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?

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What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Velocity Functions & Instantaneous Acceleration concept. You can view video lessons to learn Velocity Functions & Instantaneous Acceleration. Or if you need more Velocity Functions & Instantaneous Acceleration practice, you can also practice Velocity Functions & Instantaneous Acceleration practice problems.