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Could you pease tell me why energy unit must be $Energy=Mass . \frac{Distance^2} {Time^2}$? (I tried to write general form of Energy unit)

What is the strong proof of that unit? Does it just depend on Newton's second law ($F=ma$) and Work formula. Or something else?

Newton's second law:

($F=ma$=Mass x Acceleration of the center of mass=$Mass.\frac{Distance} {Time^2}$)

$Work = Force . Displacement =Mass . Acceleration .Distance =Mass . \frac{Distance^2} {Time^2}$

Is it possible to change the Energy unit after realizing that Newton's second law and work formula are approximation? Or is it strict true in nature? Note: Einstein showed that the relativity can change our approach to the nature but he did not touch Energy unit. ($E=m.c^2$)

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Even if they are approximations, the additional (missing) terms are of the same unit. So the unit doesn't change. – Manishearth Apr 10 '12 at 8:58
up vote 3 down vote accepted

The reasoning you've used in your post is called dimensional analysis, and it's exactly why the units of energy must be $ML^2T^{-2}$. If Newton's second law was incorrect then the dimensions of energy might be different, but we don't believe the second law is wrong because too many well established bits of Physics would break if it weren't correct.

You need to be a bit careful when your geometry isn't Euclidean, e.g. in special and general relativity, and you may wish to write the second law as something like:

$$F = m_0 a + \text{higher order terms}$$

but as Manishearth says in his comment, the extra terms have to have the same dimensions as $m_0a$ because you can't add terms that have different dimensions (well, not if you want it to make physical sense).

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Use \text{..} for writing upright text in $\TeX$ and preserving the spaces. There's also \mathrm{...}(alternatively \rm{..}) which just make the font upright(killing spaces). Note that if you want to use math/other markup inside a \text block, you have to ensure that the main math is enclosed in double-dollars and use single-dollars insite the textblock. \mathrm allows math inside it wothout any fancy nesting. – Manishearth Apr 10 '12 at 11:15

As you stated, energy comes directly from the definition of work, and the derivation of formula for kinetic energy requires using Newton's second law:

$$ \textrm{d}W = F \textrm{d}s = m \frac{\textrm{d}v}{\textrm{d}t} v \textrm{d}t = m v \textrm{d}v $$

After integration

$$W = \frac{1}{2} m v^2$$

Of course, $F$, $ds$, $v$ are in fact vectors, what I've neglected for sake of simplicity.

Newton's second law and definition of work are not approximations, at least in the differential form. Of course, when you want make a derivation of kinetic energy for non-point objects, i.e. rigid body, things get complicated and you get rotational part of kinetic energy not included in expression above.

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There is a bit of philosophy here.

You think that you understand what "energy" is, but can you give a quantifiable definition?

"Energy" in physics was historically defined in the way you discuss, and the usage became fixed so all "modern" definitions are required to agree with the classical one in regions where both theories apply. So this definition is forever. So far, so much semantics.

What makes these quantities interesting is their uses which manifold and powerful.

We could just as well define a quantity "umphiness" with units $\text{mass}^*\text{time}/\text{length}^3$, but will anyone care about t, use it or remember it? Only if it adds something to the discussion.

$$ \text{Energy} = \text{mass} \left( \frac{\text{length}}{\text{time}} \right)^2$$

is useful, so people use it.

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