# Problem: A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F = Ct2.(a) Determine its velocity exttip{v}{v} as a function of time.(b) Determine its position exttip{x}{x} as a function of time.

###### FREE Expert Solution

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:

1. Use the Newton's Second Law equation ΣF=ma to find a function for the acceleration
2. Integrate the acceleration function to get velocity (don't forget about the integration constant!)
3. Integrate the velocity to get the position (if required)
4. 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:

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

Remember the power rule of integration.

To integrate,

, where C is the constant of integration.

The velocity function is the integral of the acceleration function, a(t). We're not given  a(t), but we are given a function for force, F(t)

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

So here's our process:

###### Problem Details

A particle of mass m, initially at rest at x = 0, is accelerated by a force that increases in time as F = Ct2.
(a) Determine its velocity as a function of time.
(b) Determine its position as a function of time.