What are the conditions for different types of damping in oscillations?

[kraigandrews]

Homework Statement

A mass m moves in one dimension x subject to a restoring force −kx and a damping force −γ[(x)\dot]. What are the conditions for underdamped oscillations, overdamped oscillations, and critical damping?
Now, suppose m is 0.80 kg, γ is 1.18 kg/s, and the oscillations are critically damped. What is k?

The object starts at rest, displaced by some amount a. At what time is the displacement a/2?

Homework Equations

$\beta$=$\gamma$/(2m)

The Attempt at a Solution

Ok so I know k=0.435 N/m

and I know the solution for the diff eq for critical damping is:

x(t)=(A+Bt)e^(-$\beta$t)

my problem then becomes solving for A and B.
from the given info I would think x(0)=a and x'(t)=0
giving me A=a and B=$\beta$a.

I am not sure if this because when I solve for t at x(a/2) I do not get the right answer

 

[vela]

How did you solve for t? I found t=2.28 s.