Position is a vector. Differentiate it.

In a plane, a particle needs two numbers to say where it is. Write them as one vector in $t$, and the calculus of Year 1 gives its velocity and acceleration, one component at a time.

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A particle moves so that after $t$ seconds

metres from $O$, with $\mathbf{i}$ east and $\mathbf{j}$ north. The arrow riding with it is its velocity: always along the path.

At $t = 1$ the velocity arrow points straight down the page: the particle is moving due south, parallel to $\mathbf{j}$, at 4 m s⁻¹. Its $\mathbf{i}$-component is zero.

At $t = 2$ the other component vanishes:

so the particle is moving due east at 9 m s⁻¹, at the bottom of its dip.

Differentiate once more: $\mathbf{a} = 6t\,\mathbf{i} + 4\mathbf{j}$, shown dashed. It says which way the velocity is changing, and the velocity arrow keeps turning towards it: at $t = 2$, $\mathbf{a} = 12\mathbf{i} + 4\mathbf{j}$.

Differentiate each component. Integrate back.

Constant acceleration as vectors, ships that meet, integrating with a vector constant, projectiles as vectors, and twelve graded exam questions with full worked solutions are waiting.

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