Key Stage 3 science · Physics – Motion and forces

KS3 speed and distance-time graphs: a guide for parents

In Key Stage 3, children learn to describe motion with numbers. They use the relationship between speed, distance and time to work out average speeds, draw and read distance-time graphs that show the whole story of a journey, and think about relative motion, such as what happens when a train and a car pass each other. It is maths-heavy physics, and confident handling of units and graphs matters more than memorising the formula.

Speed and distance-time graphs, explained

Speed tells you how far something travels in a given time. The relationship is speed = distance ÷ time, and it can be rearranged: distance = speed × time, and time = distance ÷ speed. Speed is measured in units such as metres per second (m/s), kilometres per hour (km/h) or miles per hour (mph). Most journeys are not at a steady speed, so what we calculate from the total distance and the total time is the average speed. A speedometer, by contrast, shows the speed at one instant.

A distance-time graph shows a journey as a picture. Time goes along the bottom and the distance from the start goes up the side. A flat, horizontal line means the distance isn’t changing, so the object has stopped. A straight sloping line means steady speed, and the steeper it is, the faster the object is going. A curved line means the speed is changing. Children learn that the gradient of the line (how much it rises for each step across) is the speed, so they can work out speeds directly from the graph.

The most common problem is that the graph looks like a landscape. A line going up and then flat is not a hill and a plateau; it is a journey and a rest. Teachers spend time getting children to “read the axes first” and to tell the story of a graph in words: “travelled 12 km in the first hour, stopped for half an hour, then travelled another 6 km”. Drawing a graph carefully, with a ruler, labelled axes and sensible scales, is a skill that carries into every science and maths course.

Relative motion is the idea that speed depends on who is watching. When two trains travel side by side at the same speed, each looks stationary to the passengers on the other. A car overtaking a slower train in the same direction gains only the difference in their speeds. Two vehicles heading towards each other close the gap at the sum of their speeds. At this stage the maths is simple addition and subtraction, but the thinking, about choosing a point of view, is new.

Words your child will hear

Speed: How far an object travels in a certain time, found by distance ÷ time.
Average speed: The total distance travelled divided by the total time taken, including any stops.
Distance-time graph: A graph of how far an object is from its start at different times.
Gradient: How steep a line is. On a distance-time graph, the gradient is the speed.
Stationary: Not moving. It shows as a flat line on a distance-time graph.
Relative motion: Motion described from the point of view of another moving object.
m/s: Metres per second, a unit of speed. 1 m/s is 3.6 km/h.

What the National Curriculum says

The programme of study for Key Stage 3 says:

Pupils should be taught about speed and the quantitative relationship between average speed, distance and time (speed = distance ÷ time)
Pupils should be taught about the representation of a journey on a distance-time graph
Pupils should be taught about relative motion: trains and cars passing one another

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A cyclist leaves home. She rides 12 km in the first hour, stops for half an hour, then rides a further 6 km in the next hour. Describe her journey as a distance-time graph, and find her speed in each part and her average speed for the whole trip.

List the points to plot, with time in hours on the bottom axis and distance from home in km up the side. Join them with a ruler.
```
time (h)   distance (km)
0          0
1          12
1.5        12
2.5        18
```
First section: the line slopes upward. Speed = distance ÷ time = 12 km ÷ 1 h = 12 km/h.
Second section: the line is flat, because her distance from home doesn’t change for 0.5 h. She is stationary, so her speed is 0 km/h.
Third section: the line slopes up again, but less steeply. Speed = 6 km ÷ 1 h = 6 km/h. The shallower slope shows that this part is slower than the first.
Average speed uses the whole journey, including the stop: 12 + 6 = 18 km in 1 + 0.5 + 1 = 2.5 hours.
```
18 km ÷ 2.5 h = 7.2 km/h
```

She rides at 12 km/h, rests for half an hour, then rides at 6 km/h. Her average speed over the whole 2.5 hours is 7.2 km/h, lower than the speed of the first section because of the rest and the slower last stage.

Common mistakes and how to help

Reading a distance-time graph as a picture of the route

A line that goes up and then down looks like a hill, so children say the object “climbed” and “came back down”.

What helps: Ask what the two axes measure. A line that goes back down means the object returned towards its start. Have your child walk the journey in the garden to see the graph appear.

Thinking a flat line means the object is moving at a steady speed

On many graphs a flat line means something is carrying on steadily, so children read it as steady movement rather than no movement.

What helps: Say it as a story: time keeps passing but the distance isn’t increasing, so the object has stopped. A steady speed is a straight sloping line.

Finding average speed by averaging two speeds

The mean of the speeds feels natural. A trip at 30 km/h out and 60 km/h back is called 45 km/h.

What helps: Always go back to total distance ÷ total time. For 60 km each way, the outward trip takes 2 h and the return 1 h, so 120 km ÷ 3 h = 40 km/h. The slower part takes longer, so it counts for more.

Mixing units, such as minutes with km/h

The question gives 15 minutes and a speed in km/h, so children multiply 12 × 15 and get 180 km.

What helps: Convert first: 15 min is 0.25 h, so 12 km/h × 0.25 h = 3 km. Write the units next to each number and check they match before calculating.

Adding speeds when vehicles move in the same direction

Children remember that closing speeds add up and apply the rule to every pair of vehicles.

What helps: Draw arrows. Same direction: subtract, so a car at 90 km/h overtaking a train at 80 km/h gains 10 km/h. Opposite directions: add, so the same two vehicles meeting head on close at 170 km/h.

How to help at home

Make your own graph. Mark 5 m intervals along a garden path or a hall with tape or chalk. Time your child walking, running and stopping, note the time at each mark, and plot distance against time. The shapes of the lines should match what happened.
Walk the graph. Draw a simple distance-time graph on paper, with a flat part and two different slopes, and ask your child to act it out on a measured track. Then swap and let them invent a graph for you to walk.
Tell the story. Sketch a graph with a few different sections and ask your child to describe the journey in words, including the speeds. Ask which section is fastest, which is slowest, and how they can tell from the slope alone.
Average speed of a trip. Take a bus or train journey and use the timetable to find how long it takes between two stops, and a map to estimate the distance. Work out the average speed and ask why it is lower than the top speed.
Relative motion spotting. As a passenger on a train or in a car, notice how a vehicle in the next lane seems to creep past slowly when it is only slightly faster than you. Ask how fast it would appear to be moving if it came the other way. Stay seated and let the driver concentrate.

Leads on to

Key Stage 4: Forces, fields, work done and pressure
Key Stage 4: Motion, acceleration and road safety