Derive E=mc^2 - Doubts at Red Place

[cupid.callin]

Hi

Is this derivation of E = mc^2 correct? ... I have some doubt at the red place ...

$\large{ F = \frac{dp}{dt} = \frac{d}{dt}(mv) }$

$\large{ F = v\frac{dm}{dt} + m\frac{dv}{dt} }$

Let this force cause a displacement dx

$\large{ dW = F \cdot dx }$

Assuming body was initially at rest and this work is converted into kinetic energy and increase it by dK

$\large{ dK = F\cdot dx }$

$\large{ dK = v\frac{dm}{dt}dx + m\frac{dv}{dt}dx }$

$\large{ dK = mvdv + v^2dm }$ --- Equation 1


Now using eqn

$\large{ m = \frac{m_o}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }$

Squaring both sides,

$\large{ m^2= \frac{{m_o}^2}{1-\frac{v^2}{c^2}} }$

$\large{ m^2c^2 - m^2v^2 = {m_o}^2c^2 }$

differentiating the expression

$\large{ 2mc^2dm - 2mv^2dm - 2vm^2dv = 0 }$

$\large{ c^2dm = mvdv - v^2dm }$

Using this in eqn 1

$\large{ dK = c^2dm }$

Integrating

$\large{ K = \int_0^K{dK} = \int_{m_o}^{m}{c^2dm} }$ < --- HERE

$\large{ K = c^2(m - m_o) }$

Total energy of body,

$\large{ E = K + m_o c^2 }$

$\large{ E = c^2(m - m_o) + m_o c^2 }$

$\large{ E = mc^2 = \frac{m_o c^2}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }$

Also $\large{ K = E - m_o c^2 = (m - m_o)c^2 }$

$\large{ \Delta E = \Delta m c^2 }$

 

[BruceW]

$\large{ c^2dm = mvdv - v^2dm }$

Using this in eqn 1

$\large{ dK = c^2dm }$

I think this step isn't right. Maybe you got the negative sign in the wrong place? Up until this bit, I'd say it was all correct.

 

[cupid.callin]

Oops ... it was a typo
It would be ...

$\large{ c^2dm = mvdv + v^2dm }$

What i was confused was that, we write E = mc² and not m_oc² because when moving mass increase, and that is a way of saying (if I'm not wrong) that KE of body adds to its mass.
So is it ok for me to take limits as m_o → m ?
I mean, when I'm treating KE as separate and not a part of addition in mass, should i really take these limits? ... Well i know my these sentences are also against some steps of my derivation above ... but ,,, ummm,,, i don't know ,,, please help me get rid of my confusion ...

$\large{ K = \int_0^K{dK} = \int_{m_o}^{m}{c^2dm} }$

 

[jedishrfu]

the m in e=mc^2 is rest mass. see wikipedia article:

http://en.wikipedia.org/wiki/Mass–energy_equivalence

 

[BruceW]

Yes, I see it was simply a typo, and the rest of the derivation is correct :) (for 1-d motion, of course).

m is the relativistic mass. m₀ is the rest mass. So you can say that the KE of a body adds to its relativistic mass, but not to its invariant (rest) mass.

And yes, those are the correct limits of integration, because when KE is zero, m=m₀ (in other words, the only energy it has is rest energy).

 

[BruceW]

the m in e=mc^2 is rest mass. see wikipedia article:

http://en.wikipedia.org/wiki/Mass–energy_equivalence

no, cupid.callin is using notation where m is relativistic mass. and m₀ is rest mass.

 

[cupid.callin]

Yes, I see it was simply a typo, and the rest of the derivation is correct :) (for 1-d motion, of course).

m is the relativistic mass. m₀ is the rest mass. So you can say that the KE of a body adds to its relativistic mass, but not to its invariant (rest) mass.

And yes, those are the correct limits of integration, because when KE is zero, m=m₀ (in other words, the only energy it has is rest energy).

So you mean that, the integrated eqn of K accounts for change in mass, ... m-m_o mass is added to the particle when it moves ...
Well is makes sense now ... I don't know why i was thinking all that before ,,, thank you for your help BruceW !

 

[BruceW]

No worries. Practice makes perfect!