The idea

Motion in two dimensions with vectors

Why the a → v → r calculus chain works component-by-component in 2D, that speed is the magnitude of the velocity vector, and how "parallel to" and "perpendicular to" a direction become simple component conditions.

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Mechanics · Kinematics

Motion in two dimensions with vectors

Why it works

The calculus chain, one component at a time

In two dimensions a particle's position, velocity and acceleration are all vectors, written in

\mathbf{i}

\mathbf{j}

components. The good news: nothing about the calculus changes. If the position is

\mathbf{r} = x(t)\,\mathbf{i} + y(t)\,\mathbf{j},

then differentiating gives the velocity and acceleration one component at a time:

\mathbf{v} = \frac{d\mathbf{r}}{dt} = \dot{x}\,\mathbf{i} + \dot{y}\,\mathbf{j}, \qquad \mathbf{a} = \frac{d\mathbf{v}}{dt} = \ddot{x}\,\mathbf{i} + \ddot{y}\,\mathbf{j}.

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