1. The problem statement, all variables and given/known data (A) The damped oscillator is described by the equation mx''+bx'+kx=0. What is the condition for critical damping expressed in terms of m,b,k. Assume this is satisfied. (B) For t<0 the mass is at rest (x=0). This mass is set in motion at t=0 by a sharp impulsive force so that the velocity is v0 at time t=0. Determin the position x(t) for t>0. (C) Suppose k/m=(2*pi rad/s)2 and v0=10 m/s. Plot an accurate graph of x(t) using an appropriate range for t. 2. Relevant equations For critical damping; B=W0 General solution for critically damped oscillator: x(t)=(C1+C2t)e-βt 3. The attempt at a solution I am running into issues at (B) where I'm not entirely sure how to find the values of the two constants in the general solution. (A) I have shown through the given relations that b2=4km which I believe is the correct answer. (B) Given the general solution for a critically damped oscillator I can take its derivative to find v(t): v(t)=e-βt(C2-C2βt-C1β) I know that at t=0 v(0)=v0 so I can solve the solution for v(t) for a constant: C2=v0+C1β I'm just not sure about how to solve for x(t). Can I consider x(0)=0 since it is set in motion at t=0 but since no time has elapsed it has not moved? If not, how do I proceed in solving for the constants? I believe the rest of the problem will fall into place without much issue after that point. Thank you for your time and help!