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Interpret distance-time and velocity-time graphs for IGCSE Edexcel Maths. Calculate speed, acceleration and distance with worked examples from grade 5 to grade 7.
You must interpret two types of graph: distance-time (where gradient = speed and horizontal = stationary) and velocity-time (where gradient = acceleration, horizontal = constant speed, and area under the graph = distance). You need to calculate average speed, acceleration from a gradient, and total distance from the area under a velocity-time graph using triangles, rectangles and trapeziums.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1 — Topic 3.6
A distance-time graph plots distance on the y-axis against time on the x-axis. Every feature of the graph tells you something about the journey:
Speed = gradient = change in distance divided by change in time. For a straight section, pick two clear points on the line and calculate (d₂ - d₁) / (t₂ - t₁). Always include units in your answer (e.g. km/h, m/s).
Average speed = total distance divided by total time. This is not the same as averaging the speeds of each section. Use the total distance from the graph (including any return sections) and the total elapsed time.
A velocity-time graph plots velocity (speed with direction) on the y-axis against time on the x-axis. The interpretation is different from a distance-time graph:
Acceleration = gradient = change in velocity divided by change in time. For a straight section, calculate (v₂ - v₁) / (t₂ - t₁). A negative gradient means deceleration. Units are typically m/s² or km/h².
The total distance travelled equals the area between the velocity-time line and the time axis. Break the area into simple shapes: triangles (area = 1/2 x base x height), rectangles (area = base x height) and trapeziums (area = 1/2 x (a + b) x h). Add the areas together to get the total distance.
| Feature | Distance-Time | Velocity-Time |
|---|---|---|
| Gradient gives | Speed | Acceleration |
| Horizontal line means | Stationary (speed = 0) | Constant speed (acceleration = 0) |
| Area under graph gives | Not applicable | Distance travelled |
| Steeper line means | Faster speed | Greater acceleration |
| Negative gradient means | Returning to start | Deceleration (slowing down) |
This comparison is essential to memorise. Examiners frequently set questions where students must switch between graph types within the same paper, and confusing the interpretations leads to losing all marks on that question.
A cyclist rides 20 km in the first hour, stops for 30 minutes, then rides another 30 km in 1.5 hours. Calculate the average speed for the entire journey.
Step 1 (Total distance): 20 km + 30 km = 50 km.
Step 2 (Total time): 1 hour + 0.5 hours + 1.5 hours = 3 hours.
Step 3 (Average speed): Average speed = total distance / total time = 50 / 3 = 16.7 km/h (to 1 decimal place).
Note: The stop counts in the total time because we are calculating average speed for the entire journey, not just the moving parts.
Answer: 16.7 km/h
A car accelerates from 10 m/s to 30 m/s in 5 seconds. Calculate the acceleration.
Step 1 (Identify values): Initial velocity = 10 m/s. Final velocity = 30 m/s. Time = 5 seconds.
Step 2 (Calculate gradient): Acceleration = (v₂ - v₁) / (t₂ - t₁) = (30 - 10) / 5 = 20 / 5 = 4 m/s².
Note: On a velocity-time graph this would be a straight line from (0, 10) to (5, 30) with a gradient of 4.
Answer: 4 m/s²
A train's velocity-time graph forms a trapezoid: it accelerates from 0 to 20 m/s in 10 seconds, travels at 20 m/s for 20 seconds, then decelerates from 20 m/s to 0 in 10 seconds. Calculate the total distance.
Step 1 (Acceleration phase — triangle): Area = 1/2 x base x height = 1/2 x 10 x 20 = 100 m.
Step 2 (Constant speed phase — rectangle): Area = base x height = 20 x 20 = 400 m.
Step 3 (Deceleration phase — triangle): Area = 1/2 x base x height = 1/2 x 10 x 20 = 100 m.
Step 4 (Total distance): 100 + 400 + 100 = 600 m.
Alternative method: Use the trapezium formula directly: Area = 1/2 x (a + b) x h = 1/2 x (20 + 40) x 20 = 600 m, where a = 20 s (top), b = 40 s (base) and h = 20 m/s.
Answer: 600 m
Sports scientists use velocity-time graphs to analyse athletic performance. A 100-metre sprinter's graph shows three phases: explosive acceleration from the blocks (steep positive gradient), a brief period of peak velocity (near-horizontal section), and slight deceleration in the final metres (shallow negative gradient). The total area under the curve equals the race distance of 100 metres.
Coaches compare these graphs between athletes and across training sessions to identify where performance gains can be made. This is an excellent exam context because it combines gradient interpretation, area calculation and real-world reasoning.
Mixing up graph types: On a distance-time graph, a horizontal line means stationary. On a velocity-time graph, a horizontal line means constant speed. Students who confuse these interpretations lose all marks on the question.
Unit conversion errors: Time is often given in minutes on the graph but the answer is required in hours. Always convert before calculating: 30 minutes = 0.5 hours, 45 minutes = 0.75 hours. A common trap is dividing by 60 when you should be dividing by 1.
Forgetting the rest period in average speed: When calculating average speed for an entire journey, include the time spent stationary in the total time. Average speed = total distance / total time (including stops).
Misinterpreting negative gradients: On a distance-time graph, a negative gradient means the object is returning towards the starting point. On a velocity-time graph, a negative gradient means the object is decelerating but still moving forward (until velocity reaches zero).
Read the axis labels first: Before answering any question, check whether the y-axis shows distance or velocity. This single check prevents the most common error in this entire topic.
Break areas into simple shapes: Split the area under a velocity-time graph into triangles, rectangles and trapeziums. Calculate each area separately, then add them together. Label each shape on the graph paper to show your method.
Always include units: Speed (m/s or km/h), acceleration (m/s²), distance (m or km), time (s or h). Missing or incorrect units cost marks even when the numerical answer is correct.
Convert time carefully: If the graph uses minutes but the question asks for km/h, convert minutes to hours first. Write the conversion step in your working so the examiner can award method marks even if you make an arithmetic slip.
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