# Find coefficient of friction given initial and final velocity?

1 vote
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I am not sure how to approach a problem involving the friction coefficient when only given initial velocity, final velocity, and distance traveled. I could combine the friction and kinematic equations, but I am only familiar with doing that for one velocity...

Here is the specific problem I had trouble with:

A cardboard box of unknown mass is sliding upon a mythical frictionless surface. The box has a velocity of 4.56 m/s when it encounters a bit of friction. After sliding 0.700m, the box has a velocity of 3.33 m/s.
What is the coefficient of friction of the surface?

How do I solve this problem in particular, and what is the general procedure to solve problems like this?

asked Aug 24, 2017
edited Jul 20, 2018
Please upvote my answer if you are happy with it. Otherwise, please post a comment.

## 1 Answer

1 vote

The general approach is to apply the Work-Energy Theorem : work done on the body by an external force = increase in KE of the body.

Friction does not depend on velocity. The friction force is the same throughout this motion, so the acceleration is uniform. Use the kinematic equations to find the uniform acceleration $a$ :
$$v^2=u^2+2as$$ where $u, v$ are initial and final velocities and $s$ is the distance which the box has slid over the rough part of the surface.

The friction force is $F=ma=\mu N$ where $m$ is the unknown mass of the box, $\mu$ is the coefficient of friction and $N=mg$ is the normal force. Then $a=\mu g$ from which you can deduce the value of $\mu$.

answered Aug 24, 2017 by (28,746 points)
edited Jul 20, 2018
How are you defining s? Also, how could I calculate a based on that equation if I don't know u?
$s$ is the distance of sliding. $u, v$ are initial and final velocities.