What the area means
- The area enclosed between a velocity-time line and the time axis equals the distance that has been covered during the interval (or, equivalently, the magnitude of the displacement if the motion runs in a single direction)
- This is true because area = velocity × time on every thin vertical strip under the line, and velocity × time = distance moved in that strip
Splitting a multi-stage motion into shapes
- A constant-velocity section is a horizontal line, so the area under it is a rectangle:
rectangle area = base × height
- A section of constant acceleration or deceleration is a straight slope, so the area under it (down to the time axis) is a triangle:
triangle area = ½ × base × height
- A section that combines a non-zero starting velocity with an acceleration gives a trapezium; split it into a rectangle and a triangle and add the two
- The total distance for a multi-stage motion is found by adding together the area of every enclosed region one stage at a time
Example — a delivery van's 60 s journey is described by the v-t graph above: 0–10 s accelerating from rest to 12 m/s, 10–40 s at a constant 12 m/s, 40–60 s decelerating from 12 m/s to rest. Calculate the total distance travelled.
- Section 1 (triangle, 0–10 s): ½ × 10 × 12 = 60 m
- Section 2 (rectangle, 10–40 s): 30 × 12 = 360 m
- Section 3 (triangle, 40–60 s): ½ × 20 × 12 = 120 m
- Total distance = 60 + 360 + 120 = 540 m