This problem requires us to find the **velocity** and **position**** **functions** **of an object given a** force **function.

This is a **Force with Calculus** type of problem since we have the force as a function of time. We'll follow these steps:

**Use the Newton's Second Law equation Σ***F*=to find a function for the acceleration**ma****Integrate**the**acceleration function**to get velocity (don't forget about the integration constant!)**Integrate the velocity**to get the position (if required)**Calculate**the position and/or velocity at a**specific time**of interest (if required)

A diagram like this one can help you remember the relationships between the variables:

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

Remember the **power rule** of integration.

To integrate,

$\overline{){\mathbf{\int}}{{\mathbf{t}}}^{{\mathbf{n}}}{\mathbf{}}{\mathbf{d}}{\mathbf{t}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{n}\mathbf{+}\mathbf{1}}{{\mathbf{t}}}^{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{+}}{\mathbf{C}}}$, where **C** is the constant of integration.

The **velocity** function is the **integral** of the acceleration function,** a(t)**. We're not given

The object's **position **function, **x****( t)**, is the integral of the velocity function

So here's our process:

A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F = Ct^{2}.**(a)** Determine its velocity as a function of time.**(b)** Determine its position as a function of time.

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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 Forces with Calculus concept. You can view video lessons to learn Forces with Calculus. Or if you need more Forces with Calculus practice, you can also practice Forces with Calculus practice problems.