The context of this problem is a table$-$top game in which a puck is shot at a target, up a slope $\backslash (\backslash$theta$=13^\{\backslash$circ$\}\backslash ),$ and length $\backslash ($ d$=1\mathrm{.}1$ m $\backslash )$ with a spring, as shown. The slope has friction and the coefficient of kinetic friction is equal to $\backslash (\backslash$mu$\_\{$k$\}=0\mathrm{.}15\backslash )$

The puck is initially at rest and has a mass $\backslash (\backslash$mathrm$\{$m$\}=0\mathrm{.}12\backslash$mathrm$\{$~kg$\}\backslash )).$ The spring, with spring constant $\backslash (\backslash$mathrm$\{$k$\}=203\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ),$ is compressed by a distance x $.$

The puck has to hit the target with a certain velocity to win, and that has been set at $\backslash (2\mathrm{.}69\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$($a$)$ What is the minimum compression of the spring to achieve this?

$($b$)$ The puck will bounce off the target, back down the slope. Let's assume the speed of the puck after the bounce is the same just before the bounce $[$it will be in the opposite direction of course; means no energy lost during the bounce$].$

The puck will start coming back down $[$across the rough surface$]$ and will eventually hit the spring again.

What is the final compression of the spring?