In this problem, we are required to determine the **magnitude **of the force on an object given its 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:

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

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) **expression 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 straightforward:

**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)**.**Use the equation***F*=to find a function for the**ma**__force__**Calculate the force**at a specific time of interest (if required).

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

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

The position of a 2.2-kg mass is given by x(t) = (2t^{ }^{3} - 4t^{ }^{2}) m, where t is in seconds.**(a)** What is the net horizontal force on the mass at t = 0 s?**(b)** What is the net horizontal force on the mass at t = 1 s?

Frequently Asked Questions

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.