GCSE Maths Velocity-Time Graphs With Worked Examples
A velocity–time graph shows how velocity changes over time. Its gradient represents acceleration, and the signed area between the graph and the time axis represents displacement. GCSE questions often combine both ideas. Understanding which calculation the question requires prevents the classic mistake of using gradient to find distance or area to find acceleration.
Read the axes and units
Velocity is measured in metres per second when the vertical axis is labelled m/s; time may be measured in seconds. A horizontal line at 5 m/s means constant velocity, not that the object has stopped. A sloping line shows changing velocity. If values are positive, the object is moving in the positive reference direction. A line below the axis represents motion in the opposite direction, not a negative amount of total distance travelled.
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Find acceleration from gradient
Suppose velocity rises from 2 m/s at time zero to 10 m/s at four seconds along a straight segment. Acceleration is change in velocity divided by time: (10 − 2)/4 = 2 m/s². The positive value means the velocity increases by two metres per second each second. If the line slopes downward, the acceleration is negative, although that does not automatically mean speed is decreasing when velocity itself is negative.
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Calculate displacement from area
Over the same four-second interval, velocity increases linearly from 2 to 10 m/s. The area under the graph is a trapezium: (2 + 10)/2 × 4 = 24 metres. Equivalently, add a rectangle of area 2 × 4 = 8 and a triangle of area (8 × 4)/2 = 16. The total is 24 metres of displacement. The height of the graph is velocity, so multiplying height by time yields metres.
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Describe a journey in sections
A graph might show acceleration for the first three seconds, constant velocity for the next five, and deceleration to zero over the final two. Divide the area into simple triangles, rectangles or trapezia and add the corresponding distances when the velocity remains positive. For gradient, use the endpoints of the particular line segment under discussion. Do not use total journey time when calculating the acceleration of only the opening section.
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Distinguish displacement from distance
If a velocity–time graph goes below the time axis, its negative signed area reduces total displacement. Distance travelled, however, adds the magnitudes of areas both above and below the axis. Imagine travelling three metres east and then two metres west: displacement is one metre east, whereas distance travelled is five metres. Exam wording determines which interpretation is required. Label the chosen positive direction clearly.
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Check units and common mistakes
Gradient units are m/s²; area units are m. A pupil who gives the gradient as 24 metres or the area as 2 m/s² has mixed the concepts. Label every segment and check the axes before drawing calculations. Read whether the graph is velocity–time or distance–time: the same-looking line can have a completely different physical interpretation. Use a sketch to identify all regions whose area contributes to the answer.
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Frequently asked questions
Is the area under a velocity–time graph speed?
No. Signed area represents displacement, while magnitude of the areas yields distance travelled.
What does a horizontal velocity–time line mean?
Constant velocity and zero acceleration during that interval.
Related GCSE revision resources
The existing guides below explore supporting skills. Choose the explanations that match the area where further understanding is required.
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