Periodically Dampened Oscillator

[Squire1514]

Homework Statement

A body with mass m is connected to a spring in 1D and is at rest at X = A > 0. For the region X > 0, the only force acting on the mass is the restoring force of the spring. For the region X < 0, a viscous fluid introduces damping into the system.
a) Find the speed of the particle at X = 0.
b) Find the speed of the particle at X = 0, as it emerges from the X < 0 region.
c) Find the maximal position D the body reaches in the positive region after having left the negative region

Homework Equations

F = -kx for x > 0
F = -kx-bv for x < 0

The Attempt at a Solution

a) x⁽²⁾ +k/m x = x⁽²⁾ + ω₀²x = 0
The characteristic equation for free oscillations:
x = x₀cos(ω₀t + δ)
where x₀ = A and δ = 0
x = Acosω₀t
From here its easy enough to find the the speed at x = 0 is Aω₀
b)x⁽²⁾+ 2βx⁽¹⁾+ ω₀²x = 0 when x < 0
The characteristic equation for this differential is:
x = Ae^-βtcos(ω₁t) where ω₁ is √(ω₀²-β²)
The period of oscillation is ∏/ω₁

My confusion at this point is how to incorporate the initial velocity into this equation.
Once I know that I can easily plug in the period into the velocity equation and find the new velocity at x = 0

 

[TSny]

b)x⁽²⁾+ 2βx⁽¹⁾+ ω₀²x = 0 when x < 0
The characteristic equation for this differential is:
x = Ae^-βtcos(ω₁t) where ω₁ is √(ω₀²-β²)

You need to reconsider how you wrote the solution here. If you redefine t = 0 to be the time the body enters the viscous region, then what should x equal when t = 0?

Also, you can't assume that the constant factor "A" in the solution for x < 0 is the same A as in part (a).