The system below is initially at rest, suppose the cart is set in motion by a unit impulsive force.

$($a$)$ Is it possible to step the motion by another impulsive force? What would be its magnitude and instance of application?

$($b$)$ If the system is set in motion by a unit step force, is there a value for its spring coefficient such that the cart would not oscillate? Explain.

$($c$)$ Add a damping element to the system, what should be the "value" for the damping coefficient such that the system becomes critically damped?coefficient such that the system becomes critically damped?