Maximum speed of overdamped oscillator without crossing origin

[MeMoses]

Homework Statement

An overdamped oscillator with natural frequency w and damping coefficient g starts out at postion x0 > 0. What is the maximum initial speed towards the origin it can have without crossing the origin?

Homework Equations

x(t) for overdamped oscillator

The Attempt at a Solution

x(t) for a general overdamped oscillator has been solved already. However if I only know x0 > 0 and nothing else, how do I go about determining the maximum speed. I can't see how this problem will workout and I can't get it started. Thanks for any help

 

[tiny-tim]

Hi MeMoses!

x(t) for a general overdamped oscillator has been solved already. However if I only know x0 > 0 and nothing else, how do I go about determining the maximum speed. I can't see how this problem will workout and I can't get it started.

Just find the answer as a function of x_o

 

[MeMoses]

Edit: My idea made no sense. So how do find this maximum v0?

 

[tiny-tim]

Edit: My idea made no sense. So how do find this maximum v0?

what is your x(t) equation?

 

[MeMoses]

I get x(t) = Ae**((-g-z)t) + Be(-(g-z)t) with z=sqrt(g**2 - w**2). At t=0 I get x0 = A + B

 

[tiny-tim]

Hi MeMoses!

I get x(t) = Ae**((-g-z)t) + Be(-(g-z)t) with z=sqrt(g**2 - w**2). At t=0 I get x0 = A + B

(hmm … I'm not sure how that's supposed to be read … but anyway …)

So what is the maximum value of x(t) (presumably at t = ∞) ?

And what is the intitial speed?

 

[MeMoses]

Sorry i type that from my phone. The lim x(t) as t approaches infinity is 0, it will always go to 0 eventually but how do I know if it crosses the origin or not and how can I find the initial speed using what you told me? Edit: also how can I solve the coefficients?