Velocity-Time Graphs and Displacement
A velocity-time graph plots velocity against time; its slope gives the acceleration and the area between the line and the time axis gives the displacement. In brief: a horizontal line means constant velocity, a straight sloping line means constant acceleration, and a line crossing the time axis means that the object has reversed its direction. Areas above the axis count as positive displacement and areas below the axis as negative.
The velocity-time graph is the most useful graph in kinematics, because it contains information about both acceleration and displacement. This guide explains how to read the slope and the area, how to deal with graphs that go below the axis, compares graph features in a table, works through an example with a trapezoid-shaped graph and a graph with reversal, and corrects common mistakes.
What does the slope show?
The slope Δv ÷ Δt is the acceleration. A steeper line means a larger acceleration. A positive slope shows increasing velocity (in the positive direction), a negative slope shows decreasing velocity, and a horizontal line shows zero acceleration. A curved line shows changing acceleration, and the acceleration at one moment is the slope of the tangent.
The position of the line relative to the time axis shows the direction of motion: above the axis the velocity is positive, below it is negative. A line that crosses the time axis shows that the object stopped for an instant and reversed its direction.
What does the area show?
The area between the line and the time axis is the displacement, because velocity × time = displacement for each narrow strip. The common shapes are:
- rectangle: area = base × height (constant velocity);
- triangle: area = ½ × base × height (uniform acceleration from or to rest);
- trapezoid: area = ½ × (sum of parallel sides) × width, which gives s = ½(u + v)t.
Area below the axis is negative displacement. The sum of the signed areas is the net displacement, and the sum of the absolute values of the areas is the total distance traveled.
Features of velocity-time graphs
| Feature of the graph | What it means |
|---|---|
| Horizontal line | Constant velocity, acceleration zero |
| Straight line sloping up | Constant positive acceleration |
| Straight line sloping down | Constant negative acceleration |
| Line on the time axis | At rest |
| Crossing the time axis | Reversal of direction |
| Area under the line | Displacement (signed) |
Worked example: reading a velocity-time graph
Part A. A car’s velocity increases uniformly from 0 to 8 m/s in 4 s, stays at 8 m/s for 6 s, then decreases uniformly to 0 in 4 s.
- Accelerations: 0 to 4 s: (8 − 0) ÷ 4 = +2 m/s2. 4 to 10 s: 0. 10 to 14 s: (0 − 8) ÷ 4 = −2 m/s2.
- Displacements (areas): first triangle ½ × 4 × 8 = 16 m; rectangle 6 × 8 = 48 m; last triangle ½ × 4 × 8 = 16 m.
- Total displacement: 16 + 48 + 16 = 80 m. As a check, the trapezoid formula with parallel sides 14 s and 6 s and height 8 m/s gives ½ × (14 + 6) × 8 = 80 m.
- Average velocity: 80 ÷ 14 = 5.7 m/s.
Part B. A cart has a velocity that changes uniformly from +4 m/s at t = 0 to −4 m/s at t = 4 s.
- Acceleration: (−4 − 4) ÷ 4 = −2 m/s2.
- Time at which v = 0: 4 − 2t = 0, so t = 2 s. The cart reverses at that time.
- Displacements: 0 to 2 s: ½ × 2 × 4 = +4 m. 2 to 4 s: −4 m.
- Net displacement: +4 − 4 = 0, and the distance is 4 + 4 = 8 m. The cart returns to the start.
How did the area rule develop?
Nicole Oresme proved in the fourteenth century that the distance traveled in uniformly accelerated motion equals the distance with the average speed, using a geometric argument about areas. The result, also called the Merton rule, was used by Galileo in his analysis of falling bodies. Integration in calculus makes the idea general: displacement is the integral of velocity over time.
Which mistakes are common?
Mistake 1: Confusing the slope with the area
The slope gives the acceleration. The area gives the displacement.
Mistake 2: Treating areas below the axis as positive in the displacement
Areas below the axis are negative displacements. Add them with their signs for the net displacement.
Mistake 3: Reading the graph as a path
A graph that goes down does not mean that the object is going downhill, only that the velocity is decreasing.
Mistake 4: Saying that a crossing of the time axis means that the object stops permanently
It means that the velocity is zero at that instant and then changes sign.