Work and Energy Can Reveal Distance, Too

We know that once the person stops pushing, friction will bring the package to rest. In terms of energy, friction does negative work until it has removed all of the package’s kinetic energy. The work done by friction is the force of friction times the distance traveled times the cosine of the angle between the friction force and displacement; hence, this gives us a way of finding the distance traveled after the person stops pushing.

The normal force and force of gravity cancel in calculating the net force. The horizontal friction force is then the net force, and it acts opposite to the displacement, so $\theta =\text{180º}$. To reduce the kinetic energy of the package to zero, the work $W_{\text{fr}}$ by friction must be minus the kinetic energy that the package started with plus what the package accumulated due to the pushing. Thus $W_{\text{fr}}=-\text{95}\text{.}\text{75 J}$. Furthermore, $W_{\text{fr}}=fd\prime \text{cos}\theta \text{= –}fd\prime$, where $d\prime$ is the distance it takes to stop. Thus,

and so

This is a reasonable distance for a package to coast on a relatively friction-free conveyor system. Note that the work done by friction is negative (the force is in the opposite direction of motion), so it removes the kinetic energy.