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:

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

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

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

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

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

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

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

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

**(a)** The first part of the problem asks us to calculate the position vector as a function of time. Since the problem states that the bird starts at the origin at *t* = 0, the initial position ${\stackrel{\rightharpoonup}{r}}_{0}=\stackrel{\rightharpoonup}{0}$.

$\begin{array}{rcl}\stackrel{\rightharpoonup}{r}\left(t\right)& =& \int \stackrel{\rightharpoonup}{v}\left(t\right)dt+{\stackrel{\rightharpoonup}{r}}_{0}\\ & =& \int 2.4+1.6{t}^{2}dt\hat{\mathrm{i}}+\int 4.0tdt\hat{\mathrm{j}}+{\stackrel{\rightharpoonup}{r}}_{0}\\ & =& \mathbf{(}\mathbf{2}\mathbf{.}\mathbf{4}\mathit{t}\mathbf{+}\frac{8}{15}{\mathit{t}}^{\mathbf{3}}\mathbf{)}\mathbf{}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{0}{\mathit{t}}^{\mathbf{2}}\mathbf{)}\mathbf{}\hat{\mathbf{j}}\mathbf{+}\mathbf{0}\end{array}$