$4.(20$ points$)$ A child riding a bicycle can be modeled as a spring$-$mass$-$damper system with an equivalent weight, stiffness, and damping constant of $\backslash (800\backslash$mathrm$\{$~N$\},50,000\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ),$ and $\backslash (1,000\backslash$mathrm$\{$~N$\}-\backslash$mathrm$\{$s$\}/\backslash$mathrm$\{$m$\}\backslash ),$ respectively. The poor construction of the concrete blocks on the sidewalk cause there to be a sudden $5$ cm drop, as indicated in the figure below. If the speed of the bicycle is $\backslash (18\backslash$mathrm$\{$~km$\}/\backslash$mathrm$\{$hr$\}\backslash )$ to the right, determine the displacement of the boy in the vertical direction for all time. Assume that the bicycle is free of vertical vibration before encountering the step change in the vertical displacement. Hint: Consider when the bicycle is at the edge of the $\backslash (\backslash$operatorname$\{$drop$\}(\backslash )$ point $\backslash ($ A$)\backslash )$ as time $\backslash ($ t$=0\backslash$mathrm$\{$~s$\}\backslash ).$