We are asked to find the *position*, *acceleration*, and a *specific coordinate* for a bird in flight, given its velocity as a 2D function of time and initial position.

Anytime we're given a **position**, **velocity**, or **acceleration** function and asked to find one or more of the others, we know it's a **motion problem with calculus**. A PVA diagram like the one below can help remind you of the relationships between the three functions:

$\mathit{P}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\underset{\frac{\mathit{d}}{\mathit{d}\mathit{t}}}{\overset{{\mathbf{\int}}{\mathit{d}}{\mathit{t}}}{\mathit{V}}}\begin{array}{c}{\mathbf{\leftarrow}}\\ {\mathbf{\to}}\end{array}\mathit{A}$

To get from velocity to position, we integrate the velocity function. Make sure to add the initial position as a constant of integration!

$\overline{)\stackrel{\mathbf{\rightharpoonup}}{\mathit{r}}{\mathbf{\left(}}{\mathit{t}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{\int}}\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}{\mathbf{\left(}}{\mathit{t}}{\mathbf{\right)}}{\mathbf{}}{\mathit{d}}{\mathit{t}}{\mathbf{+}}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{r}}}_{{\mathbf{0}}}}$

To get from velocity to acceleration, we take the derivative of the velocity function:

$\overline{)\stackrel{\mathbf{\rightharpoonup}}{\mathit{a}}{\mathbf{\left(}}{\mathit{t}}{\mathbf{\right)}}{\mathbf{=}}\frac{\mathit{d}\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}\mathbf{\left(}\mathit{t}\mathbf{\right)}}{\mathit{d}\mathit{t}}}$

We'll often use the __power rule of ____integration__ and the __power rule of ____derivation__, which are:

$\overline{){\mathbf{\int}}{{\mathit{x}}}^{{\mathit{n}}}{\mathbf{}}{\mathit{d}}{\mathit{t}}{\mathbf{=}}\frac{\mathbf{1}}{\mathit{n}\mathbf{+}\mathbf{1}}{{\mathit{x}}}^{\mathit{n}\mathbf{+}\mathbf{1}}}$ (for example, $\mathbf{\int}{\mathit{x}}^{\mathbf{2}}\mathit{d}\mathit{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}{\mathit{x}}^{\mathbf{3}}$)

$\overline{)\frac{\mathit{d}}{\mathit{d}\mathit{t}}\mathbf{\left(}{\mathit{x}}^{\mathit{n}}\mathbf{\right)}{\mathbf{=}}{\mathit{n}}{{\mathit{x}}}^{\mathit{n}\mathbf{-}\mathbf{1}}}$ (for example, $\frac{\mathit{d}}{\mathit{d}\mathit{t}}{\mathit{x}}^{\mathbf{3}}\mathbf{=}\mathbf{3}{\mathit{x}}^{\mathbf{2}}$)

Whenever you take the derivative or integral of a vector, **make sure to do the operation on each component (î, ĵ, and k̂) separately**—they're independent of each other and shouldn't get mixed up!