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My questions mostly concern the history of physics. Who found the formula for kinetic energy $E_k =\frac{1}{2}mv^{2}$ and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way.

My guess is that someone thought along the following lines:

Energy is conserved, in the sense that when you lift something up you've done work, but when you let it go back down you're basically back where you started. So it seems that my work and the work of gravity just traded off. But how do I make the concept mathematically rigorous? I suppose I need functions $U$ and $V$, so that the total energy is their sum $E=U+V$, and the time derivative is always zero, $\frac{dE}{dt}=0$.

But where do I go from here? How do I leap to either

  • a) $U=\frac{1}{2}mv^{2}$
  • b) $F=-\frac{dV}{dt}$?

It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.

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Related (but not a duplicate): Why does kinetic energy increase quadratically, not linearly with speed? –  ACuriousMind Aug 27 at 20:39
We can thank Émilie du Châtelet for the modern, unified understanding Kinetic energy. You are right that the full theory of Energy conservation was difficult. It started with Liebniz and didn't end until Noether. –  user121330 Aug 27 at 21:02
There is no "energy formula." Two important names in the discovery of conservation of energy are Joule and Helmholtz. –  Ben Crowell Aug 27 at 23:54
Addem (and others), you may also be interested in this SE proposal: History of Science –  Danu Aug 28 at 13:54

5 Answers 5

As you probably know, Newton thought that energy is linearly proportional to velocity. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "any change of motion (velocity) is proportional to the motive force impressed".

This law, which nowadays is wrongly interpreted as: $F = ma$ (there is no reference to mass here) simply states states: $[\delta] v \propto [\delta] Vis_ {motrix}$ and in modern terms can only be interpreted correctly as: $$\delta v \propto J [/m]$$ which when multiplied by mass becomes momentum, whereas 'motive force' is used, like all other scholars of the time, referring to the yet unknown kinetic 'force'. The interpretation of this formula as the definition of F in modern usage is an ex post facto historical manipulation.

It was Gottfried Leibniz, as early as 1686, one year before the publication of the Principia, who first affirmed that kinetic energy is proportional to squared velocity (or that velocity is proportional to the square root of energy): $$ v \propto \sqrt{V_{viva}} [/m]$$. He called it, a few years later, vis viva = 'a-live' force in contrast with vis mortua = 'dead' force: (Cartesian) momentum (mass/weight * speed: $m *|v|$). This was accompanied by a first formulation of the principle of conservation of kinetic energy, as he noticed that in many mechanical systems of several masses $m_i$ each with velocity $v_i$,

$\sum_{i} m_i v_i^2$

was conserved so long as the masses did not interact. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction or in elastic collisions. Many physicists at that time held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: $\,\!\sum_{i} m_i v_i$ was the conserved kinetic energy.

The concept of PE played no role, it did not exist yet, nor did the concept of mechanical energy to which you refer (E = U + V), but Leibniz, in this first paper, uses the term potentia motrix ('motive power') to refer both to (K)energy a body acquires falling from an altitude and to the force necessary to lift it to the same altitude (mass/weight * space: $F*s$) which are considered equal. Some scholars see, wrongly, here a first definition of PE, but this is just one of the axioms of Galileo.

The principle to which you refer: $E_{mech} [KE + PE] = k$ in astrodynamics is called vis viva equation in his honour. Leibniz stated the conservation of KE per se besides the conservation of all (kinds of) energy in the whole universe. We need to underline this amazing stroke of genius.

His theory was strongly adversed by Newton[ians] and DesCartes-ians because it seemed to contrast, to be incompatible with the conservation of momentum. In Newton there was no distinction (as shown above) between speed, motion, momentum and energy but quantitas motus (momentum) was the prevailing concept and it was proven to be conserved in all situations, therefore Leibniz' vis viva was considered a threat to the whole system. Only later (in 1745), it was proven that both energy and momentum, being different entities, could be conserved (by Bošković and later by d'Alembert, etc.)

We can thank Émilie du Châtelet for the modern..understanding of kinetic energy – user121330

There is no energy formula ..in the discovery of conservation of energy are Joule and... – Ben Crowell

That's overlooking historical facts (Joule was not concerned with KE): soon after Leibniz' death, the quadratic relation was confirmed by experiments independently by the Italian Poleni in 1719 and the Dutch Gravesande in 1722, who dropped balls from varying heights onto soft clay and found that balls with twice speed produced and indentation four times deeper. The latter informed M.me du Châtelet of his results and she publicized them. Two centuries later, after Joule had shown that mechanical work can be transformed in heat, Helmholz suggested that the lost energy, in inelastic collisions, might have been transformed in heat.

