A recoil mechanism of a gun is designed as a spring and viscous damper in parallel such that the system has critical damping. A \(52-\mathrm{kg}\) cannon has a maximum recoil of \(50 \mathrm{~cm}\) after firing. Specify the stiffness and damping coefficient of the recoil mechanism such that the mechanism returns to within \(5 \mathrm{~mm}\) of firing position within \(0.5 \mathrm{~s}\) after firing.