determine velocity using the gradient of a displacement–time graph
Equations of Motion: Determining Velocity from a Displacement–Time Graph 🚀
Objective: Learn how to find the velocity of an object by looking at the slope (gradient) of its displacement–time graph.
Think of the graph as a map of a road. The steeper the slope, the faster you’re going. That’s exactly what we’ll use to calculate velocity.
1️⃣ What is a Gradient?
The gradient (or slope) of a line is the “rise over run” – the change in vertical distance divided by the change in horizontal distance. In a displacement–time graph, the vertical axis is displacement s (meters), and the horizontal axis is time t (seconds). The gradient therefore gives us s/t, which is velocity v.
Mathematically: v = \frac{ds}{dt}.
2️⃣ How to Read the Graph
- Pick two points on the straight line segment of the graph.
- Write down their coordinates: (t_1, s_1) and (t_2, s_2).
- Calculate the change in displacement: Δs = s_2 - s_1.
- Calculate the change in time: Δt = t_2 - t_1.
- Divide: v = Δs / Δt. This gives the average velocity over that interval.
If the line is perfectly straight, the average velocity is also the instantaneous velocity at any point on that line.
3️⃣ Example Problem
A car moves along a straight road. Its displacement–time graph is a straight line that passes through the points (0 s, 0 m) and (10 s, 200 m).
- Δs = 200 m – 0 m = 200 m
- Δt = 10 s – 0 s = 10 s
- v = 200 m / 10 s = 20 m s⁻¹
So the car’s velocity is 20 m s⁻¹.
4️⃣ Quick Reference Table
| Symbol | Meaning | Units |
|---|---|---|
| s | Displacement | m |
| t | Time | s |
| v | Velocity (average or instantaneous) | m s⁻¹ |
| a | Acceleration | m s⁻² |
5️⃣ Examination Tips 📚
Tip 1: Always check the units. If you get a slope in m s⁻¹, you’ve found velocity.
Tip 2: For a straight line, the gradient is the same everywhere. You can pick any two points, even the endpoints.
Tip 3: If the graph is curved, remember that the gradient at a point gives the instantaneous velocity. Use a small segment around that point.
Tip 4: When the question asks for “average velocity,” use the overall change in displacement divided by the total time.
Tip 5: Practice sketching graphs from equations like s = vt + ½at² to see how the slope changes with acceleration.
Revision
Log in to practice.