Distance-Time Graphs

A distance-time graph, also called a travel graph, shows a journey. Time goes along the bottom. Distance from a starting place, such as home, goes up the side.

In the exam you are usually asked to read the graph: how far from home someone was at a certain time, or how many minutes they stopped for. These parts are worth 1 mark each.

You may also be asked to work out a speed from the graph, for 2 marks, or to complete the graph from a description of the rest of the journey, for 1 to 3 marks.

• A line sloping up means moving away from home.
• A flat line means stopped: time passes, but the distance from home does not change.
• A line sloping down means coming back towards home. When it reaches 0 km, the person is home.

On this ride, the cyclist is 6 km from home at 09:30. The flat line runs from 09:30 to 10:30, so the stop lasts 1 hour. The cyclist is home again at 11:30.

Before you read anything, work out what one small square is worth on each axis. Count the small squares between two labels, then divide.

For example, if 1 hour is split into 6 small squares, each small square is 60\div6=10 minutes. If 4 km is split into 4 small squares, each small square is 1 km.

To read a value, draw a guideline with a ruler: from the axis to the line, then across or down to the other axis.

Times on the graph are clock times, and an hour has 60 minutes. From 10:45 to 11:20 is 15 minutes to 11:00, then 20 more, so 35 minutes.

The steepness of the line shows the speed. The speed is the gradient of the line: the distance it goes up or down for each hour along.

To work out the speed for one part of a journey, read the distance travelled and the time taken for that part. Then divide:

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

Distance in km and time in hours give a speed in km/h. Miles and hours give miles per hour (mph).

The distance travelled is how far the line goes up or down for that part, not where it ends. A part that goes from 20 km up to 35 km is 35-20=15 km travelled.

If the time is in minutes, scale up to 60 minutes. For example, 5 km in 20 minutes is 5\times3=15 km in 60 minutes, so the speed is 15 km/h. Dividing 5 by 20 gives kilometres per minute, not km/h.

Compare the parts of a journey by how steep they are. The steeper the line, the faster the journey. A flat line has a speed of 0

On the bike ride above, going away covers 6 km in 30 minutes, which is 12 km/h. Coming back covers 6 km in 1 hour, which is 6 km/h. The going-away line is the steeper one.

A line sloping down can be the steepest part too. Sloping down only means the direction is towards home; it does not mean slower.

Some questions describe the rest of a journey and ask you to complete the graph. Start each new part where the last part ends.

• For a stop, draw a flat line for the length of the stop.
• For a part at a steady speed, work out where it ends, then join the two points with a ruled straight line.
• For a journey home, the line must end on the time axis, at 0 km.

To find where a part ends, use one of these:

\text{distance}=\text{speed}\times\text{time}

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

For example, at 40 km/h for 30 minutes, which is half an hour, a car travels 40\div2=20 km.

Then turn the time into small squares. On a grid where each small square is 5 minutes, 20 minutes is 20\div5=4 small squares.

The average speed for a whole journey uses the total distance travelled and the total time, including any stops:

\text{average speed}=\frac{\text{total distance}}{\text{total time}}

For a journey there and back, the total distance is there and back. A walk of 5 km to a shop and 5 km home is 10 km, even though the graph ends at 0 km from home.

Read the total time from the start of the graph to the end. A stop adds time but no distance, so it makes the average speed lower.

• Misreading the scale. Count the small squares between two labels first; on a 10-minute scale, one small square is not 1 minute.
• Dividing time by distance. Speed is distance divided by time, so 6 km in 2 hours is 6\div2=3 km/h.
• Dividing by the minutes for a speed in km/h. Scale up to 60 minutes: 6 km in 20 minutes is 18 km/h, not 0.3
• Using 100 minutes in an hour. An hour has 60 minutes, so 45 minutes is 0.75 hours, not 0.45 hours.
• Using where the line ends instead of how far it goes. For one part, subtract: from 20 km to 35 km is 15 km travelled.
• Leaving out the stop or the way back in an average speed. Use the total distance there and back, and the total time including stops.