use graphical methods to represent distance, displacement, speed, velocity and acceleration
Equations of Motion: Graphical Methods
📚 In this lesson we’ll learn how to use simple graphs to understand how far an object travels, how fast it moves, and how its speed changes over time. These skills are essential for the Cambridge A‑Level Physics 9702 exam.
1️⃣ Distance vs. Time
Distance is the total length travelled, regardless of direction. On a graph, distance is plotted on the vertical axis (y) and time on the horizontal axis (x). The slope of the line tells you the speed.
| Time (s) | Distance (m) |
|---|---|
| 0 | 0 |
| 2 | 10 |
| 4 | 20 |
The graph is a straight line with a constant slope. The slope (rise/run) is the speed:
$$\text{speed} = \frac{\Delta \text{distance}}{\Delta \text{time}} = \frac{20-0}{4-0} = 5 \text{ m/s}$$
2️⃣ Displacement vs. Time
Displacement is the straight‑line distance from the start point to the end point, including direction. On a graph, the vertical axis shows displacement (positive or negative). The slope still represents velocity.
| Time (s) | Displacement (m) |
|---|---|
| 0 | 0 |
| 3 | -9 |
| 6 | -18 |
The negative values mean the object is moving in the opposite direction (e.g., left). The slope is:
$$\text{velocity} = \frac{\Delta \text{displacement}}{\Delta \text{time}} = \frac{-18-0}{6-0} = -3 \text{ m/s}$$
3️⃣ Speed vs. Time
Speed is the magnitude of velocity (always positive). On a speed‑time graph, the vertical axis shows speed. A horizontal line means the speed is constant; a sloping line means the speed is changing.
| Time (s) | Speed (m/s) |
|---|---|
| 0 | 0 |
| 2 | 4 |
| 4 | 8 |
The slope of this graph gives the acceleration:
$$\text{acceleration} = \frac{\Delta \text{speed}}{\Delta \text{time}} = \frac{8-0}{4-0} = 2 \text{ m/s}^2$$
4️⃣ Velocity vs. Time
Velocity includes direction. A velocity‑time graph can cross the horizontal axis, indicating changes in direction. The slope of the graph gives the acceleration.
| Time (s) | Velocity (m/s) |
|---|---|
| 0 | -2 |
| 3 | 1 |
| 6 | 4 |
The average acceleration over the whole interval is:
$$a_{\text{avg}} = \frac{4-(-2)}{6-0} = \frac{6}{6} = 1 \text{ m/s}^2$$
5️⃣ Acceleration vs. Time
Acceleration is the rate of change of velocity. On an acceleration‑time graph, the vertical axis shows acceleration. A horizontal line indicates constant acceleration; a sloping line means the acceleration itself is changing.
| Time (s) | Acceleration (m/s²) |
|---|---|
| 0 | 2 |
| 4 | 2 |
Because the acceleration is constant, the velocity changes linearly. The area under the acceleration‑time graph between two times gives the change in velocity:
$$\Delta v = \int a \, dt = a \times \Delta t = 2 \times 4 = 8 \text{ m/s}$$
📌 Quick Summary
- Distance‑time: Slope = speed (always positive).
- Displacement‑time: Slope = velocity (can be negative).
- Speed‑time: Slope = acceleration.
- Velocity‑time: Slope = acceleration.
- Acceleration‑time: Area under curve = change in velocity.
🎓 Final Exam Tip: Always label your axes, check units, and remember that the slope of a graph is a ratio of the vertical change to the horizontal change. Practice sketching graphs from data and vice versa – this will save you time during the exam!
Revision
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