Q$4[6$ points$]$: Consider the figure on the right. A box, with mass $\backslash ($ m$=1\backslash$mathrm$\{$~kg$\}\backslash ),$ is hanging from the ceiling by a spring with a spring constant $\backslash ($ k$=4\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ).$ It is also tied to the ground by a rope. The force of tension on the rope is $\backslash ($ T$=8\backslash$mathrm$\{$~N$\}\backslash ).$ Both the spring and the rope can be considered massless. Gravitational acceleration has a magnitude of $\backslash ($ g$=9\mathrm{.}8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash )$ and is directed downward. The box is at rest, and its distance to the ground is $1$ m $.$

At $\backslash ($ t$=0\backslash ),$ the rope is cut. a$)$ Find the velocity of the mass as a function of time after the rope is cut; and b$)$ find the maximum speed of the mass.

Q$5[8$ points$]$: Again consider the system above. The spring now additionally has internal friction that causes some dissipation. Specifically, it exerts a force of friction on the box's motion that can be described as $\backslash (-$b $\backslash$dot$\{$z$\}\backslash ),$ where $\backslash (\backslash$dot$\{$z$\}\backslash )$ is the vertical velocity of the box, and $\backslash ($ b$=4\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Again, at $\backslash ($ t$=0\backslash ),$ the rope is cut. a$)$ Find the velocity of the mass as a function of time after the rope is cut; and b$)$ find the maximum speed of the mass.