PROBLEM $13\mathrm{.}68$

A spring is used to stop a $\backslash (50-\backslash$mathrm$\{$kg$\}\backslash )$ package which is moving down a $\backslash (20^\{\backslash$circ$\}\backslash )$ incline. The spring has a constant $\backslash ($ k$=30\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ and is held by cables so that it is initially compressed $50$ mm $.$ Knowing that the velocity of the package is $\backslash (2\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ when it is $8$ m from the spring and neglecting friction, determine the maximum additional deformation of the spring in bringing the package to rest.

Hint: Don't forget that when the spring is compressing, the package will lose a bit more height. You may have to use the quadratic formula to solve for $\backslash ($ x $\backslash )$ once you develop an equation based on the conservation of energy.