1. The problem statement, all variables and given/known data A mass on the end of a spring is released from rest at position x0. The experiment is repeated, but now with the system immersed in a fluid that causes the motion to be critically damped. Show that the maximum speed of the mass in the first case is e times the maximum speed in the second case. This is question is out of Morin 4.28 3. The attempt at a solution So for the initial case, we have x(t) = Acost(wt) => v(t) = -Awsin(wt). From initial conditions, we have x(0) = x0 and v(0) = 0. Using this, we find that A=x0. Thus v_max = +/- x0*w. The next part is more confusing. Taking the solution to the critically damped case, we have x(t) = e^(-γt) * (A+Bt) Then solving for the initial conditions, I have A = x0 and B = γ*x0. Taking the derivative of x, we have v(t) = e^(γt) * (-γ^2 * x0 * t) . Note γ = w. But you can't solve for v_max from there. So I'm kinda stuck. Any advice?