prove-that-E-mc-2-

Question Number 98003 by M±th+et+s last updated on 11/Jun/20

p r o v e t h a t

E = {m c}^{2}

Answered by Rio Michael last updated on 11/Jun/20

Let me begin with {Plank}^{'} s equation

E = h f

\Rightarrow E = h (\frac{c}{λ}) c = f λ

\Rightarrow E = \frac{h c}{λ}

also E = p c as we know momentum = \frac{h}{λ}

\Rightarrow E = m c (c), momentum = m v in this case v = c

\Rightarrow E = {m c}^{2}

let me use other methods to prove this

Answered by Rio Michael last updated on 11/Jun/20

P_{p h o t o n} = \frac{E}{c} \dots . . (i)

P = m v \dots . . (i i)

v = \frac{Δ x}{Δ t} \dots . . (i i i)

by conservation of momentum,

M \frac{Δ x}{Δ t} = \frac{E}{c}

Δ t = \frac{L}{c} \dots \dots (i v)

\Rightarrow M Δ x = \frac{E L}{c^{2}}

also \bar{x} = \frac{{M x}_{1} + {m x}_{2}}{M + m}

\Rightarrow \frac{{M x}_{1} + {m x}_{2}}{M + m} = \frac{M (x_{1} - Δ x) + m L}{M + m}

\Rightarrow M L = M Δ x

\Rightarrow M L = \frac{E L}{c^{2}}

E = {M c}^{2}

Answered by Rio Michael last updated on 11/Jun/20

K E = \int_{0}^{s} F d s

where F = \frac{d (m v)}{d t}

\Rightarrow K E = \int_{0}^{s} \frac{d (m v)}{d t} d s = \int_{0}^{m v} v d (m v) = \int_{0}^{v} v d [\frac{m_{0} v}{\sqrt{1 - v^{2} / c^{2}}}]

integrating by parts

K E = \frac{m_{0} v^{2}}{\sqrt{1 - v^{2} / c^{2}}} - m_{0} \int_{0}^{v} \frac{v d v}{\sqrt{1 - v^{2} / c^{2}}}

= \frac{m_{0} v^{2}}{\sqrt{1 - v^{2} / c^{2}}} + ∣ m_{0} v^{2}

= \frac{m_{0} c^{2}}{\sqrt{1 - v^{2} / c^{2}}} + m_{0} c^{2}

= {m c}^{2} - m_{0} c^{2}

\Rightarrow {m c}^{2} = m_{0} c^{2} + K E

\Rightarrow E = E_{0} + K E

where E_{0} = m_{0} c^{2}

also see that E = {m c}^{2} = \frac{m_{0} c^{2}}{\sqrt{1 - v^{2} / c^{2}}} .

Commented by M±th+et+s last updated on 11/Jun/20

w e l l d o n e t h a n k y o u

Commented by Rio Michael last updated on 11/Jun/20

welcome sir .

Answered by smridha last updated on 11/Jun/20

l e t m e s h o w h o w I u s u a l l y l i k e t o

d o t h i s \dots t h i s i s n o t t h e m e t h o d

o f t h e g o d o f p h y s i c s [EINSTEIN] .

w i t h d u e r e s p e c t t o h i m {l e t}^{'} s s t a r t .

★ {l e t}^{'} s d o s o m e t h o u g h t e x p t : ★

c o n s i d e r t w o i n e r t i a l f r a m e o f r e f e r e n c e

s a n d s^{'} . n o w a n a t o m i s p l a c e d

i n t o t h e s^{'} w h i c h i s m o v i n g v e r y

v e r y f a s t a t a s p e e d u w i t h r e s p e c t

t o s . n o w c o n s i d e r t w o o b s e r v e r s

A_{s} a n d A_{s^{'}} . n o w s u p p o s e t h e a t o m

e m i t s t w o p h o t o n o f f r e q u e n c y

ν, o n e i s g o i n g t o w a r d s b o t h t h e

o b s e r v e r s a n d a n o t h e r i s g o i n g

f a r t h e r a w a y f r o m t h e m .

