We're asked for the **x-coordinate (position)** where an object acted on by friction *comes to **rest* and the

This is a **Friction** type of problem with kinematics. When we're tackling a problem that has **forces** and** kinematics**, we'll follow these steps:

- Draw
**free body diagrams (FBDs)** - Set up
**Newton's 2nd Law**equations (**∑***F*=)**ma** - Set up (choose)
**kinematics equations** **Solve**for the target variables

Remember that we calculate kinetic friction using

$\overline{){{\mathit{f}}}_{{\mathit{k}}}{\mathbf{=}}{{\mathit{\mu}}}_{{\mathit{k}}}{\mathit{N}}}$

The four UAM (kinematics) equations are:

$\overline{)\mathbf{}{{\mathit{v}}}_{{\mathit{f}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathbf{}}{\mathbf{+}}{\mathit{a}}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{x}}{\mathbf{=}}{\mathbf{}}\mathbf{\left(}\frac{{\mathit{v}}_{\mathit{f}}\mathbf{+}{\mathit{v}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{\right)}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{x}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathit{t}}{\mathbf{+}}{\frac{1}{2}}{\mathit{a}}{{\mathit{t}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{}}{{{\mathit{v}}}_{{\mathit{f}}}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{}}{{{\mathit{v}}}_{{\mathbf{0}}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{2}}{\mathit{a}}{\mathbf{\u2206}}{\mathit{x}}}$

A box is sliding with a constant speed of 3.80 m/s in the +*x*-direction on a horizontal, frictionless surface. At *x* = 0 the box encounters a rough patch of the surface, and then the surface becomes even rougher. Between *x* = 0 and *x* = 2.00 m, the coefficient of kinetic friction between the box and the surface is 0.200; between *x* = 2.00 m and *x* = 4.00 m, it is 0.400.

(*a*) What is the *x*-coordinate of the point where the box comes to rest?

(*b*) How much time does it take the box to come to rest after it first encounters the rough patch at *x* = 0?

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 Friction concept. If you need more Friction practice, you can also practice Friction practice problems.