1. quick proof
Word Summary without Math
Physicists define kinetic energy as mass times velocity squared. Since nothing moves at velocity squared, the equation cannot represent real energy. Energy is real as gallons of fuel and watts of electricity. It must therefore be represented as something that actually exists.
Origins of the error
The origins of the error are in a premise stated by Gottfried Leibniz in 1686. Scientists were studying motion using pendulums, because they could determine velocity from height of the swing. Christian Huygens noticed that when hard and elastic objects collided, both mass times velocity and mass times velocity squared were conserved. Conserved means the totals stay the same, even while the component quantities are rearranged. Mass times velocity squared was ignored as a strange artifact. So Rene Descartes published a paper around 1635 stating that there is a fixed amount of motion in the universe, because it is always conserved during interactions. By motion he meant mass times velocity nonsquared. Leibniz then published a paper in 1686 stating that Descartes was wrong, it is not mass times velocity which is the conserved quantity of motion but mass times velocity squared.
Leibniz conserved force times distance
The basis of the Leibniz argument was that force times distance is conserved, and it is equal to mass times velocity squared, not mass time velocity nonsquared. He said that if a four kilogram object is dropped one meter, it will do the same thing as one kilogram object dropped four meters. He then showed the direct proportionality between force times distance and mass times velocity squared.
It should have been force times time
Leibniz's error was in his starting point. There was no evidence for his claim that rearranging force and distance would produce the same total. Pendulums could not show the relationship between force and distance. He should have instead combined force and time stating that a four kilogram object dropped one second would produce the same result as a one kilogram object dropped four seconds. This result would be proportional to mass times velocity nonsquared, not mass times velocity squared, and it would have shown Descartes to be correct.
Joule claimed to have measured the result
Scientists argued the subject for two hundred years, until it was supposedly settled by James Joule in 1845. Joule claimed he stirred water in a wooden bucket to show the transformation of force and distance into heat. Joule did not have the slightest ability to make such a measurement. He was copying physicists who were trying to make the measurement but failing. Weights on cords were dropped to turn paddles which stirred the water. The force times distance could be determined from the weight and height. One of the problems was that the weights would go faster and faster, which altered the force due to acceleration. Joule said he solved that problem by using more floats on top. Floats would have been worthless. If they were the answer, other experimenters could have determined so as easily as Joule could.
Joule lied about environmental effects
Joule never published a complete description of his experiment. Instead he sent a note to a publisher describing a few elements of it saying he would publish more later, which he never did. What he described was a total fraud. He said he dropped the weights 12 yards (meters) 16 times. It would have taken two athletes about an hour to wind them back up. He said he got 0.5°F (0.3°C) temperature increase. The heat would have dissipated into the metal and wood in about 30 seconds. There were no thermal conductivity coefficients. So he said he eliminated the effects of the "atmosphere" by dropping the weights an extra time. There is no method of eliminating environmental effects. He was totally lying. If the first 16 times he dropped the weights said something about the water, the 17th one isn't going to say something about the atmosphere. Switching from water to atmosphere is a fraud problem. Science doesn't come out of such a fraud well.
The modern number has no visible origins
Yet Joule was only off by 3 parts per thousand from the number being used now days. Joule's number was 4.2 Newton-meters per calorie, while the most recent number is 4.1868 N-m/calorie. Supposedly, it means Joule was a wizard. But the definition of kinetic energy can be mathematically proven to be incorrect, which means the modern number cannot be correct. It has to be a synthesis. In fact, there is no imaginable way to measure the number at more than two significant digits. There is no accessible evidence of how the number is measured.
Mathematical proof of the error
The mathematical proof is to show that one of the Leibniz objects has twice as much energy as the other. This can be done by mathematically replacing gravity with a rocket. The amount of fuel that a rocket requires to accelerate the objects to the same velocity can easily be calculated with no error. The burn-time for the rocket can represent the amount of fuel use, when the rocket is constant powered. When doing this, the burn-time for the 4kg object is exactly twice that of the 1 kg object.
Rockets use energy in proportion to force times time
But when force times time replaces force times distance, the rocket uses exactly the same amount of fuel for both objects. This result is quite predictable, because rockets transform energy in proportion to force times time, not force times distance. They burn fuel at a constant rate and produce a constant force. Constant force means unchanging through time. So force times time is proportional to fuel use.