n o w {l e t}^{'} s s e e h o w t h e e n e r g y a n d

m o m e n t u m c h a n g e d w i t h r e s p e c t

t o t w o o b s e r v e r s .

f o r t h e o b s e r v e r A_{s^{^{'}}} t h e c h a n g e o f

m o m e n t u m i s {(Δ p)}_{s^{^{'}}} = \frac{h ν}{c} - \frac{h ν}{c} = 0

b e c a u s e {i t}^{'} s r e m a i n s t a n d i n g

w i t h r e s t w i t h r e s p e c t t o A_{s^{^{'}}} (a c c

t o 1 s t p o s t u l a t e o f R e l a t i v i t y)

n o w c h a n g e o f o f e n e r g y

{(Δ E)}_{s^{^{'}}} = 2 h ν .

n o w {l e t}^{'} s s e e w h a t i s g o i n g o n w i t h

A_{s} . w e f a s t c o n s i d e r t h e a t o m e m i t s

t w o p h o t o n f r o m t h e m o v i n g f r a m

o f r e f e r e n c e s^{^{'}} (r e s p e c t t o s) . .

a n d t h e s^{^{'}} f r a m e m o v i n g v e r y v e r y

f a s t s o n o w R e l a t i v i s t i c d o p p l e r

e f f e c t c o m e s w i t h a m a g i c a l

s p e l l o n o b s e r v e r A_{s} .

t h e c h a n g e f r e q u e n c y o f p h o t o n

w h i c h c o m e s t o w a r d s A_{s}

ν_{1} = ν \sqrt{\frac{1 + \frac{u}{c}}{1 - \frac{u}{c}}} (b l u e s h i f t)

a n d w h i c h i s m o v i n g a w a y f r o m

A_{s}, ν_{2} = ν \sqrt{\frac{1 - \frac{u}{c}}{1 + \frac{u}{c}}} (r e d s h i f t)

s o t h e c h a n g e e n e r g y i n

f r a m e s w i t h r e s p e c t t o s^{^{'}} i s

{(Δ E)}_{s} = h ν_{1} + h ν_{2} = \frac{2 h ν}{\sqrt{1 - \frac{u^{2}}{c^{2}}}} = γ {(Δ E)}_{s^{^{'}}}

w h e r e γ = {[1 - \frac{u^{2}}{c^{2}}]}^{- \frac{1}{2}}

n o w c h a n g e i n m o m e n t u m

{(Δ p)}_{s} = \frac{h ν_{1}}{c} - \frac{h ν_{2}}{c} = \frac{γ {(Δ E)}_{s^{^{'}}}}{c^{2}} u = \frac{{(Δ E)}_{s}}{c^{2}} u \dots . (a)

n o w c o n s i d e r t h e m a s s o f t h e

a t o m i s (Δ m) v e r y s m a l l . a n d

f o r o v s e r v e r A_{s} t h e m a s s i s m o v i n g

s o i t h a s m o m e n t u m s o n o w f r o m

{e q}^{n} (a) w e g e t

{(Δ m)}_{s} u = \frac{{(Δ E)}_{s}}{c^{2}} u [c s p e e d o f l i g h t]

n o w o u r w o r l d s f a m o u s {e q}^{n} c o m e

w h i c h r e a l l y c h a n g e d t h e w o r l d

i s \dots

{(Δ E)}_{s} = {(Δ m)}_{s} c^{2}

b u t a c c t u a l l y m o s t l o g i c a l

{e q}^{n} i s E = \sqrt{p^{2} c^{2} + m^{2} c^{4}}

i n

o r i g i n a l p a p e r o f E i n s t e i n

t h e e q u a n t i o n i s l i k e t h a t

m = \frac{L}{v^{2}}

^{″} d o e s t h e i n e r t i a o f b o d y d e p e n d

u p o n {i t}^{'} s e n e r g y {c o n t e n t}^{″}

t h e t i t e l o f t h i s p a p e r .

Commented by M±th+et+s last updated on 11/Jun/20

n i c e w o r k t h a n k y o u s i r

Commented by smridha last updated on 11/Jun/20

w e l c o m e .

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