determine acceleration using the gradient of a velocity–time graph

Equations of Motion: Acceleration from a Velocity–Time Graph 🚀

What is Acceleration?

Acceleration is the rate at which an object's velocity changes with time. Think of it as the “speed‑up” or “slow‑down” of a moving car. If a car’s velocity increases from 0 m/s to 10 m/s in 5 s, it’s accelerating.

Why the Gradient Matters 📈

On a velocity–time graph, the gradient (slope) of a line tells you how fast velocity changes. A steeper slope means a larger acceleration. For a straight line, the gradient is constant, giving a constant acceleration. For a curved line, the gradient changes, so the acceleration changes too.

Calculating Acceleration from a Straight Line

1. Pick any two points on the straight part of the graph.
2. Compute the change in velocity: Δv = v₂ – v₁.
3. Compute the change in time: Δt = t₂ – t₁.
4. Divide: a = Δv / Δt.
5. Units: m s⁻².

Mathematically: $$a = \frac{\Delta v}{\Delta t}$$

Example: A Car Starting from Rest

Suppose a car starts from rest and accelerates uniformly to 20 m/s in 10 s. On the graph, the line goes from (0 s, 0 m/s) to (10 s, 20 m/s).

Δv = 20 m/s – 0 m/s = 20 m/s Δt = 10 s – 0 s = 10 s So, $$a = \frac{20}{10} = 2 \,\text{m/s}^2$$ The car’s acceleration is 2 m/s².

Curved Graphs: Instantaneous Acceleration 📉

When the velocity–time graph is curved, the acceleration changes over time. The instantaneous acceleration at a particular time is the slope of the tangent line at that point.

Imagine driving up a hill that gets steeper and steeper. The steeper the hill (tangent slope), the more you accelerate. To find the exact value, you’d take the derivative: $$a(t) = \frac{dv}{dt}$$

Quick Check Questions ??

• What is the gradient of a horizontal line on a velocity–time graph?
• How would you describe acceleration if the graph is a straight line with a negative slope?
• Why does a curved line require a tangent to find acceleration?

Key Takeaways

• The gradient of a velocity–time graph is acceleration.
• For straight lines, acceleration is constant; for curves, it varies.
• Use Δv/Δt for straight lines, and dv/dt (tangent slope) for curves.
• Always check units: m s⁻².