$3.$ A packaged product is dropped from a height of $38$ inches and is found to have rebounded $12$ inches using a high speed camera. What was the coefficient of restitution of the impact?

A product weighing $12$ lb $.$ can sustain a maximum shock of $80$ g$\text{'}$s$.$ Using a shock theory approach, find the spring coefficient for a linear spring which will limit the acceleration of the product to $80$ g $\text{'}$s in a $36$ inch drop. What is the maximum deflection of the product on the cushion and find the duration of the shock pulse sustained by the product. Assume a COR of $0\mathrm{.}5.$