- #1
cupid.callin
- 1,132
- 1
Hi
Is this derivation of E = mc^2 correct? ... I have some doubt at the red place ...
[itex]\large{ F = \frac{dp}{dt} = \frac{d}{dt}(mv) }[/itex]
[itex]\large{ F = v\frac{dm}{dt} + m\frac{dv}{dt} }[/itex]
Let this force cause a displacement dx
[itex]\large{ dW = F \cdot dx }[/itex]
Assuming body was initially at rest and this work is converted into kinetic energy and increase it by dK
[itex]\large{ dK = F\cdot dx }[/itex]
[itex]\large{ dK = v\frac{dm}{dt}dx + m\frac{dv}{dt}dx }[/itex]
[itex]\large{ dK = mvdv + v^2dm }[/itex] --- Equation 1
Now using eqn
[itex]\large{ m = \frac{m_o}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }[/itex]
Squaring both sides,
[itex]\large{ m^2= \frac{{m_o}^2}{1-\frac{v^2}{c^2}} }[/itex]
[itex]\large{ m^2c^2 - m^2v^2 = {m_o}^2c^2 }[/itex]
differentiating the expression
[itex]\large{ 2mc^2dm - 2mv^2dm - 2vm^2dv = 0 }[/itex]
[itex]\large{ c^2dm = mvdv - v^2dm }[/itex]
Using this in eqn 1
[itex]\large{ dK = c^2dm }[/itex]
Integrating
[itex]\large{ K = \int_0^K{dK} = \int_{m_o}^{m}{c^2dm} }[/itex] < --- HERE
[itex]\large{ K = c^2(m - m_o) }[/itex]
Total energy of body,
[itex]\large{ E = K + m_o c^2 }[/itex]
[itex]\large{ E = c^2(m - m_o) + m_o c^2 }[/itex]
[itex]\large{ E = mc^2 = \frac{m_o c^2}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }[/itex]
Also [itex]\large{ K = E - m_o c^2 = (m - m_o)c^2 }[/itex]
[itex]\large{ \Delta E = \Delta m c^2 }[/itex]
Is this derivation of E = mc^2 correct? ... I have some doubt at the red place ...
[itex]\large{ F = \frac{dp}{dt} = \frac{d}{dt}(mv) }[/itex]
[itex]\large{ F = v\frac{dm}{dt} + m\frac{dv}{dt} }[/itex]
Let this force cause a displacement dx
[itex]\large{ dW = F \cdot dx }[/itex]
Assuming body was initially at rest and this work is converted into kinetic energy and increase it by dK
[itex]\large{ dK = F\cdot dx }[/itex]
[itex]\large{ dK = v\frac{dm}{dt}dx + m\frac{dv}{dt}dx }[/itex]
[itex]\large{ dK = mvdv + v^2dm }[/itex] --- Equation 1
Now using eqn
[itex]\large{ m = \frac{m_o}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }[/itex]
Squaring both sides,
[itex]\large{ m^2= \frac{{m_o}^2}{1-\frac{v^2}{c^2}} }[/itex]
[itex]\large{ m^2c^2 - m^2v^2 = {m_o}^2c^2 }[/itex]
differentiating the expression
[itex]\large{ 2mc^2dm - 2mv^2dm - 2vm^2dv = 0 }[/itex]
[itex]\large{ c^2dm = mvdv - v^2dm }[/itex]
Using this in eqn 1
[itex]\large{ dK = c^2dm }[/itex]
Integrating
[itex]\large{ K = \int_0^K{dK} = \int_{m_o}^{m}{c^2dm} }[/itex] < --- HERE
[itex]\large{ K = c^2(m - m_o) }[/itex]
Total energy of body,
[itex]\large{ E = K + m_o c^2 }[/itex]
[itex]\large{ E = c^2(m - m_o) + m_o c^2 }[/itex]
[itex]\large{ E = mc^2 = \frac{m_o c^2}{ \Large{ \sqrt{1-\frac{v^2}{c^2}} } } }[/itex]
Also [itex]\large{ K = E - m_o c^2 = (m - m_o)c^2 }[/itex]
[itex]\large{ \Delta E = \Delta m c^2 }[/itex]