I have a system with a mass $m$ attached to the end of a cable. The cable mass is assumed negligible.  The cable is attached to the ground at the one end while the other, with the attached mass $m$, is moving vertically with some known velocity $v$.  The cable is modelled as a 2nd order system with known values for the damping $b$ and spring $k$ coefficients.  I'm trying to use the conservation of energy principle to determine the ***impulse force*** resulting in the cable (at the moving end specifically) as the cable becomes taut, but I'm having trouble accounting for the energy dissipated due to the cable damping.  The cable is well damped and this significantly reduces the impulse force generated.  I've setup up the following system of equations, does anyone have any advice on how I can go about solving them? I'm guessing that I need to follow some sort of iterative process? Or is there an alternative method that I could follow that would be simpler?

$$ E_1 = E_2 $$
$$E_1 = \frac{1}{2}mv_1^2$$
$$E_2 = \frac{1}{2}kx^2 +mgx + \int_0^t bv(t)^2 dt$$
$$v(t) = v_0 + \int_0^ta(t) dt$$
$$a(t) = F(t)/m$$
$$F(t) = kx(t) + bv(t)+mg$$

$E_1$ is the kinetic energy the instant before the cable begins to stretch, while $E_2$ is the energy of the system when the mass has come to rest with the cable having stretched vertically some distance $x$.  I'm trying to solve **Equation 3** to calculate the change in cable length $x$, which I'll then use to calculate the impulse force from,
$$F_{impulse} = kx$$

Does the answer perhaps lie with using these equations with the conservation of momentum principle?

$$mv_1 = F_{impulse}*t$$

The mass is also assumed to not rotate on impact and I neglect the energy dissipation due to the propagation of lateral or longitudinal waves in the cable.