Problem $1($similar to the Birdcage problem from video$)$

In an imagined new Olympic event, a runner starts from rest a distance of $50$ m from a $25$ kg sled that waits at rest at the top of a $10$ m $-$long $30$ degree snow$-$covered incline. The runner accelerates as he or she runs toward the sled. They jump horizontally onto the sled and then travel down the incline and across the snow. They are brought to rest by a strong spring of spring constant $\backslash (500\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )$ embedded in very light padding. The distance from the bottom of the incline to the left edge of the spring is $8$ m $.$ The coefficient of friction between the sled and the snow is $0\mathrm{.}17.$

One athlete, whose mass is $52$ kg accelerates at a rate of $\backslash (0\mathrm{.}5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ while running. By how much will this athlete compress the spring? Show your solution neatly on the next page, including ALL steps of any protocol you use.