Overshooting in a critically damped oscillator

[Arijun]

How can we tell whether a given v₀ will cause an oscillator to overshoot the equilibrium? If the velocity high enough, we know the oscillator will overshoot and return to equilibrium. But if v₀ is low, the system would act like it came from a point a bit farther out and not overshoot (right?). So how would we figure out which it will be?

 

[Antiphon]

The response to a step of function of any magnitude will be the same shape but scaled to the size of the input step.

 

[Arijun]

I don't understand your response at all. Could you clarify?