Topic 4 of 6
Speed-time graphs
Height gives speed. Gradient gives acceleration. Area gives distance.
Here a gradient measures change in velocity per second. It has a different meaning from the gradient of a distance-time graph because the vertical quantity has changed.
The vertical axis now shows speed at each time. A horizontal line on this graph means constant speed. The object is at rest only if that line is at zero.
| Graph shape | Motion |
|---|---|
| Line at zero | At rest throughout that interval. |
| Horizontal line above zero | Constant speed; zero acceleration. |
| Straight sloping line | Uniform acceleration. Its gradient is constant. |
| Curve | Non-uniform acceleration. Its gradient changes. |
A curve becoming less steep still shows a changing speed, but the magnitude of acceleration is getting smaller. For a sloping straight segment, calculate acceleration using the change in speed divided by the elapsed time. In the one-direction journey below, forwards is positive.
A curve means changing acceleration
This example moves forwards throughout. Its graph becomes steeper: the speed increases by a larger amount each second.
| Time / s | Speed / (m/s) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
Over the successive one-second intervals, the changes are 1, 3 and 5 m/s. The average accelerations over those intervals are therefore 1, 3 and 5 m/s2. The acceleration is non-uniform. A straight rising line would instead have equal changes in equal times.
Why area gives distance
At constant speed, distance = speed x time. On this graph, the height is speed and the width is time, so their product is the area of a rectangle. When speed changes uniformly, use the area under the sloping line, splitting it into rectangles and triangles where needed.
Motion and graphs
Moving in one direction
A trolley moves along a straight track in one direction. It travels 18 m in 6 s.
Move the slider to see the trolley's position and the values on both graphs at the selected time. You can also use the arrow keys.
Moving forwards at a constant 4 m/s.
- Distance
- 6 m
- Speed
- 4 m/s
- Acceleration
- 0 m/s2
View the data table
| Time / s | Distance / m | Speed / (m/s) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 2 |
| 2 | 4 | 4 |
| 3 | 8 | 4 |
| 4 | 12 | 4 |
| 5 | 16 | 4 |
| 6 | 18 | 0 |
Worked example
Find the distance and average speed
The trolley reaches 4 m/s in 2 s, stays at 4 m/s for 3 s, then brakes to rest in 1 s. Use the one-direction example above. Split the area under the graph into two triangles and a rectangle.
- 0 to 2 s, triangle: (1/2) x 2 s x 4 m/s = 4 m.
- 2 to 5 s, rectangle: 3 s x 4 m/s = 12 m.
- 5 to 6 s, triangle: (1/2) x 1 s x 4 m/s = 2 m.
Total distance = 4 + 12 + 2 = 18 m
Average speed = 18 / 6 = 3 m/s
Multiplying time by speed gives metres: s x m/s = m. The area counts distance throughout the one-direction journey.
Find the accelerations from the gradients
First interval: (4 - 0) / 2 = 2 m/s2.
Middle interval: (4 - 4) / 3 = 0 m/s2.
Final interval: (0 - 4) / 1 = -4 m/s2.
The trolley loses 4 m/s of velocity each second while braking, compared with a gain of 2 m/s each second at the start.
Check the vertical axis. Area under a distance-time graph does not give distance. The area rule here works because the vertical axis is speed.