Problem $1.[50$ Points$].$ The following figure shows a SDOF dynamic system with the weight $w=100lb,$ natural period $T_n=1\mathrm{.}0sec$ and the critical damping ratio ${}_n=5\%.$ This system is subjected to the pulse force $P(t).$

For this dynamic system, perform the following tasks:

$($a$)$ Find the displacement response for $P=P_1(t)$ for $t_d=T_n$ and $P_0=50$ Kips. $[10$ Points$]$

$($b$)$ Find the displacement response for $P=P_2(t)$ for $t_t=T_n$ and $P_0=50$ Kips. $[10$ Points$]$

$($c$)$ Use MATLAB $($or similar plotting tools$)$ to plot the displacement responses in $($a$)$ and $($b$)$ and compare the maximum displacements for the two cases with $t_d=T_n$ and $t_d=0\mathrm{.}1T_n.$ Discuss your observations. $[30$ Points$]$