🛑 Horizontal line
The distance is not changing as time passes, so the object is stationary. The skateboarder stopped between 10 s and 20 s.
Movement and Position » Distance-Time Graphs
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 1.3
- How to plot a distance-time graph from a table of data, with sensible scales
- What a horizontal line, a straight sloping line, a steeper line and a curve each tell you
- How to find a speed from the gradient of a straight section
- How to describe a whole journey in words from its graph
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Unlock This CourseWhat is a distance-time graph?
A table of numbers can record a journey, but it is hard to picture. A distance-time graph turns the same numbers into a line, and the shape of the line shows at a glance how the object moved.
Time always goes along the bottom (the horizontal axis). Distance always goes up the side (the vertical axis). Each point on the line answers one question: how far had the object moved by this time?
Key terms:
- Distance-time graph: a graph with time on the horizontal axis and distance moved on the vertical axis, showing how the distance changes during a journey.
- Stationary: not moving. The distance stays the same as time passes.
- Constant speed: a steady speed. The object moves the same distance in every second.
- Gradient: how steep a line is. It is the change up the graph divided by the change along the graph.
Plotting a graph from a table
A girl rides a skateboard along a straight path. A friend records how far she has moved every 5 seconds.
| Time (s) | 0 | 5 | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|---|
| Distance (m) | 0 | 15 | 30 | 30 | 30 | 55 | 80 |
Follow these steps to plot the graph. They are the same steps for any table.
- Draw the axes with a ruler. Time goes along the bottom and distance goes up the side.
- Label each axis with its quantity and unit: "Time (s)" and "Distance (m)".
- Choose the scales. Look at the largest value in each row: 30 s and 80 m. Pick scales that go up in equal steps and let the points fill more than half of the grid in each direction. Here, 5 s for each large square along the bottom and 10 m for each large square up the side work well.
- Plot each pair of values as a small, neat cross. For the pair (5 s, 15 m), go along to 5 s, then up to 15 m.
- Join the points. Where the points lie in a straight line, join them with a ruler. Where they follow a bend, draw one smooth curve through them.
Choosing a good scale
Make each square worth an easy number such as 1, 2, 5 or 10. Scales that go up in 3s or 7s make points very hard to plot and to read. The scale must go up evenly: the gap from 0 to 10 must be the same size as the gap from 10 to 20. The two axes do not need the same scale as each other.
For the skateboard data you get three straight sections. The line rises steadily from (0, 0) to (10, 30). It is then flat from (10, 30) to (20, 30). Finally it rises more steeply from (20, 30) to (30, 80).
What the shape of the line tells you
You can describe a journey just by looking at the shape of each part of the line.
🚶 Straight sloping line
The distance goes up by the same amount every second, so the object is moving at a constant speed.
🚀 Steeper line
More distance is covered in each second, so the object is moving faster. The skateboarder was faster after her stop than before it.
📈 Curve
The steepness is changing, so the speed is changing. A curve that gets steeper shows speeding up. A curve that gets less steep shows slowing down.
Put the parts together and you can describe a whole journey in words. For the skateboarder: she moves at a constant speed for the first 10 s, is stationary from 10 s to 20 s, then moves at a faster constant speed from 20 s to 30 s.
Here is the data for a sledge sliding from rest down a snowy hill.
| Time (s) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Distance (m) | 0 | 1 | 4 | 9 | 16 |
You can spot the changing speed in the table before you plot anything. The sledge moves 1 m in the first second, then 3 m, then 5 m, then 7 m. It covers more distance in each second, so it is speeding up. When plotted, the points do not fit a straight line. They make a smooth curve that starts almost flat and gets steeper and steeper.
The same check works for any table. Equal distances in equal times means constant speed and a straight line. No change in distance means stationary and a flat line.
Finding speed from the gradient
The gradient of a distance-time graph is the speed. This is because the gradient is a distance divided by a time, which is exactly what speed is.
Speed from a straight section
speed = gradient = change in distance ÷ change in time
Pick two points on the straight section. Subtract the distances, subtract the times, then divide.
Worked example 1: the skateboarder
First section (0 s to 10 s):
change in distance = 30 − 0 = 30 m
change in time = 10 − 0 = 10 s
speed = 30 ÷ 10 = 3 m/s
Last section (20 s to 30 s):
change in distance = 80 − 30 = 50 m
change in time = 30 − 20 = 10 s
speed = 50 ÷ 10 = 5 m/s
The steeper section gives the bigger speed, as expected. The flat section has a gradient of zero, so the speed there is 0 m/s.
Worked example 2: reading points from a line
The distance-time graph for a ferry is one straight line. It passes through the points (40 s, 320 m) and (160 s, 1280 m). Calculate the speed of the ferry.
change in distance = 1280 − 320 = 960 m
change in time = 160 − 40 = 120 s
speed = 960 ÷ 120 = 8 m/s
The two points are far apart on the line. A large gap makes any small error in reading the graph matter less.
The gradient of one section gives the speed for that section only. For the average speed of a whole journey, use the equation from the lesson Speed, Distance and Time with the total distance and the total time. For the skateboarder that is 80 ÷ 30 = 2.7 m/s. It is lower than both of her moving speeds because the total time includes her stop.
Common mistakes
- Thinking a flat line means constant speed. On a distance-time graph a flat line means the object is not moving at all.
- Reading one point instead of a change. For the skateboarder's last section, 80 ÷ 30 is wrong. Use the change in distance (50 m) and the change in time (10 s).
- Calling the line a hill. A steep line does not mean the object is going uphill. It only means it is moving fast.
- Uneven scales. Writing the table values along the axis as they come (0, 15, 30, 55, 80) gives an uneven scale and the wrong shape.
- Joining a curve with a ruler. Points that follow a bend need one smooth curve, not a set of straight lines.
- Missing labels and units. Both axes need a quantity and a unit.
Exam-style question
A tram travels along a straight track. The table shows the distance it has moved at different times.
| Time (s) | 0 | 20 | 40 | 60 | 80 | 100 |
|---|---|---|---|---|---|---|
| Distance (m) | 0 | 200 | 400 | 400 | 400 | 700 |
A student plots a distance-time graph using this data and joins the points with straight lines.
(a) Describe the motion of the tram between 40 s and 80 s. [1 mark]
(b) Use the gradient of the graph to calculate the speed of the tram during the first 40 s. [3 marks]
(c) Without doing a calculation, state in which part of the journey the tram is moving fastest. Explain how the graph shows this. [2 marks]
(d) Calculate the average speed of the tram for the whole 100 s. [2 marks]
Model answer
(a) The tram is stationary. (1)
(b) change in distance = 400 − 0 = 400 m and change in time = 40 − 0 = 40 s (1)
speed = gradient = 400 ÷ 40 (1)
speed = 10 m/s (1)
(c) Between 80 s and 100 s. (1) The line is steepest in this part, so it has the largest gradient. (1)
(d) average speed = distance moved ÷ time taken = 700 ÷ 100 (1)
average speed = 7 m/s (1)
Exam tip
Part (c) says "without doing a calculation", so the marks are for linking steepness to speed, not for numbers. When a question asks how the graph shows something, always name the feature of the line: flat, straight, steeper or curved.