14.15 Kinematics Graphs

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Main Content

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Quick Recap: The Key Relationships

Before drawing any graph, you need to remember these three relationships from topic 14.14:

These relationships are the engine behind all five types of graphs. Once you have the displacement function $s(t)$, you can differentiate to find $v(t)$, then differentiate again to find $a(t)$. From there, you can sketch every graph.

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Understanding the Five Graph Types

There are five graphs you need to be able to draw and use. Here is a clear overview of each one:

Important distinction — scalar vs vector:

• Displacement and velocity are vectors — they can be positive or negative depending on direction.
• Distance and speed are scalars — they are always zero or positive. They never go below zero.

This means:

• The distance–time graph is never below the x-axis.
• The speed–time graph is never below the x-axis.
• The displacement–time and velocity–time graphs can go below the x-axis (when the particle moves in the negative direction).

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How to Sketch Each Graph: Step-by-Step

The process always follows the same logical order. Given a displacement function $s(t)$:

Step 1: Find $v = \frac{ds}{dt}$ by differentiating $s$.

Step 2: Find $a = \frac{dv}{dt}$ by differentiating $v$.

Step 3: Find the key values — starting values (at $t = 0$), turning points, zeros, and end values.

Step 4: Use these values to sketch each graph.