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## Homework Statement

A critically damped oscillator with natural frequency [tex]\omega[/tex] starts out at position [tex]x_0>0[/tex]. 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,

[tex]x(t)=e^{(-\gammat)}(A+Bt) where \gamma=\omega=\sqrt{k/m}[/tex]

## The Attempt at a Solution

Well first I derived the above equation (and verified it with my textbook). I then evaluated the initial position:

[tex]x(0)=A[/tex].

I then took the derivative of the position function to get velocity:

[tex]v(t)=(e^{-\gamma t}) ( B-Bt\gamma -A\gamma)[/tex]

Setting t=0 I obtained

[tex]v(0)=B-A\gamma[/tex] recalling that x

_{0}=A, [tex]v(0)= B-x_0\gamma[/tex]. I then solved for Beta: [tex] B=v_0 +\gamma x_0. [/tex]

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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