$($a$)$ With $m=10kg,C=300k\frac{g}{s},$ and $k=1000\frac{N}{m},$ determine the displacement $x(t)$ of the mass for the initial conditions given by: $x(0)=10mm,$ and $v(0)=0$

$($b$)$Determine the displacement $x(t)$ of the mass with the previous damper replaced by one with $C=3k\frac{g}{s},$ using the same initial conditions. $($The mass and stiffness for the stay the same.$)$

${}_n=\sqrt[\phantom{2}]{K_m}=\sqrt[\phantom{2}]{\frac{1000}{10}}=10,z_1=4_{cc}=\frac{c}{2\sqrt[\phantom{2}]{k_n}}=\frac{300}{2\sqrt[\phantom{2}]{100010}}=1\mathrm{.}5$

$x=A_e(-Z_{{}_n}+u_n\sqrt[\phantom{2}]{Z^2-1})$