Dynamics of slinky?

Question

Semi-infinite spring of specific stiffness s = 8N and
linear density q = 0.5 kg/m originally rests on frictionless
horizontal surface. Force F = 6N is applied at moment
t=0 to the hook at free end of the spring and maintained
constant thereafter.

<=== F ===C-wwwwwwwwwwwwwww......


What is terminal speed of the hook?

Kirchwey is right in that a wave will travel along the spring with certain velocity, setting new and new portions of the spring in motion. We either need to find the speed of sound and use one conservation law, or write two conservations laws.

Kirchwey's conjecture could be proved as follows:

The equation of motion of the spring is
s ∂²u/∂x² + q ∂²u/∂t² = 0

This is pure 1D wave equation with speed of longitutional sound wave √s/q.

At this speed during time ΔT a portion of spring of length Lo = ΔT√s/q is set in motion, its momentum increasing by vqLo = vΔT√(sq).

Momentum gained by the spring is equal to FΔT, and using conservation of momentum one concludes that
F = v√(sq),

v = F/√(sq)
6N/√(8 N x 0.5 kg/m) =
3 (kg m/s²)/√(kg m/s² kg/m) =
3 m/s <--- answer

Answer 1

For materials, specific stiffness is defined as Young's modulus/density, so the units are (N/m^2)/(kg/m^3), or N-m/kg. I haven't found a definition of specific stiffness (S) for a spring, but I have an idea what it should be, stiffness k (=F/x) times length L0 of the relaxed spring, This means S = F*L0/x with units of N-m/m or N as you have given it. This reflects that fact that stiffness of a spring of given design (material and coil size) is inversely proportional to length. S is analogous to Young's modulus but for a specific spring design, and like Young's modulus is independent of length. A tensile force F will stretch such a spring by F*L0/S; spring constant K = S/L0.
In steady state, with the force applied, a portion of the spring moves at terminal velocity V and has 6 N tension throughout; the semi-infinite portion is motionless, forceless and relaxed; and a transition portion between them is both expanding and accelerating with tension increasing from 0 to 6 N. The transition portion can be envisioned as a black box which eats zero-velocity relaxed spring at a given mass rate and expels it at the same mass rate but with added potential energy (stretching) and kinetic energy (velocity). Assume it consumes L0 relaxed spring in unit time. It adds both kinetic and potential energy to that parcel of spring.
Potential energy PE = .5*K*X^2, where K = S/L0 = 8/L0, and X = F/K = 6*L0/8.
Kinetic energy KE = .5*Mass*V^2, where Mass = Q*L0 and V = L1/time = L1 assuming unit time.
The sum of PE+KE should = total energy input to the system by the moving force TE = F*L1.
Rather than try to solve this, I wrote a quick program using the above equations to converge on the value of V. The result is as follows:
V=5.7446 m/s; L1=5.7446 m; L0=3.2826 m; Mass=1.6413 kg; K=2.4371 N/m; X=2.4620 M; TE=34.467 J; PE=7.3859 J; KE=27.0815 J.