determine displacement from the area under a velocity–time graph

Equations of Motion: Finding Displacement from a Velocity–Time Graph 🚀

1️⃣ What is a Velocity–Time Graph?

A velocity–time graph shows how an object's speed (and direction) changes over time.

The vertical axis is velocity v (m/s) and the horizontal axis is time t (s).

Think of it like a road map where the height of the line tells you how fast you’re going at each moment.

2️⃣ The Area Under the Curve = Displacement 📈

The key rule:

\$\text{Displacement } s = \int v(t)\,dt\$

In a graph, this integral is simply the area between the curve and the time axis.

• If the curve lies above the axis, the area is positive (forward motion).

• If it lies below, the area is negative (backward motion).

Analogy: Imagine a water tank. The area is the amount of water that flows in (positive) or out (negative) over time.

3️⃣ Step‑by‑Step: Calculating Displacement

1. Identify the segments of the graph where the velocity is a simple shape (rectangles, triangles, trapezoids).
2. Calculate the area of each segment using the appropriate formula:

   • Rectangle: A = width × height
   • Triangle: A = ½ × base × height
   • Trapezoid: A = ½ × (top + bottom) × height

3. Assign a positive sign to areas above the axis and a negative sign to areas below.
4. Sum all signed areas to get the total displacement.

4️⃣ Example Problem

A car starts from rest, accelerates at a constant rate of 2 m/s² for 5 s, then travels at a constant speed of 10 m/s for 8 s, and finally stops over 4 s.

Sketch the velocity–time graph and find the total displacement.

The car travels a total of 85 m forward. Notice how the negative area from the deceleration segment reduces the total.

5️⃣ Common Pitfalls & Exam Tips 🧠

• Mixing distance and displacement: Always check the sign of the area.
• Ignoring units: Velocity is in m/s, time in s, so area (m/s × s) gives metres.
• Overlooking small segments: Even a brief negative area can change the final answer.
• Sketching first: A quick sketch helps identify shapes and signs.
• Check your arithmetic: A single mis‑calculated area can throw off the whole answer.