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I stumbled across the following quote and couldn't understand how one wouldn't use the factor of 1/2 without completely disrupting the work-energy principle. Though, informal, energy is defined as the ability to do work, it still captures the essence of definition and the relationship. You remove the factor of 1/2 from kinetic energy formula, you wrongly end up with more work.

Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed c. Each of these particles has a kinetic energy of mc^2 up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics.[74] By assuming that every particle has a mass that is the sum of the masses of the ether particles, the authors concluded that all matter contains an amount of kinetic energy either given by E = mc^2 or 2E = mc^2 depending on the convention.

Source: https://en.wikipedia.org/wiki/Mass–energy_equivalence#Others

Both Preston and De Pretto used the kinetic energy formula without the 1/2 factor in 1875 and 1903 respectively.[1]

Leibniz introduced the vis viva concept back in 1686.

Source: The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering 2013, Page #5

It was Coriolis who introduced the kinetic energy formula with factor 1/2 and the notion of work around 1829.

Coriolis introduces the word work (travail) to indicate what Carnot called moment of activity and others moment, mechanical power, quantity of action, energy, or even simply force.

These various and quite vague expressions were not suitable to spread easily. We propose the appellation dynamical work, or simply work, for the quantity ∫Pds [...]. This name will not be confused with any other mechanical denomination. It seems suitable to give the right idea of the thing, by maintaining its common meaning of physical work [...] this name is then suitable to designate the union of these two concepts, displacement and force [78].10 (A. 16.8)

Coriolis uses the term work also in subsequent studies, particularly in the Mémoire sur la manière d’differéns établi les principes pour des systèmes de mécanique des corps, comme en des assemblages de considérant the molecules of 1835 [79]. It is a use that he definitely will consolidate with his work Mécanique des corps solides of 1844 where, in the preface, he writes:

I employed in this work some new nomenclature: I name work the quantity usually named puissance mécanique, quantité d'action ou effet dynamique, and I propose the name dy- namode for the unity of measure of this quantity. I introduced also one more little innovation by naming living force the product of the weight times the height associated to the height. This living force is one half of the product that today is associated to this name, i.e. the mass times the square of speed [80].11 (A. 16.9)

Notice Coriolis is introducing the factor 1/2 in front of the expression of living force (i.e. kinetic energy), because he suggests measuring the living force of a body of mass m and velocity v with the product mh, with h the height the body can reach if thrown upward with an initial velocity v (h = v^2/2).

In a note to the passage quoted above, Coriolis writes:

This term work is so natural in the sense that I use it, which, though it has never been either proposed or approved as a technical expression, nevertheless it was used accidentally by Mr. Navier in his notes on Belidor and Prony in his Mémoire sur les expériences de la machine du Gros-Caillou [80].12 (A. 16.10) Although Coriolis’s texts were fundamental to the spread of the term work, again, at the end of the XIX century propositions like: principle of virtual velocities, principle of moments and principle of virtual work, co-existed.

Source: History of Virtual Work Laws, A History of Mechanics Prospective By Danilo Capecchi • 2012, Page #368

My question is that how come scientists, like Preston and De Pretto, were using the 'wrong' formula for kinetic energy without any 1/2 factor. Was work-energy relationship not really established until the beginning of 20th century, or did they not really care about the relationship? Personally, I doubt that the relationship wasn't well establish or they didn't care about it.

I'd appreciate it if you could help me to find the answer.

By the way, Wikipedia cites this article Prentiss, J.J. (August 2005), "Why is the energy of motion proportional to the square of the velocity?", American Journal of Physics but unfortunately it's behind the paywall and I cannot pay for it.

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    $\begingroup$ The Prentiss papers would be worthwhile to get from your local library. It starts from Johann Bernoulli, Discours sur les lois de la Communication du Mouvement (Chez Claude Jombert, Paris, 1724) and builds from there. $\endgroup$ – Jon Custer Aug 26 '20 at 15:24
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    $\begingroup$ The definition of work is as conventional as the definition of kinetic energy, if one does not wish to have 1/2 in the energy they can stick 2 into the definition of work. Or simply scale the units. $\endgroup$ – Conifold Aug 26 '20 at 16:48
  • $\begingroup$ @JonCuster Thank you for the suggestion but it won't be possible for me. $\endgroup$ – PG1995 Aug 26 '20 at 22:48
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    $\begingroup$ It is not that it was "considered" or specific to this. All proportionality factors are conventional, it is only the proportionality itself that is substantive. Factors are typically matched to units without any special notice. "Work" came to stand for the change in energy, whichever way one chose to scale energy. $\endgroup$ – Conifold Aug 27 '20 at 4:30
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    $\begingroup$ Given that "nice things" happen when you allow the 1/2 factor, like differentiating $\frac{d(KE)}{dt}$ , You can see why it's preferable to include it. ( $\frac{dv^2}{dt} = 2*v*\frac{dv}{dt}$ and so on) $\endgroup$ – Carl Witthoft Aug 27 '20 at 12:05

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