Final answer:
The position function for the critically damped harmonic motion of the mass is x(t) = (7 - 22t)e^(-5t), derived using the given initial conditions and values for mass, spring constant, and damping constant.
Step-by-step explanation:
The question pertains to the topic of damped harmonic motion in classical mechanics, where the goal is to determine the position function x(t) of a mass attached to a spring and a dash-pot. Given values are mass m = 5 kg, spring constant k = 320 N/m, damping constant c = 80 N·s/m, initial position x₀ = 7 m, and initial velocity v₀ = -57 m/s. Since the problem states that the motion is critically damped, we use the formula x(t) = (C1 + C2t)e^(-βt), where β is the damping ratio β = c/(2√(mk)), and C1 and C2 are constants determined by the initial conditions.
To solve for these constants, we first calculate β:
β = 80/(2√(5*320)) = 80/(2*8) = 5 s⁻¹.
This gives us a differential equation which we solve using the initial conditions:
x(0) = C1 = 7 m,
x'(0) = -βC1 + C2 = -57 m/s.
So C2 = x'(0) + βC1 = -57 + (5*7) = -22 m/s.
The final position function is therefore:
x(t) = (7 - 22t)e^(-5t).