The most fundamental equation in mechanics ,
derived on the basis of absolute truths (derivation shown below), makes
it really inevitable to recognize the fundamental discrepancy hampering
clasical physics, deeply penetrating it and causing the unnecessary appearance
of the physically deformed view of nature brought about by quantum mechanics.
Here, the word is about the fundamental misunderstanding in classical
mechanics of the concept of motion.
Thus, if the three laws of Newton are called laws
of motion, then Newton's second and third laws being in fact laws of
rest are in contradiction with Newton's first law, which requires an
uncompensated force to be applied to the free body if it is to change
its state of rest or uniform rectilinear motion.
Indeed, Newton's second law ,
which is an expression of Newton's third law, especially if it's written
as
to reflect the fact that the force
invokes a compensating equal in magnitude but opposite in direction
inertial counterforce .
The application of such force
on the body cannot change the state of the body as required by Newton's
first law and invoke motion. Force
only causes matching
to appear.
Furthermore, the free body under the action of the constant force
cannot undergo real displacement
and a claim that work A has been done on the body is unsustained:
.
Another direct demonstration that mechanics treats incorectly as equations
of motion equations describing rest is seen in the most general equations
of mechanics -- Hamilton's equations. Not only is Newton's second law
derived from Hamilton's equations but they serve as the basis to form
the quantum mechanical operator called the Hamiltonian. Therefore, quantum
mechanics possesses the flawed birthmarks of classical mechanics. They
are passed on to quantum mechanics.
and
(7)
Now, if conservation of energy is in effect, as physics
accepts, then, .
Therefore,
and
(8)
Hence,
and
(9)
In other words, if conservation of energy is to be
obeyed then the body must be at rest.
Physics based on absolute truths, not on postulates
The Most Fundamental Equation in
Mechanics
After the obviously necessary abandonment
of quantum mechanics, the comporting of classical mechanics with the scientific
method and its further development along the road of truth and reason
must begin with recognizing the imprtance of an equation, elevated by
this author as the most fundamental equation of mechanics when treating
a free body under the action of a constant force.
It is not appreciated so far that an equation, which sometimes is used
in university physics exercises and which stems from the absolute truths
of physics, is in fact the most fundamental equation of mechanics, inherently
containing the so far neglected proper treatment of motion when a constant
force acts on a free body.
That most fundamental equation shows directly that when a real displacement
of a free body occurs under the action of a constant force, there must
necessarily be a change of velocity of that body, an inescapable fact
blanketly disregarded in physics. This follows from the
absolute truths of physics. Absolute truths of physics are, for
instance, its definitions.
Thus, it's an absolute truth that velocity
is
(10)
Also, it's an absolute truth that acceleration
a is
(11)
From these two definitions we can express
dt in two ways
(12)
and
(13)
Now, these two expressions being equal,
allows us to write
(14)
or, reordering it
(15)
which may also be written as
(16)
Now, we can integrate both sides of this
equality
(17)
factoring out of the integral the constant
acceleration a as well as the constant one half. From where we get
(18)
Finally, we have
(19)
This equalion of a parabola, has come
about from absolute truths of physics. Therefore, the connection between
the velocity v, acceleration a and the position x is set in stone and
is also an absolute truth. We now can plot it and see what conclusions
we can get from it.
For simplicity, we will take the acceleration to be .
Thus, the parabola becomes: and
the plot appears as follows:
Classical Velocity Limit
It is seen from the above that the graph of
as a function of square root of x is only an idealized plot, which
considers that acceleration is valid for any velocity, no matter how high.
However, as seen, as the velocity gets higher, the constant acceleration
becomes less and less significant and for some high velocities it becomes
so small in comparison that it can be neglected. In other words, the velocity
attains a constant value, reaching the clasical velocity limit.
Now, one may wonder, didn't I say that kinetic energy is the measure of
motion at high velocities but, on the other hand, didn't I say that when
at these high velocities the velocity becomes constant, that means that
the body is in uniform rectilinear motion, which I also said, is akin
to rest.
There's no contradiction.at all because from the point of view of the
body (for an observer at rest with the body), its state will indeed be
akin to rest. There's no experiment that can be performed in the coordinate
system attached to the body to prove that the body is not at rest but
is in uniform rectilinear motion. The coordinate system attached to the
body will be an inertial system. It will have no acceleration.
On the other hand, for a stationary observer outside of the body, with
respect to whom the body is moving at the said constant velocity, the
body will be in motion, specifically in uniform rectilinear motion, and
it will possess kinetic energy dependent on its mass and its constant
velocity.
Some Interesting Relations
When one looks at the above-shown graph of ,
one may wonder how is the velocity reaching a plateau. A parabola, such
as the one shown, doesn't have an asymptote. This is one of these moments
when we should remember that mathematics is only a helping tool in physics.
It's the physical reality which determine the conclusions. Thus, in this
case we should come to realize that numerically, when the velocities become
very high, they dwarf the constant acceleration. Therefore, acceleration
can be neglected and, therefore, force loses meaning. This can be seen
immediately from the complete expression of Newton's second law:
(20)
where the negligible acceleration makes the first
term after the equality disappear. The second term after the equality,
being equal to the first term (following from the most fundamental equation
of mechanics), is also negligible. Therefore, the whole expression for
the real force vanishes. Force has no meaning at high velocitites attained
when a free body initally at rest is acted upon by a constant force.
On the other hand, if we use the complete Newton's second law to express
the work needed to drive into motion the free body acted upon by a constant
force:
(21)
we see that none of the terms on the right side of
the equality can be neglected -- firstly, because the second term contains
the ever increasing positive value of velocity, squared at that. Secondly,
because the first term after the equality sign is equal (following from
the most fundamental equation of mechanics) to the non-neglectable second
term .
Further, that sameness of the two terms after the equality sign allows
us to write the above equation as:
(22)
which yields
(23)
or finally
(24)
As for the equation ,
eq.(22), it shows that even at low velocities the motion of a free body
under the action of a constant force can be expressed in energy terms;
that is, two times kinetic energy. Similar to the finding thst the complete
Newton's second law is
and not just .
Of course, by utilizing the fact that the two terms after the equality
sign of
are equal we may write it this way (considering also , for simplicity,
that )
(25)
or
(26)
This way of expressing energy conceals the origin
of the motion causing the displacement and only states that had there
been such displacement of the mass m, it would have required energy equal
to the product of the mass and the displacement itself. So, again, we
must consider the physical meaning of what was derived mathematically.
This also demonstrates the equivalency rather than transformation of other
quantities into energy. Thus,
only means the energy has its equivalent expression in terms of mass and
not that mass can be transformed into energy. The same way,
shows another equivalent expression of energy for low values of ,
where acceleration is
still not negligible.
Author's
email:
qmiscm@hotmail.com
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