Distance-time graphs: gradient shows speed
What you will learn
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
On distance-time graphs: gradient = speed
Identify the three stages from the description
Stage 1: 0 to 2 h, distance 0 to 120 km
Calculate speed for the third stage
Distance travelled = 120 − 40 = 80 km
Note the direction of travel
The line slopes downward in stage 3, showing the car is returning toward the start
Watch out
Students read the total distance from the graph rather than the change in distance for the stage
A flat line means the object has stopped.
A car travels 120 km in 2 hours, stops for 30 minutes, then travels 80 km back toward the start in 1 hour. Find the speed during the third stage.
Identify the three stages from the description: Stage 1: 0 to 2 h, distance 0 to 120 km. 5 h, stationary at 120 km.5 h, distance falls from 120 km to 40 km.
Calculate speed for the third stage: Distance travelled = 120 − 40 = 80 km. 5 = 1 hour.Speed = distance/time = = 80 km/h.
Note the direction of travel: The line slopes downward in stage 3, showing the car is returning toward the start.
80 km/h
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
WorkedreasoningA car travels 120 km in 2 hours, stops for 30 minutes, then travels 80 km back toward the start in 1 hour. Find the speed during the third stage.
3 marks4 minsreal-life-graphs-workedShow solution
A car travels 120 km in 2 hours, stops for 30 minutes, then travels 80 km back toward the start in 1 hour. Find the speed during the third stage.
- 1.Identify the three stages from the description: Stage 1: 0 to 2 h, distance 0 to 120 km. 5 h, stationary at 120 km.5 h, distance falls from 120 km to 40 km.
- 2.Calculate speed for the third stage: Distance travelled = 120 − 40 = 80 km. 5 = 1 hour.Speed = distance/time = = 80 km/h.
- 3.Note the direction of travel: The line slopes downward in stage 3, showing the car is returning toward the start.
80 km/h
- M1: identify the three stages from the description
- M1: calculate speed for the third stage
- M1: note the direction of travel
- A1: 80 km/h
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
DiagnosticrecallA distance-time graph has gradient 25. What does this mean?
1 mark2 minsreal-life-graphs-q1Show solution
A distance-time graph has gradient 25. What does this mean?
- 1.Spot the skill: On distance-time graphs: gradient = speed.
- 2.Use the identify the three stages from the description stage first, then calculate speed for the third stage.
- 3.Keep the final answer visible: The object is moving at 25 m/s (or in the units of the graph).
The object is moving at 25 m/s (or in the units of the graph)
- M1: use the correct on distance-time graphs: gradient = speed.on velocity-time graphs: gradient = acceleration, area under graph = distance. a horizontal section means stationary.steeper line means faster speed.
- A1: The object is moving at 25 m/s (or in the units of the graph)
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
EasyprocedureA cyclist is stationary for 10 minutes on a distance-time graph. What does that section look like?
2 marks3 minsreal-life-graphs-q2Show solution
A cyclist is stationary for 10 minutes on a distance-time graph. What does that section look like?
- 1.Spot the skill: On distance-time graphs: gradient = speed.
- 2.Use the calculate speed for the third stage stage first, then note the direction of travel.
- 3.Keep the final answer visible: A horizontal (flat) line.
A horizontal (flat) line
- M1: use the correct on distance-time graphs: gradient = speed.on velocity-time graphs: gradient = acceleration, area under graph = distance. a horizontal section means stationary.steeper line means faster speed.
- A1: A horizontal (flat) line
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
MediumreasoningA velocity-time graph rises from 0 to 30 m/s in 6 seconds. Find the acceleration.
3 marks4 minsreal-life-graphs-q3Show solution
A velocity-time graph rises from 0 to 30 m/s in 6 seconds. Find the acceleration.
- 1.Spot the skill: On distance-time graphs: gradient = speed.
- 2.Use the note the direction of travel stage first, then identify the three stages from the description.
- 3.Keep the final answer visible: 5 m/s2.
5 m/s2
- M1: use the correct on distance-time graphs: gradient = speed.on velocity-time graphs: gradient = acceleration, area under graph = distance. a horizontal section means stationary.steeper line means faster speed.
- A1: 5 m/s2
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
Hardproblem solvingOn a velocity-time graph, the area of a trapezium has parallel sides 10 and 20 m/s and width 5 s. Find the distance.
3 marks5 minsreal-life-graphs-q4Show solution
On a velocity-time graph, the area of a trapezium has parallel sides 10 and 20 m/s and width 5 s. Find the distance.
- 1.Spot the skill: On distance-time graphs: gradient = speed.
- 2.Use the identify the three stages from the description stage first, then calculate speed for the third stage.
- 3.Keep the final answer visible: 75 m.
75 m
- M1: use the correct on distance-time graphs: gradient = speed.on velocity-time graphs: gradient = acceleration, area under graph = distance. a horizontal section means stationary.steeper line means faster speed.
- A1: 75 m
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
Exam-stylemulti-stepA trip: 60 km in 1.5 h, rest 30 min, return 60 km in 2 h. What is the average speed for the whole journey?
4 marks6 minsreal-life-graphs-q5Show solution
A trip: 60 km in 1.5 h, rest 30 min, return 60 km in 2 h. What is the average speed for the whole journey?
- 1.Spot the skill: On distance-time graphs: gradient = speed.
- 2.Use the calculate speed for the third stage stage first, then note the direction of travel.
- 3.Keep the final answer visible: Average speed = 120 km / 4 h = 30 km/h.
Average speed = 120 km / 4 h = 30 km/h
- M1: use the correct on distance-time graphs: gradient = speed.on velocity-time graphs: gradient = acceleration, area under graph = distance. a horizontal section means stationary.steeper line means faster speed.
- A1: Average speed = 120 km / 4 h = 30 km/h
Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
Grade 9 stretchproblem solvingA cyclist travels 30 km in 40 minutes, rests for 20 minutes, then travels 45 km in 30 minutes. Find the average speed for the whole journey.
4 marks7 minsjourney-g9Show solution
A cyclist travels 30 km in 40 minutes, rests for 20 minutes, then travels 45 km in 30 minutes. Find the average speed for the whole journey.
- 1.Add the total distance.
- 2.Convert the whole journey time into hours, including the rest.
- 3.Divide distance by time.
50 km/h
- M1: total distance 75 km
- M1: total time 1.5 hours
- A1: 50 km/h
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Real-life graphs - 2 marksA distance-time graph has gradient 25. What does this mean?Mark answer
The object is moving at 25 m/s (or in the units of the graph)
2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
−4
3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
3x²y + xy
4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
3x(2x + 3)
- I can explain the method for real-life graphs.
- I can show clear working without skipping key steps.
- I can avoid this mistake: Students read the total distance from the graph rather than the change in distance for the stage.Always find distance as (final distance − initial distance) for each stage, and always check whether the line is going up (away) or down (returning).
This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.