# Critically damped oscillator: Classical mechanics help!

Theorem.

## Homework Statement

A critically damped oscillator with natural frequency $$\omega$$ starts out at position $$x_0>0$$. What is the maximum initial speed (directed towards the origin) it can have and not cross the origin?

## Homework Equations

For the case of critical damping,
$$x(t)=e^{(-\gammat)}(A+Bt) where \gamma=\omega=\sqrt{k/m}$$

## The Attempt at a Solution

Well first I derived the above equation (and verified it with my textbook). I then evaluated the initial position:
$$x(0)=A$$.
I then took the derivative of the position function to get velocity:
$$v(t)=(e^{-\gamma t}) ( B-Bt\gamma -A\gamma)$$
Setting t=0 I obtained
$$v(0)=B-A\gamma$$ recalling that x0=A, $$v(0)= B-x_0\gamma$$. I then solved for Beta: $$B=v_0 +\gamma x_0.$$
I then tried substituting this back into the position equation, and solving for the initial velocity, the program is there is always time dependency that i cant get rid of.... what am i doing wrong? any advice would be much appreciated!
Thanks

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