Calculus with vectors can be used to solve problems involving motion in two dimensions with variable acceleration.

Differentiation

If \(\bold{r} = x\bold{i} + y\bold{j}\), then:

\(\bold{v} = \dfrac{d\bold{r}}{dt} = \bold{\dot{r}} = \dot{x}\bold{i} + \dot{y}\bold{j} \)

\(\bold{a} = \dfrac{d\bold{v}}{dt} = \dfrac{d^2\bold{r}}{dt^2} = \bold{\ddot{r}} = \ddot{x}\bold{i} + \ddot{y}\bold{j} \)

Dot notation is shorthand for differentiation with respect to time:

\(\dot{x} = \dfrac{dx}{dt} \) and \(\ddot{x} = \dfrac{d^2x}{dt^2} \)

Integration

Integration is the reverse process to differentiation, therefore:

\(\bold{r} = \displaystyle\int{\bold{v}}~dt\)

\(\bold{v} = \displaystyle\int{\bold{a}}~dt\)

The constant of integration \(c\) is also a vector, and should be written in the form \(p\bold{i}+ q\bold{j} \).