Chapter 4.9, Problem 17E

Consider the equation for free mechanical vibration, $m{y}^{″}+b{y}^{\prime }+ky=0$, and assume the motion is critically damped. Let $y\left(0\right)=y_0$, $y^{\prime }\left(0\right)=v_0$ and assume $y_0\ne 0$.

a. Prove that the mass will pass through its equilibrium at exactly one positive time if and only if $\frac{-2m{y}_0}{2m{v}_0+b{y}_0}>0$.

b. Use computer software to illustrate part (a) for a specific choice of $m$, $b$, $y_0$, and $v_0$. Be sure to include an appropriate graph in your illustration.