How does Einstein got equivalent mass - energy equation E=mc2? [duplicate]

Question

How does Einstein got equivalent mass- energy equation E=mc2 ?

Answer 1

Einstien got this by applying his STR to modify the classical expression of Kinetic energy. In reletvistic machenics mass is changed $m(v)=\dfrac{m_0}{\sqrt{1-\left(v/c\right)^2}}$, where $m_0$ is the rest mass and $m(v)$ is the moving mass, which is a function of $v$. Kinetic Energy as you might know in classical mechanics is defined to me $\dfrac 12 mv^2$ because defining it such way holds the work-Energy theorem, that is $\int F \cdot dx$ becomes equal to $K.E$. Now in relativistic mechanics we need to recalculate the work done keeping in miond that mass isn't constant and momentum $p=mv=\dfrac{m_0}{\sqrt{1-\left(v/c\right)^2}}v$. Force is as we know rate of change of momentum w.r.t to time, i.e. $F=\dfrac{dp}{dt}$. So, $$W = \int F \cdot dx = \int \dfrac{dp}{dt} \cdot dx = \int dp \cdot \dfrac{dx}{dt} = \int v \cdot dv$$.

From here you can carry on directly to the wikipedea article http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Background

$$E_\text{k} = \int \mathbf{v} \cdot d \mathbf{p}= \int \mathbf{v} \cdot d (m \gamma \mathbf{v}) = m \gamma \mathbf{v} \cdot \mathbf{v} - \int m \gamma \mathbf{v} \cdot d \mathbf{v} = m \gamma v^2 - \frac{m}{2} \int \gamma d (v^2)$$

Since $\gamma = (1 - v^2/c^2)^{-1/2}\!,$

\begin{align} > E_\text{k} &= m \gamma v^2 - \frac{- m c^2}{2} \int \gamma d (1 - v^2/c^2) \\ &= m \gamma v^2 + m c^2 (1 - v^2/c^2)^{1/2} - E_0 > \end{align} $E_0$ is a constant of integration for the indefinite integral. Simplifying the > expression we obtain

\begin{align} > E_\text{k} &= m \gamma (v^2 + c^2 (1 - v^2/c^2)) - E_0 \\ &= m \gamma (v^2 + c^2 - v^2) - E_0 \\ &= m \gamma c^2 - E_0 > \end{align} $E_0$ is found by observing that when $\mathbf{v }= 0 , \ \gamma = 1\!$ and $E_\text{k} = 0 \!,$ giving

$E_0 = m c^2 \,$ resulting in the formula

$E_\text{k} = m \gamma c^2 - m c^2 = \frac{m c^2}{\sqrt{1 - v^2/c^2}} - m c^2$ This formula shows that the work expended accelerating an object from rest >approaches infinity as the velocity approaches the speed of light. Thus it is >impossible to accelerate an object across this boundary.

The mathematical by-product of this calculation is the mass-energy equivalence >formula—the body at rest must have energy content

$E_\text{rest} = E_0 = m c^2 \!$ At a low speed

$E=m_0c^2$ is called the rest mass energy because it is the K.E of a partilce at rest.

Another related article on wikipedea might also be of interest to you: http://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies