed with a stop
A cyclist rides \(30\) km in \(1\) hour, rests for \(30\) minutes, then rides a further \(20\) km in \(1\) hour. Find the average speed for the whole trip, including the rest.
\(\text{total distance}=30+20=50\text{ km}\)
\(\text{total time}=1+0.5+1=2.5\text{ h}\)
\(\text{average speed}=\dfrac{50}{2.5}=20\text{ km/h}\)
What is happening?
Average speed uses totals, not the speed of any one leg. The rest counts because the question says to include it. Change \(30\) minutes to \(0.5\) hours first.
💡 Example 3: acceleration from a speed-time graph
A train speeds up steadily from rest to \(24\) m/s in \(8\) seconds. Find its acceleration.
\(\text{acceleration}=\dfrac{\text{change in speed}}{\text{time}}\)
\(\text{acceleration}=\dfrac{24-0}{8}\)
\(\text{acceleration}=3\text{ m/s}^2\)
What is happening?
Acceleration is the gradient of the speed-time line. The speed rises by \(24\) m/s over \(8\) s. A downward slope would give a negative answer, meaning deceleration.
💡 Example 4: distance as the area under a speed-time graph
A drone accelerates from rest to \(20\) m/s in \(5\) s, holds \(20\) m/s for \(20\) s, then slows steadily to rest in \(5\) s. Find the total distance travelled.
Split the area into a triangle, a rectangle, and a triangle.
\(\text{rise triangle}=\tfrac{1}{2}\times 5\times 20=50\text{ m}\)
\(\text{rectangle}=20\times 20=400\text{ m}\)
\(\text{fall triangle}=\tfrac{1}{2}\times 5\times 20=50\text{ m}\)
\(\text{total distance}=50+400+50=500\text{ m}\)
What is happening?
The distance is the area under the line. The trapezium breaks into three simple shapes. Work out each area, then add them. The units stay in metres because speed is in m/s and time in seconds.
🔑 Key points
- On a distance-time graph the gradient is the speed; a flat line means stationary.
- On a speed-time graph the gradient is the acceleration; the area underneath is the distance.
- Average speed is total distance over total time. Read whether stops should be included.
- Split a speed-time area into rectangles and triangles, then add the pieces.
- Keep units consistent: m/s with seconds and metres, or km/h with hours and kilometres.
⚠️ Common pitfalls
- Confusing the two graphs: a flat line means stopped on a distance-time graph, but steady speed on a speed-time graph.
- Forgetting to convert minutes to hours (or hours to seconds) before dividing.
- Using one leg's speed as the average speed for a whole journey.
- Reading the distance off the top of a speed-time graph instead of finding the area.
- Leaving out a triangle when finding the area under a trapezium.