This problem requires us to find the **velocity** function** **and **evaluate its magnitude **after a very long time** **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!)**Calculate the velocity**at a specific time of interest.

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

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

In __step 2__, we need to recall how to integrate an exponential.

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

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

So here's our process:

At t = 0, an object of mass m is at rest at x = 0 on a horizontal, frictionless surface. Starting at t = 0, a horizontal force F_{x} = F_{0}e^{−t/T} is exerted on the object.**(a)** Find an expression for the object's velocity at an arbitrary later time t.**(b)** What is the object's velocity after a very long time has elapsed?￼

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.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Wallace's class at UNC.