Consider a damped mass$-$spring system described by the equation:

$r(t)=1if1$

$(1)$ Using Laplace transforms, determine the system's transfer function $Q(s)$ and find its inverse Laplace transform $to$ obtain the system's impulse response $q(t)$

$(2)$ Using the convolution integral formula:

$y(t)={\displaystyle \underset{0}{\overset{t}{}}}q(t-)r()d$

compute the system response $y(t)$ for $1t<mn>2</mn>$

$(3)$ Compute the system response $y(t)$ for $t>2$

$(4)$ Sketch the graphs $ofy(t)$ and $r(t).$ Analyze the behavior $of$ the system's output, particularly why the response approaches zero for $t>2$