13.4 Velocity Vectors

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📖 Main Content

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Part 1: What is Velocity?

Velocity is a vector quantity — this means it has both a size (how fast something is moving) and a direction (which way it is moving).

Speed is just the size (magnitude) of the velocity. Speed tells you how fast something is moving, but not which direction. Speed is a scalar — it has size only, no direction.

We write velocity vectors using i and j notation:

• i = the unit vector pointing in the positive x-direction (to the right / east)
• j = the unit vector pointing in the positive y-direction (upward / north)

So a velocity like v = 4i − 2j ms⁻¹ means the object moves 4 units per second to the right and 2 units per second downward.

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Part 2: Finding Speed from a Velocity Vector

Since velocity is a vector, its magnitude (size) gives you the speed.

$$\text{speed} = |\mathbf{v}| = \sqrt{x^2 + y^2}$$

Where v = xi + yj.

$$\text{speed} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \text{ ms}^{-1}$$

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Part 3: Finding Velocity from Displacement and Time

If you know how far an object has travelled (its displacement vector) and how long it took, you can find the velocity:

$$\mathbf{v} = \frac{\text{displacement}}{\text{time taken}}$$

This works because the object is moving at constant velocity — meaning it does not speed up or slow down and it keeps moving in the same direction.

$$\text{speed} = \sqrt{(8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ ms}^{-1}$$