6. A spring with a spring constant of k = 2 is placed in a medium with a damping force numerically equal to 4 times the instantaneous velocity. If an object of mass m is suspended from the spring, determine all values of m for which the subsequent free motion will be non-oscillatory.

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When an object with a mass of 5 kg is connected to the end of a spring, the spring extends 10 meters and reaches equilibrium. This spring-mass system is placed in an environment with a damping constant of 5 kilograms/second, and the mass is pulled down √ 3 meters from the equilibrium position and released without initial velocity. Find the function that shows the time-dependent change of the position of the given damped spring-mass system. Determine the period and amplitude of the movement. Determine how many seconds it takes for the system to first pass the maximum position (in the positive or negative direction) and the equilibrium position after t = 0. (gravity = 10 meters/second) Then transform the equation of the mass-spring system into a system of equations and draw a sample orbit by creating the vector field of the system and interpret the shape of this orbit.

The maximum permissible recoil distance of a gun is specified as 0.5 m. If the initial recoil velocity is to be between 8 m/s and 10 m/s, find the mass of the gun and the spring stiffness of the recoil mechanism. Assume that a critically damped dashpot is used in the recoil mechanism and the mass of the gun has to be at least 500 kg.