Slope Is Rate, Area Is Total: Read Every Graph That Way
A motion graph encodes one quantity in its height, another in its slope and a third in the area beneath it. Reading any graph correctly means remembering which is which.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Slope and area for each graph
On a displacement-time graph, the slope is the velocity, so two straight lines at 30° and 60° to the time axis have velocities in the ratio tan 30° : tan 60° = 1 : 3. On a velocity-time graph, the slope is the acceleration and the area under the curve is the displacement. On an acceleration-time graph, the area under the curve is the change in velocity. A particle starting from rest with an acceleration that falls steadily from 5 m/s² to zero reaches its maximum speed when the acceleration reaches zero, and the speed is the area under the a-t graph up to that moment.
Maximum values and turning points
The maximum instantaneous velocity on a distance-time curve is at the point of steepest slope. The maximum acceleration on a velocity-time graph is the steepest segment, found by taking the largest rise over run. For a rocket launched upward, the speed is positive while it ascends, so the maximum height is the area under the velocity-time graph up to the moment the velocity reaches zero, not to the end of the graph. Area beyond that point, when the velocity is negative, subtracts displacement as the rocket comes back down.
Bouncing ball and graph shapes
For a ball dropped onto a glass surface that bounces elastically back to its initial height, the velocity changes sign at every bounce but its magnitude is the same at the same height, and the slope of the velocity-time graph is a constant -g throughout (the acceleration is always g downward), so the graph is a series of parallel straight segments.
OlympIQ exam lens
- ✓Slope of x-t = velocity; slope of v-t = acceleration; area under v-t = displacement; area under a-t = change in velocity
- ✓Straight x-t lines at angles θ have velocities in the ratio of tan θ
- ✓Maximum speed occurs when the acceleration reaches zero (a-t graph) and is the area up to that moment
- ✓Maximum height of a rocket = area under v-t up to the point where v = 0
- ✓An elastic bounce keeps the same slope (-g) in the v-t graph; only v changes sign at the bounce
Do not confuse the area under a velocity-time graph with the distance travelled when the velocity changes sign. Areas below the axis count as negative displacement; the total distance adds all areas as positive.
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