A velocity-time graph plots speed on the y-axis against time on the x-axis. The gradient of any straight section gives the acceleration, while the area trapped between the graph and the x-axis gives the total distance travelled. These two facts unlock almost every velocity-time graph question at GCSE.
How does a velocity-time graph differ from a distance-time graph?
The two graphs look similar but tell you different things:
| Feature | Distance-time graph | Velocity-time graph |
|---|---|---|
| y-axis | Distance (m or km) | Velocity/speed (m/s or km/h) |
| Gradient means | Speed | Acceleration (m/s²) |
| Horizontal line means | Stationary (not moving) | Constant speed (not accelerating) |
| Area under graph means | Not directly useful | Distance travelled |
| Sloping down means | Returning towards start | Decelerating (slowing down) |
At GCSE, velocity-time graphs are used in the context of objects moving in a straight line. Velocity is speed with a direction, but for most GCSE questions the object travels in one direction only, so "velocity" and "speed" are used interchangeably.
How do you calculate acceleration from a velocity-time graph?
Acceleration = gradient of the line
Gradient = (change in velocity) ÷ (change in time) = Δv ÷ Δt
Worked example 1: A car accelerates uniformly from rest (0 m/s) to 20 m/s in 5 seconds.
- Change in velocity: 20 − 0 = 20 m/s
- Change in time: 5 − 0 = 5 s
- Acceleration = 20 ÷ 5 = 4 m/s²
A positive gradient means the object is speeding up (accelerating). A negative gradient means the object is slowing down (decelerating). Deceleration is negative acceleration — for example, −3 m/s² means the object loses 3 m/s every second.
How do you find the distance from a velocity-time graph?
Distance = area between the graph and the x-axis
For straight-line sections this means calculating areas of rectangles and triangles.
Worked example 2: A cyclist travels at 6 m/s for 10 seconds, then decelerates uniformly to rest in a further 4 seconds.
Phase 1 (rectangle): area = base × height = 10 × 6 = 60 m
Phase 2 (triangle): area = ½ × base × height = ½ × 4 × 6 = 12 m
Total distance = 60 + 12 = 72 m
Always split the graph into simple shapes — rectangles for constant speed sections, triangles for uniform acceleration or deceleration from/to zero, and trapeziums when the speed changes between two non-zero values.
How do you find the area of a trapezium section?
When a section starts at one non-zero velocity and ends at a different non-zero velocity, the shape is a trapezium.
Area of trapezium = ½ × (sum of parallel sides) × height = ½ × (v₁ + v₂) × t
Worked example 3: A train accelerates from 10 m/s to 30 m/s in 8 seconds.
Area = ½ × (10 + 30) × 8 = ½ × 40 × 8 = 160 m
This is the same formula as the trapezium area rule used in statistics; at GCSE it comes up in both contexts.
How do you read a multi-stage journey?
Most GCSE questions show a graph with three or four distinct stages. Work through each stage in turn.
Worked example 4: The graph shows:
| Stage | Time (s) | Velocity (m/s) | Shape | Distance |
|---|---|---|---|---|
| A: accelerate | 0 → 6 | 0 → 12 | Triangle | ½ × 6 × 12 = 36 m |
| B: constant speed | 6 → 14 | 12 → 12 | Rectangle | 8 × 12 = 96 m |
| C: decelerate | 14 → 20 | 12 → 0 | Triangle | ½ × 6 × 12 = 36 m |
Total distance = 36 + 96 + 36 = 168 m
Always label each stage before calculating — it prevents skipping a section.
What does a curved velocity-time graph mean?
A straight line on a velocity-time graph means uniform (constant) acceleration. A curve means the acceleration is changing — the object is speeding up or slowing down at a non-constant rate.
At Higher GCSE, you may be asked to estimate the area under a curved graph using the trapezium rule (dividing the area into narrow strips and summing them). This is the same technique as the area-under-a-graph topic in pure maths.
What common errors should you avoid?
- Confusing gradient and area. The gradient gives acceleration; the area gives distance. Mixing these up is the most common error on GCSE papers.
- Reading the y-axis as distance. The y-axis is velocity, not distance. A horizontal line at v = 10 m/s means constant speed, not that the object has stopped 10 m from the start.
- Forgetting units. Acceleration is in m/s², distance in metres. Check your units match the axes of the graph you are given.
- Missing a phase. In a multi-stage graph, find the total number of sections before you start — it is easy to ignore the final decelerating triangle.
Frequently asked questions
What is the difference between velocity and speed?
Speed is how fast an object is moving; velocity is speed in a specified direction. On a velocity-time graph that shows an object going backwards, the velocity would be negative. At Foundation GCSE the distinction is usually simplified — you will mostly see speed on the y-axis and treat all motion as positive.
Can the gradient be calculated between any two points?
Yes, but only if the line between them is straight (uniform acceleration). For a curved graph, the gradient at a single instant is found by drawing a tangent — a straight line that just touches the curve at that point — and then calculating the gradient of the tangent.
If the line goes below the x-axis, what does that mean?
Negative velocity means the object is moving in the opposite direction to its original motion. The area below the x-axis still represents distance, but in the reverse direction. At GCSE Higher, displacement (distance in a particular direction) may then require you to subtract this area from the positive area.
How do I check my answer for the total distance?
A useful check: the average velocity of a uniformly accelerating section is (start velocity + end velocity) ÷ 2. Multiplying by the time gives the same area as the trapezium formula. If these two methods agree, your calculation is almost certainly correct.
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