Leibniz invented the concept of KE, prefigured/discovered its real formula: $E_k = mv^2$ suggested the right way (on time) of integration and established the principle of conservation of KE as prevailing on/independent from conservation of momentum.

He engaged in passionate controversies until his death but was opposed and overwhelmed by obtuse/ignorant Newtonian contemporaries. He was vulnerable as he could not account for the loss of energy in inelastic collisions. He lost and 'newtonian integration on space' produced: $\frac{1}{2}$ [mv2] which is not the formula, but just one of the possible formulas of KE: the newtonian formula. Had he won, instead of the joule, now we would use the 'leibniz' (= 1/2J) and would have a different, probably deeper, insight into the laws of the world

History, as we know, is written by the victors.

You can find additional information here

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Momentum and energy are two ways of quantifying motion. I don't think it's correct to say that Newton was talking about energy at all; he simply defined a momentum as a convenient measure of the movement power of motion. –  Larry Harson Aug 28 at 16:16
@LarryHarson: " I don't think it's correct to say that Newton was talking about energy at all" Right, and who said that? I wrote 'he made no disctinction' –  bobie Aug 28 at 16:19
So he just noticed a pattern in a bunch of examples? So he did something like: Work out the masses and velocities of this and that system, and just by chance he realized that in each example, if you square the velocities and dot product it with the masses that you'd get a constant? –  Addem Aug 29 at 20:27
Let's see if other members wish this post to be expanded. –  bobie Sep 3 at 5:04
A really interesting post, bobie! –  Charo 8 hours ago

You've already got some answers, but nobody mentioned Noether's Theorem yet. Noether's theorem maps a conserved quantity to each continuous symmetry. The relevant continuous symmetry needed to prove the conservation of energy is the one that leaves the laws of nature invariant, meaning the laws of physics don't change with time. Each continuous symmetry implies a certain function and the time derivative of that function must be zero. If you wanna read more about it, check out the wikipedia entry or any book about classical mechanics! Noether's Theorem on wikipedia.

Note: from the invariance of space (the laws of physics are the same everywhere in space) momentum conservation follows!

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Noether's theorem was way way after Leibniz. –  Love Learning Aug 28 at 9:43
I'm aware of that, but the OP asked for a mathematically rigorous proof, and Noether's theorem is the closest thing to that. –  user17574 Aug 28 at 10:33
OK, but adding some history would make your answer richer. –  Love Learning Aug 28 at 12:35
@user17574 There are much much simpler proofs, all of these conserved quantities, momentum, energy, angular momentum could be proved rigorously just from Newton's laws. Also considering the Lagrangian involves the kinetic energy, I don't think this helps. –  JLA Aug 30 at 7:17

The master of the law of energy conservation was Hermann von Helmholtz (1821-94). See his classic 1847 paper "Über die Erhaltung der Kraft," translated into English as "On the Conservation of Force." (He called energy force.)

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I suspect, though I'm not sure, that a nineteenth century French mathematician and scientist, Gaspard-Gustave Coriolis, is your man. He was the first to define the notion of "work done" and even kinetic energy. His wiki reads:

In 1829, Coriolis published a textbook, Calcul de l'Effet des Machines ("Calculation of the Effect of Machines"), which presented mechanics in a way that could readily be applied by industry. In this period, the correct expression for kinetic energy, ½mv2, and its relation to mechanical work, became established.

from which I guess that many mathematicians in that era independently found a ½mv2 formula based on Coriolis's work, though it seems likely that Coriolis was the first.

I would guess that the derivations used Coriolis's work-energy theorem: $$ \textrm{d}W = F \textrm{d}x $$ Subsituting $F=ma$, one quickly finds that $W=\frac{1}{2} mv^2$: $$ W = \int ma\,\textrm{d}x = \int m \frac{dv}{dt}\,\textrm{d}x = m\int v \,\textrm{d}v = \frac{1}{2} mv^2. $$

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The responses given so far are fairly accurate however, the question you should be asking goes to the experimental proof for the kinetic energy formula. Mathematically, the formulas for work and kinetic energy seem to function perfectly as taught. Unfortunately, there are at least 2 or 3 situations where it does not. No physics teacher ever looks at these and so, physics students never get the full story.

I'll give you one scenario that no one will argue against. Imagine you are performing a space walk and you throw a wrench. Because the wrench is far less massive than you, it has far more work done to it than you. If instead of a wrench you threw a small satellite that had the same amount of mass as you did and threw it with the same effort as the wrench, the tossed item would have the same amount of work done to it as was done to you. The change in kinetic energy for both you and the thrown items would be different; the total ke for you and the wrench would be far more than the ke for you and the satellite even though you used the same amount of biological energy in both cases.

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