# Problem: An object with mass m moves along the x-axis. Its position as a function of time is given by x(t)=At - Bt3, where A exttip{A}{A}and B exttip{B}{B}are constants.Find an expression for the x-component of the force acting on the object as a function of time.

###### FREE Expert Solution

In this problem, we are required to determine the function for the force on an object given the position as a time-varying function

Since we have the position as a function of time, we know this is a calculus problem.

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

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

Since we're looking for the magnitude of the force exerted on the object, we'll have to find an expression for acceleration, a(t). The a(t) function is obtained by differentiating the position function, x(t), twice.  Then, we'll multiply the a(t) function by the mass, m, to get the force expression.

The steps needed to solve this problem are simple and straight forward:

1. Differentiate the position function, x(t), twice to get the acceleration function a(t). Remember, differentiating the position function, x(t), once gives the velocity function, v(t).
2. Use the equation ma to find a function for the force
3. Calculate the force at a specific time of interest (optional).

In Step 1, we'll need to remember the power rule of differentiation:

$\overline{)\frac{\mathbf{d}}{\mathbf{dt}}\mathbf{\left(}{\mathbf{t}}^{\mathbf{n}}\mathbf{\right)}{\mathbf{=}}{{\mathbf{nt}}}^{\mathbf{n}\mathbf{-}\mathbf{1}}}$

###### Problem Details

An object with mass m moves along the x-axis. Its position as a function of time is given by x(t)=At - Bt3, where A and B are constants.
Find an expression for the x-component of the force acting on the object as a function of time.