Slinky as Resolution of Zeno’s Arrow Paradox

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Abstract


The motion of a slinky suggests a resolution of Zeno’s Arrow Paradox as a combination of compression-release and switch of stability, where the the slinky appears as a soliton wave, which itself generates the medium through which it propagates.


The Question

Zeno of Elea (490-430 BC), member of the pre-Socratic Eliatic School founded by Parmenides, questioned the concept of change and motion in his famous arrow paradox:

In the knol Resolution of Zeno’s Paradox of Particle Motion we showed that the paradox still after 2.500 years lacks a convincing resolution, and we suggested a resolution based on wave motion. 

A fundamental question of wave propagation is the nature of the medium through which the wave propagates: Is it material as in the case of sound waves in air, or is it immaterial as in the case of light waves in vacuum?  If the flying arrow is a wave, which is the medium through which it propagates? It is not enough to say that it is air, because an arrow can fly also in vacuum.
We are led to the following basic question:
  • can a wave itself act as the medium through which it propagates?

An Answer

It turns out that a slinky can serve as an answer! To see this take a look at this movie . We see that the motion of a slinky can be described as follows:
  • oscillation between two forms of energy: elastic energy and kinetic energy
  • compression stores elastic energy 
  • elastic energy is transformed into kinetic energy when the slinky expands
  • there is a critical moment with the slinky fully compressed in which the downward forward motion of the top ring is reflected in upward forward (and not upward backward motion which would lead to motion on the spot)  
  • the slinky forms itself the medium through which it as a wave propagates
  • the slinky acts like a soliton wave.

We understand that the slinky offers a model for resolution Zeno’s paradox as a wave which itself generates the medium through which it propagates. 

What is Mass?

You can take this model one step further, and view the work required to compress the slinky from an uncompressed rest state, as an investment into kinetic energy of motion, just as a body can be accellerated from rest by the action of a force and gain kinetic energy. 
This would mean that the slinky has inertial mass and that it can move with different velocities depending on the amount of work invested in the initial compression. We may compare with the propagation of massless electromagnetic waves with given fixed speed of light.  This connects to the knol Does the Earth Rotate?  
suggesting to define mass as inertial mass M in terms of kinetic energy K and velocity V from the formula 
                                                         K = 1/2 x M x V x V