(Solved): This problem is an example of critically damped harmonic motion. A mass m=3kg is attached to bot ...

This problem is an example of critically damped harmonic motion. A mass $m=3\mathrm{kg}$ is attached to both a spring with spring constant $k=147\mathrm{N}/\mathrm{m}$ and a dash-pot with damping constant $c=42\mathrm{N}?\mathrm{s}/\mathrm{m}$. The ball is started in motion with initial position $x_0=5\mathrm{m}$ and initial velocity $v_0=?39\mathrm{m}/\mathrm{s}$. Determine the position function $x(t)$ in meters. Graph the function $x(t)$. Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected ( so $c=0$ ). Solve the resulting differential equation to find the position function $u(t)$. In this case the position function $u(t)$ can be written as $u(t)=C_0\cos \left({?}_{0}t?{?}_0\right)$. Determine $C_0,{?}_0$ and ${?}_0$. $\begin{array}{ll}{C}_0=1& \\ {?}_0=& \\ {?}_0=& \text{ (assume }\mathrm{v}?{?}_0<<?\text{ ) }\end{array}$ Finally, graph both function $x(t)$ and $u(t)$ in the same window to illustrate the effect of damping.