Homework H$\mathrm{.}6.$G

Given: Particle A $($having a mass of $m)$ is attached to a grounded spring of stiffness $k$ and a dashpot with a damping coefficient $c.$ Let $x$ represent the motion of A $,$ with $x=0$ when the spring is unstretched. At an instant when A is at rest and with the spring being unstretched$/$uncompressed $($that is$,$ when $x=0),$ particle B $($of mass $m),$ traveling with a speed of $v_0,$ strikes A$.$ On the impact with A$,$ block B immediately sticks to block A$.$

Find: For this problem:

$($a$)$ Determine the speed of A immediately after B sticks to it$.$

$($b$)$ Derive the dynamical equation of motion $($EOM$)$ of $A+B$ in terms of the coordinate $x$ describing the motion after the two blocks stick together;

$($c$)$ Determine the undamped natural frequency ${}_n,$ the damping ratio $$ and the damped natural frequency ${}_d$ for the system;

$($d$)$ Determine the response $x(t)$ of the system after A and B stick together.

Use the following parameters in your analysis: $m=10kg,k=3000\frac{N}{m}$ and $c=360k\frac{g}{s}.$