In science, kinetic energy is the quantity of work:

$\tfrac{1}{2} mv^2$

that must be done to bring a body of mass m, initially at rest, to a velocity v. [1]

History
In 1689, German mathematician Gottfried Leibniz had defined the following quantity, determined previously to be conserved in perfectly elastic collistions in 1669 by Dutch physicist Christiaan Huygens, as vis viva or living force:

Vis viva = m

In 1807, English physicist Thomas Young defined the above quantity as "energy":

E = m

In 1811, Italian mathematician Joseph Lagrange used calculus to show that a factor of two is involved in the relationship “potential” (potential energy) and “vis viva” (kinetic energy). [2] As defined via the symbols used by Lagrange, i.e. T as kinetic energy, in his 1788 Analytical Mechanics, kinetic energy was thus defined as:

$T =\tfrac{1}{2} mv^2$

where 2T denotes the whole "living force of the system". [3]

Etymology
In 1853, or earlier, the term vis viva eventually was abandoned in favor of the term "actual energy" introduced by William Rankine.

In Oct 1862, William Thomson and Peter Tait, in their article entitled “Energy” in the latitudinarian Church of Scot magazine Good Words, introduced the term “kinetic energy”. [9]

In 1871, James Maxwell cited William Thomson and Peter Tait as having introduced "kinetic energy". [4]

Thomson, to note, is assumed to be the lead or primary author, at least as surmised by John Tyndall (“Remarks on the Dynamical Theory of Heat” to William Thomson, 1863), whom the article seems to be directed at. [7] Also, of note, as Crosbie Smith discusses in more detail (1998), there is a bit of a religious issue historical agenda to the publication of this article. [8] In any event, in regards to the etymology of the term “kinetic energy”, as G.C. Foster comments on the matter (1863): [8]

“The distinction here drawn between "motion" and "falling force" is the same as that made by Helmholtz (Die Erhaltung der Kraft, 1847) between "vis viva" (lebendige Kraft) and "tension" (Spankraft). The English expressions "dynamical energy" and "statical energy " were used by Prof. W. Thomson (Phil. Mag. S. 4. vol. iv. p. 304, 1852) in the same sense, but were afterwards abandoned by him in favor of the terms "actual energy" and "potential energy" introduced by Prof. Rankine. More recently (Good Words for October 1862) Professors Thomson and Tait have employed the expression "kinetic energy" in place of "actual energy".”

Thomson, in retrospect, stated the following on the matter in an 1882 footnote: [6]

“The name kinetic energy, which I subsequently gave as seeming preferable to ‘actual energy’, has been generally adopted; but Rankine’s name ‘potential energy’ remains to this day, and is universally used to designate energy of the static kind.”

In short, the etymology of the modern term "kinetic energy" is a convoluted one, similar to the intricate etymology to the term thermodynamics.

Kinetic theory

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics, Oxford: Oxford University Press.
2. Rayner, John, N. (2000). Dynamic Climatology: Basis in Mathematical Physics, (pg. 94). Blackwell Publishing.
3. (a) Hamilton, W.R. (1834). “On a general method in dynamics by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.” Philos. Trans. R. Soc. London, 124:247-308.
(b) Hamilton, W.R. (1835). “A second essay on a general method in dynamics.” Philos. Trans. R. Soc. London, 125:95-144.
4. Maxwell, James. (1871). Theory of Heat (pg. 91). Longmans.
6. Thomson, William. (1882). “On the Mechanical Values of Distributions of Electricity, Magnetism, and Galvanism”, Mathematical and Physical Papers, Volume 1 (pg. 523). Jul 18. Glasgow University Press.
7. Mayer, J. Robert. (1863). “Remarks on the Mechanical Equivalent of Heat” (translated by G.C. Foster, B.A., from a pamphlet intended for popular circulation, published in 1851 under the title Bemerkungen uber das Mechanische Aequivalent der Warme. Von J.R. Mayer. Heilbronn und Leipzig, 1851), Philosophical Magazine (kinetic energy, pgs. 375, 513), Series 4, No. 171, Vol. 25; 493-.
8. Smith, Crosbie. 1998). The Science of Energy (pg. 184). The University of Chicago Press.
9. (a) Thomson, William and Tait, Peter. (1862). “Energy”, Good Words (“kinetic energy”, 5+ pgs). Publisher.
(b) Smith, Crosbie and Wise, M. Norton. (1989). Energy and Empire: A Biographical Study of Lord Kelvin (pg. 378). Cambridge University Press.