Lay Another Egg...

From the post "What Change KE?" dated 30 Jun 2016,

$\cfrac{d\,KE}{d\,t}=constant$

for a particle in a field.

$\cfrac{1}{2}m.v\cfrac{dv}{dt}=\cfrac{1}{2}v.m\cfrac{dv}{dt}=constant$

$\cfrac{1}{2}v.F=constant$

$v.F=B$  where $B$ is a constant.

The Newtonian force $F$ is inversely proportional to $v$.  But the drag force due to massive entanglement, sharing energy is proportional to $v^2$.

$F_{drag}=A.v^2$

where $A$ is a constant of proportionality.  For a particle driven in a field,

As such, the resultant force on the furthest point in the direction of travel on the particle is,

$F_c=F_{drag}-F$

$F_c=Av^2-\cfrac{B}{v}$

where $A$ and $B$ are both constants.

$F_c$ act as a centripetal force moving $\psi$ away from the furthest point in the direction of travel on the particle.  The particle has symmetry along $F$.  This movement of $\psi$ is in a plane perpendicular to the direction of travel, intersecting $F_c$ at the tip of the particle.  The result is to distort $\psi$ from a sphere into an egg shape.  This distortion by $F_c$ is along an infinitesimal circular arc centered at the $C.G$ of the particle as the forces involved passes through the $C.G$ of the particle.

This egg has no spin yet...

But more importantly, $F$ increases with higher concentration of $\psi$ at its rear.  $B$ increases with increased $\psi$.  The one with a bigger buttock gets a bigger kick in the rear.

Distortion stops when $F_c=0$, and $B=B^{'}$

$F_c=Av^2-\cfrac{B^{'}}{v}$

$v^3=\cfrac{B^{'}}{A}$

$v=\sqrt[3]{\cfrac{B^{'}}{A}}$ --- (*)

It is expected that $F$ is the force that causes the most distortion on the particle.  On any other point on the surface of the particle, the force components involved are less than $F$ and causes less distortion to $\psi$.

On a distorted egg,

$F_{drag}$ is a virtual force that always act through the $C.G$.  The force $F$ due to the field is the same.  These forces are of the same magnitude, but at a point not at the front tip of the particle, $F_{drag}$ they are not co-linear.

$\overset{\rightarrow }{F}+\overset{\rightarrow }{F_{ drag }}=\overset{\rightarrow }{F_{ R }}$

The resultant force, $F_{\small{R}}$ that develops can be resolved into two directions.

$\overset{\rightarrow }{ F_{ R }}=\overset{\rightarrow }{F^{ * }_{ D }}+\overset{\rightarrow }{F^{ * }_{ c }}$

The normal component of $F^{*}_{D}$ to the surface pulls the particle outwards and is countered by internal forces in the particle that holds $\psi$ together.  The tangential component of $F^{*}_{D}$ resistance the movement of $\psi$ from the front tip of the particle.

$F^{*}_c$ spins the particle perpendicular to the axis along its velocity.

So, when not spinning, the particle elongates from the front tip and its surface collapses inwards.  When the particle starts to spin, it elongates less and bulges outwards.

The bulge behind the particle causes $\psi$ to increase and so $F.c$ increases.  But at maximum distortion, the drag force and the force in the field is equal, at the front tip of the particle.

$F_{drag}=F$

When the field is attractive, spinning causes the particle to lower velocity, then the decrease in $F_{drag}$ ($\propto v^2$) and the increase in $F$ ($\propto \frac{1}{v}$) will cause the particle to increase in speed again, along $F$.

When the field is repulsive, spinning causes the particle to increase velocity, then the increase in $F_{drag}$ ($\propto v^2$) and the decrease in $F$ ($\propto \frac{1}{v}$) will increase $F_c$, the force that move $\psi$ away from the front tip of the particle and causes the particle to distort further and spin faster.  The particle does not destroy itself because the maximum linear speed that spinning can bring the particle to is at $c_{38.5}$.  There is no wave with an oscillating $B$ when the field is repulsive.

What happened to expression (*), that seems to suggest another speed limit.  (*) allows $B^{'}$ to be found.  $B$ changes to $B^{'}$ as the result of the particle spinning and distorts $\psi$.  The speed attained with spin is fixed; the maximum velocity of the particle is a constant at $c_{38.5}$ when the field is repulsive.

There is still no explicit mechanism by which $B$ oscillates when $EB$ is in an attractive field.  $B$ may not be sinusoidal.  But $c_{38.5}$ is another candidate for light speed when the field is repulsive.

$EB$ remains extraterrestrial and unreachable.