We're asked for the *maximum distance* the crate gets from its starting point and the *time* it takes to __return__ to the starting point.

For inclined-plane problems that involve friction, here's what we'll do:

- Draw a
**free body diagram (FBD)**using tilted coordinate axes - Apply
**Newton's 2nd Law**(**∑***F*=)**ma** - Apply
**kinematics**(if needed) **Solve**for the target variable

Problems like this can look complicated at first, but once you break them down, they're just like other problems we've worked on before!

The kinetic friction force is calculated using

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

The four kinematics equations are

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

Also remember that when we use tilted coordinate axes for inclined plane problems like this, the weight * mg* makes a

$\overline{){\mathit{m}}{{\mathit{g}}}_{{\mathit{x}}}{\mathbf{=}}{\mathit{m}}{\mathit{g}}{\mathbf{}}{\mathbf{sin}}{\mathit{\theta}}\phantom{\rule{0ex}{0ex}}{\mathit{m}}{{\mathit{g}}}_{{\mathit{y}}}{\mathbf{=}}{\mathit{m}}{\mathit{g}}{\mathbf{}}{\mathbf{cos}}{\mathit{\theta}}}$

A crate is given an initial speed of 3.0 m/s up the 28° plane shown in the figure. Assume *μ _{k}* = 0.16.

(*a*) How far up the plane will it go?

(*b*) How much time elapses before it returns to its starting point?

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 Inclines with Friction concept. You can view video lessons to learn Inclines with Friction. Or if you need more Inclines with Friction practice, you can also practice Inclines with Friction practice problems.