(a) Derive the equation of motion for a mass-springdamper system in free vibration. Solve for the transient response \(x(t)\) that is driven by the initial conditions \(x(0)=x_{0}\) and \(\dot{x}(0)=v_{0}\).

(b) Assume parameter values: \(k=1 \mathrm{lb} /\) in, weight \(m g=100 \mathrm{lb}\), \(x_{0}=0, v_{0}=10 \mathrm{in} / \mathrm{s}\). Vary the damping constant \(c\) so that both underdamped and overdamped responses can be demonstrated. Try values of \(c\) such that the cases \(\zeta=0.1\) and \(\zeta=0.9\) are obtained.

(c) For the case with \(\zeta=0.1\), vary the initial velocity \(v_{0}\) and study the variation of the first intercept (zero displacement) as a function of initial velocity.