You Are Given a Spring-Mass System with a Mass of 1

Essay

You are given a spring-mass system with a mass of 1 slug, a damping constant 8 lb-sec/foot, and a spring constant of 16 lbs/foot. Suppose the mass is released from rest 1.5 feet below equilibrium, and after 5π seconds the system is given a sharp blow downward which imparts a unit impulse.
(i) Write down a second-order initial value problem whose solution x(t) is the equation of motion for this system.
(ii) Find the Laplace transform X(s) of the solution x(t) of the initial value problem you formulated in part (i).
(iii) Compute the inverse Laplace transform of your function in part (ii).