Final answer:
A damped mass-spring system with a spring constant of 4, mass of 1, and damping coefficient of 4 is critically damped. The position of the object over time, assuming an initial displacement of 2 and initial velocity of 0, is given by the equation y(t) = (2 + 2t)e^(-2t).
Step-by-step explanation:
A damped mass-spring system is critically damped when the damping coefficient is equal to the square root of four times the product of the mass and the spring constant. In this case, the damping coefficient y is equal to 4, and the square root of four times the product of the mass and the spring constant (√4mk) is also equal to 4. Therefore, the system is critically damped.
The equation of motion for a critically damped system is given by y(t) = (C1 + C2t)e^(-bt/2m), where C1 and C2 are constants that depend on the initial conditions. In this case, if the initial displacement y(0) = 2 and the initial velocity y'(0) = 0, the position of the object over time can be represented by the equation y(t) = (2 + 2t)e^(-2t).