How was the formula for kinetic energy found, and who found it?

[Ganesh Ujwal]

My questions mostly concern the history of physics. Who found the formula for kinetic energy

[latex]E_k =\frac{1}{2}mv^{2}[/latex]

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 [latex]U[/latex] and [latex]V[/latex], so that the total energy is their sum [latex]E=U+V[/latex], and the time derivative is always zero, [latex]\frac{dE}{dt}=0[/latex].

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

a) [latex]U=\frac{1}{2}mv^{2}[/latex]

b) [latex]F=-\frac{dV}{dt}?[/latex]

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.

Edited December 20, 2014 by Ganesh Ujwal

[swansont]

I don't know if it was the original formulation, but if an object is subject to a constant force, that force will do work such that W = 1/2 mv²

 

F dx = ma dx = m(dv/dt) dx = m (dx/dt) dv = mv dv

 

Integrate that using v₀ = 0 and you get 1/2 mv²

[imatfaal]

Hermann von Helmholtz - I think it was Helmholtz who did much to formalise our modern conception of energy and he was able to look at preservation (what we would call conservation) of energy after he had put things in an consistent algebraic form.

[elfmotat]

I don't know if it was the original formulation, but if an object is subject to a constant force, that force will do work such that W = 1/2 mv²

 

F dx = ma dx = m(dv/dt) dx = m (dx/dt) dv = mv dv

 

Integrate that using v₀ = 0 and you get 1/2 mv²

 

I think this is probably the best way to do it because it's general and doesn't depend on some arbitrary choice for a potential. Bonus: the same approach also works for finding KE in Special Relativity, where:

 

F dx = (dp/dt) dx = v dp = mv d(γv)

 

which you can integrate by parts to get (γ-1)mc².

[ajb]

Willem Jacob's Gravesande, Émilie du Châtelet and Giovanni Poleni (1720's) showed experimentally that the kinetic energy goes like velocity squared.

 

Energy as we know it is a new concept only really formulated 1800's with the development of thermodynamics and a modern formulation of mechanics. From what I have read Newton would not have made a distinction between 'kinetic energy' and momentum.

Edited December 21, 2014 by ajb

[ydoaPs]

From what I have read Newton would not have made a distinction between 'kinetic energy' and momentum.

Well, isn't matter-energy in GR just a measure of the 4-momentum? The stress-energy tensor is just the 4-momentum 'flow', yes?

[ajb]

Well, isn't matter-energy in GR just a measure of the 4-momentum? The stress-energy tensor is just the 4-momentum 'flow', yes?

So today you would make even less of a distinction that Newton possibly could have, true.

[Eise]

Willem Jacob's Gravesande, Émilie du Châtelet and Giovanni Poleni (1720's) showed experimentally that the kinetic energy goes like velocity squared.

 

I was very surprised when I first heard about Émilie du Châtelet and her experiments. Especially that obviously many Newtonian physicists defended that the energy of a moving object would be [math]mv[/math]:

 

Émilie du Châtelet contributed towards this resolution when, inspired by the theories of Gottfried Leibniz, she repeated and publicized an experiment originally devised by Willem 's Gravesande in which balls were dropped from different heights into a sheet of soft clay. Each ball's kinetic energy - as indicated by the quantity of material displaced - was shown conclusively to be proportional to the square of the velocity. Earlier workers like Newton and Voltaire had all believed that "energy" (so far as they understood the concept at all) was indistinct from momentum and therefore proportional to velocity.

So what was in those days meant with 'energy'? Leibniz obviously had some ideas in the right direction:

 

The concept of energy emerged from the idea of vis viva (living force), which Leibniz defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved.

 

Was it originally derived from work? Force times distance? But if that is true, then a simple dimensional consideration should already lead in the right direction?

 

[math]F = ma[/math]

 

[math]W = F.d = m.a.d = m.d.d/s^2 = mv^2[/math]

 

i.e. the dimension of energy is mass time velocity squared. Or did physicists in those days not have such a clear understanding of dimensions?

 

[ajb]

I am not an expert on the historical development. However, at some point people did not know what the units of energy are as they did not properly known energy. It took a long while to get from the original Greek idea to what we mean by energy today and part of that must have been to realise the importance of momentum.

[Eise]

The point is: I do not quite get why people got interested in something that later was called energy.

 

Hmm... googled a bit around, and found this. Is still not completely satisfying for me, but at least clarifies a little.

 

Energy is the fuel that powers momentum, and velocity is merely the basis for measuring its quantity relative to one's position.

 

Sounds nearly poetic...

[Fuzzwood]

Well why are people interested in studying anything at all? Because we want to know. At least the ones with more than 2 brain cells, that is.

[imatfaal]

The point is: I do not quite get why people got interested in something that later was called energy.

 

...

 

Because energy is the quantity that is conserved through time. We know that linear momentum is conserved because otherwise a translation in space would lead to different answers - and it is axiomatic that an experiment carried out in one part of space should mirror that carried out elsewhere. For physics to be symmetrical in time then energy must be conserved.

 

Have a look at Noether's Theorem

[Eise]

Well why are people interested in studying anything at all? Because we want to know. At least the ones with more than 2 brain cells, that is.

 

You can keep your sneers.

 

Energy is an abstract concept, and I am interested in the historical question why people got interested in it. If that does not interest you then...

 

Because energy is the quantity that is conserved through time. We know that linear momentum is conserved because otherwise a translation in space would lead to different answers - and it is axiomatic that an experiment carried out in one part of space should mirror that carried out elsewhere. For physics to be symmetrical in time then energy must be conserved.

 

Have a look at Noether's Theorem

 

That also misses the historical question. Historically the concept of conservation of energy was a slow discovery, smeared out over centuries, with Noether's Theorem as possibly the highest insight. I am interested to know how it began, and why there was so much confusion in the beginning between people like Leibniz, Chatelet, 's Gravesande on one side, and the Newtonists on the other side.

[elfmotat]

I've thought about this some more, and I have an answer more along the lines of what the OP is asking for, granted that dV/dx=-ma is taken as a postulate (time-translation symmetry). Say for simplicity the forces acting on a particle are conservative, and can therefore be modeled by a scalar potential V. If we posit that there is a quantity E which is constant over time consisting of two terms, V and another we'll call T, then we know:

 

[math]\frac{dE}{dt} = \frac{dT}{dt} + \frac{dV}{dx} \frac{dx}{dt} = \frac{dT}{dt} - mav = 0.[/math]

 

So we have:

 

[math]dT = m v dv[/math]

 

Or:

 

[math] T = \frac{1}{2} mv^2 + T_0[/math]

 

where [math]T_0[/math] is a constant of integration which can be taken to be zero.

[Ilinca Sergiu]

Who found the formula for kinetic energy?

That's a good question. I think anyone who could not prove the law of conservation of power. Specify not the law of conservation of momentum,work and energy, but POWER .

[Maged_Saeed]

$$k=\frac{m\timesv^2}{2}$$

Edited December 26, 2014 by Maged_Saeed