The formula representing kinetic energy is incorrect. Energy is not transformed in proportion to velocity squared but in proportion to momentum. Because energy is so basic, a large part of the science of physics, beyond Newton's laws, is in error.
Contradictions appear throughout energy applications due to the error. Rockets show the most obvious evidence, because they transform energy without mechanical motion machinery. But there is also evidence in mechanical engines, which produce abnormal rates of energy addition, called power. Collision analysis shows abnormal transfers of energy, which creates a confusing basis for thermodynamics.
The contradictions in the definition of energy are most identifiable in the original concept created by Gottfried Leibniz (1646-1716). It was preceded by a correct concept derived by René Descartes (1596-1650) who stated that there is a fixed amount of motion (as momentum) in the universe, because it is always conserved through interactions.
At that time, pendulums were being used to study collisions, because velocity could be determined from height, and friction was minimal. Christian Huygens (1629-1695), in Holland, followed with extensive mathematical analysis, and his notes included mv² as a conserved quantity during elastic collisions. He apparently ignored mv² as the coincidence that it is; but then Leibniz, in Germany, picked up on it and turned it into the erroneous concept of energy in 1686.
Leibniz attempted to show that mv², which he called "vis viva", meaning living force, is the conserved quantity of motion, not momentum (mv). He effectively showed the contradictions between the concepts. The contradictions are important in proving what energy is; but Leibniz did not properly prove what energy is, because he did not consider both alternatives with a complete analysis.
Leibniz's error was in picking the wrong starting point for his analysis. It guaranteed an erroneous end point for a concept of energy. He arbitrarily started with a force-distance analysis, which mathematically conserves mv² under certain conditions. He chose an example of such conditions by relating to falling objects. Falling objects create a parallel to inelastic collisions and energy applications (because the force moves with the mass), where mv and mv² have different dynamics and cannot both be conserved.
This point is not recognized in modern physics, because mv and mv² are supposedly always conserved as momentum and energy under all conditions. But in fact, Leibniz showed very clearly that they are not both conserved outside elastic collisions.
There are numerous conceptual errors in the logic of energy, but they seem to be too etheric to be weighed in modern physics. It's too easy to deny them and state a different logic. They are pointed out in this analysis not as something to base an argument on but as conceptual alignment for the correct definition of energy.
The proof for the definition of energy is based on hard mathematics, the undisputed characteristics of constant powered rockets and the Leibniz analysis of falling objects. In other words, it corrects the Leibniz analysis.
A counter-argument for correcting the definition of energy (by making energy and momentum synonyms) is that momentum is not conserved through levers including gear boxes, because amplified force amplifies momentum.
There are two major points to be made about this. One, it is not a law of nature that says energy must do something other than momentum does. In fact, the absolute conservation of energy is an assumption in conflict with evidence.
Potential energy is an example. It is not energy. If a person robs a bank and leaves potential money behind, he gets thrown in jail. The same should be true of energy. (How long the jail term should be goes beyond the scope of this subject.)
If the arbitrariness of the conservation of energy can be considered, then a second major point is that the logic based on unquestionable mathematics must take precedent over the assumptions including the observations (which must have assumptions with them). The assumptions with observations are very tricky in not being recognized as assumptions at all.
Logic over Perception.
There is a tendency to put perception above logic, as if perception were proven fact. Supposedly, seeing is believing. But perception is the basis of deception and illusion. Logic is the basis of knowledge. The purpose of science is to add logic to a subject. There are numerous problems in the logic of ½mv² being energy, as explained later.
Contradictions often stem from superficial assumptions being in error, such as levers not conserving energy as momentum. To correct the assumptions or errors, the more basic points must take precedent over less basic points. For example, force amplification is not understood well enough to draw conclusions about energy being transferred by levers. A section on levers demonstrates these points.
The conservation of energy is assumed to be an unshakable pillar of science. For certain reactions (perhaps all transformations), the conservation of energy appears to be invariable. The problem is in extending the concept infinitely to assume that there must then be energy that can be located someplace and in a certain quantity, and if it cannot be located, call it potential energy.
With these concepts, there is an assumption that energy is an entity, like phlogiston, which must always exist in a certain quantity. That's not what energy is.
What Energy Is.
So let's try to conceptualize what energy is. Energy is observed as force setting matter in motion. It was originally called living force. Is the energy therefore in the force or the motion? Since force and motion are interchangeable, it is arbitrary to say that energy is in one but not the other.
Light, for example, is energy which cannot be detected unless it sets matter in motion; and it is quantitated in proportion to the motion. But light also exists as something which is not setting matter in motion, while it is moving through space.
The peculiarities of energy indicate that there is an etheric medium which influences matter in ways which are predictable but not always quantitatively "conserved."
Supposedly, an etheric medium for conducting light waves or promoting energy does not exist, because the Michelson-Morley experiment of a century ago (since repeated) failed to find it. Failing to find it should have been of little significance. Since it interacts thoroughly with matter, its motion should be influenced by the earth, which would negate the Michelson-Morley experiment. And recently, an analysis shows that when the curvature of the earth's rotation is added to the mathematics, the result of the Michelson-Morley experiment shows precisely the opposite of previous assumptions. (See link below.)
The characteristics of light including its constant velocity and absence of matter require light to be a wave in an etheric medium.
Gravity and stress forces also indicate that an etheric medium conducts waves which influence matter by creating forces. The influence of gravity through space requires an etheric medium. But when gravity creates a force acting upon another mass, it doesn't give up something. In that sense, gravity creates and destroys energy. For this reason, its influences are often called potential energy.
For example, when an object is lifted and set at a higher location, energy is required. Where does it go? The object is said to acquire potential energy. But the so-called potential energy is the ability of gravity to resist the motion and recreate a motion later. In doing so, it destroys and creates energy.
Stress forces are similar to gravity but even more mysterious in their effects upon energy. What stress and gravity have in common is that they would instantly disappear unless they were recreated. Their ability to disappear is evident when they are accelerating masses. The force is diminished or disappears. Therefore, waves in an etheric medium appear to continuously recreate stress forces and gravity.
The ability of stress to amplify force is even more mysterious, regardless of the definition of energy. Since force is interchangeable with motion, amplifying force does not conserve energy. The force-distance analysis creates the appearance of amplified force not amplifying energy, because there is a mathematical relationship between the symbols. But the symbols (½mv² and Fs) are meaningless. Their absence of a relationship to objective reality immunizes them from contradictions with it, even with such mysterious complexities as force amplification.