Critically damped oscillator — UPSC Physics Optional CSE 2012 Paper I

The motion of a damped mechanical oscillator is represented by m\ddot{x}+\alpha\dot{x}+\beta x=0 where m, \alpha and \beta are constants. The oscillator is critically damped. The system is given an impulse at x=0 and t=0, resulting in an initial velocity v. After how much time the system experiences maximum displacement?