1. The problem statement, all variables and given/known data (A) A damped oscillator is described by the equation m x′′ = −b x′− kx . What is the condition for critical damping? Assume this condition is satisfied. (B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0. (C) Suppose k/m = (2π rad/s)2 and v0=10 m/s. Plot, by hand, an accurate graph of x(t). Use graph paper. Use an appropriate range of t. 2. Relevant equations For critically damped, β2 = w02 where β = b/(2m) and w0 = √(k/m) 3. The attempt at a solution Ok, for this problem, what I did initially was find the general form of position for a critically damped oscillator, which is: x(t) = (A + B*t)*e-β*t and the velocity function is: v(t) = -Aβe-βt + (Be-βt - Bβte-βt) Using the conditions given, I found: x(0) = A (obviously) which we don't know x(0) B = v0 + Aβ and x(t) can be rewritten as: x(t) = A(e-βt + βte-βt) + v0te-βt This is where I run into a wall. I can't seem to solve for A. I believe that x(0) should also be the max displacement since there is no driver for the impulse force, so A should be the max displacement, but this doesn't seem to get me anywhere. Any help on solving for A? I know how to do the rest other than that.