The mass is 0.5 meters from its equilibrium position after 1 second.
The determine the exact position of the mass after one second, we can use the equation for critically damped motion:
x(t) = A * e^(-bt) + C
where:
x(t) represents the position of the mass at time t
A represents the initial amplitude
b represents the damping coefficient
C represents the equilibrium position
Since the mass comes to rest 0.5 meters below the release point, we know C = -0.5. Additionally, critical damping implies b = √(4mk), where m is the mass and k is the spring constant.
The initial velocity of the mass is zero, which allows us to determine the initial amplitude A. Setting x'(t) = 0 in the equation for velocity, we get:
x'(t) = -bA * e^(-bt) = 0
Solving for A, we find A = 0.
Substituting the known values into the position equation, we obtain:
x(t) = e^(-√(4mk)t)
Evaluating x(t) at t = 1, we get:
x(1) = e^(-√(4mk)) ≈ 0
Therefore, after one second, the mass is very close to its final resting position of 0.5 meters below the release point, indicating the effectiveness of critical damping in minimizing oscillations and quickly returning the system to equilibrium.