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Old Oct 22nd 2017, 09:07 AM   #1
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Was Einstein inspired?

I was noticing that there is a lot of similarity between the formula of kinetic energy 1/2 mv2 and Einstein's E=mc2. Was Einstein inspired from this kinetic energy formula when he came up with E=mc2?
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Old Oct 22nd 2017, 11:46 AM   #2
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Originally Posted by avito009 View Post
I was noticing that there is a lot of similarity between the formula of kinetic energy 1/2 mv2 and Einstein's E=mc2. Was Einstein inspired from this kinetic energy formula when he came up with E=mc2?
The short answer is "no."

The longer answer is that $\displaystyle E = mc^2$ (where m is the rest mass, sometimes written as $\displaystyle m_0$) is only true for an object at rest. The full equation for a moving object is $\displaystyle E^2 = (mc^2)^2 + (pc)^2$, where p is the momentum of the object. The total energy, E, can also be shown to be $\displaystyle E = \gamma ~ mc^2$, where I have defined $\displaystyle \gamma$ below.

The formula for kinetic energy in Special Relativity is $\displaystyle E = ( \gamma - 1)mc^2$. To make contact with non-relativistic theories we can expand the $\displaystyle \gamma = \frac{1}{\sqrt{1 - \left ( \frac{v}{c} \right ) ^2}}$ for v << c which gives

$\displaystyle KE = ( \gamma -1 )mc^2 = \left ( \frac{1}{\sqrt{1 - \left ( \frac{v}{c} \right ) ^2}} - 1 \right ) mc^2 \approx \frac{1}{2} mv^2 + \frac{3}{8} m \frac{v^4}{c^2} + \text{ ...}$

You can see that the largest term is the usual kinetic energy and that the extra terms are fairly small and can be ignored for Classical Physics.

-Dan
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Old Oct 22nd 2017, 12:29 PM   #3
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It should be remembered that energy is not an invariant quantity in relativity.

Further the velocity in topsquark's formulae may not be the same sort of the velocity in other mechanics. It is the relative velocity.
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Old Oct 23rd 2017, 07:19 AM   #4
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Originally Posted by topsquark View Post
The short answer is "no."

The longer answer is that $\displaystyle E = mc^2$ (where m is the rest mass, sometimes written as $\displaystyle m_0$) is only true for an object at rest.
And that is only true if you define m as rest mass (aka proper mass) which not every author in the physics literature does. In fact a large percentage of textbooks define m as relativistic mass rather than proper mass.
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Old Oct 23rd 2017, 11:16 AM   #5
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Of course, any formula for energy will have to have the correct units. Kinetic energy is, as you say, $\displaystyle (1/2)mc^2$, so have units of mass times distance squared divided by time squared. In, MKS units, $\displaystyle kg m^2/s^2$. ANY formula for energy will have to have those units so that "$\displaystyle E= mc^2$" is not at all surprising.
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Old Oct 27th 2017, 09:36 PM   #6
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It's useful to know that Einstein wasn't the first to write the expression relate mass to energy. That honor goes to Poincare who wrote about it in 1900. Electromagnetic energy has momentum. Poincare argued that it can be represented as a "fictitious fluid" having a mass density J/c^2 here J = energy density. Even earlier, in 1885 the mass of a moving charged conductor increases in mass.

There's an article on this subject in the American Journal of Physics

Did Einstein really discover "E = mc^2"? by W.L. Fadner, Am. J. Phys., 56(2), Feb (1988)

The abstract is here: http://aapt.scitation.org/doi/10.1119/1.15713
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