We are asked to find the *position* of an object at a specific time, given an acceleration function and some initial conditions.

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:

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

We need to get from an acceleration function to a position function, which means we need to integrate the acceleration twice.

First, we integrate the acceleration function to get to velocity. **Make sure** to remember the **constant of integration**, which is going to be the velocity at *t* = 0 (initial velocity, *v*_{0}).

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

Then, to get from velocity to position, we integrate the velocity function. (Again, make sure to add the initial position as a constant of integration!)

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

We'll also use the __power rule of integration__:

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

A small object moves along the *x*-axis with acceleration *a*_{x}(*t*) = −(0.0320 m/s^{3})(15.0 s−*t*). At *t* = 0 the object is at *x* = −14.0 m and has velocity = 6.70 m/s.

What is the *x*-coordinate of the object when *t* = 10.0 s?