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## 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)}+ ω

_{0}

^{2}x = 0

The characteristic equation for free oscillations:

x = x

_{0}cos(ω

_{0}t + δ)

where x

_{0}= A and δ = 0

x = Acosω

_{0}t

From here its easy enough to find the the speed at x = 0 is Aω

_{0}

b)x

^{(2)}+ 2βx

^{(1)}+ ω

_{0}

^{2}x = 0 when x < 0

The characteristic equation for this differential is:

x = Ae

^{-βt}cos(ω

_{1}t) where ω

_{1}is √(ω

_{0}

^{2}-β

^{2})

The period of oscillation is ∏/ω

_{1}

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