Can you replace c with velocity in $E = m c^{2}$ $E = mc^2$ ?

Question

If the question is about the energy of a neutron moving at a certain fraction of the speed of light. mass of neutron is given.

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Answer 1 · 2018-06-23T20:34:38

see below

It really depends upon what you mean by "certain fraction".

Generalising that, the relativistic expressions for energy and momentum of a free particle moving in 1-D are:

${(E = mc^2 = gamma m_o c^2),(p = mv = gamma m_ov qquad square):} qquad qquad {(m_o = "rest mass"),(gamma = 1/sqrt(1-v^2/c^2) = "Lorentz factor") :}$

So the additional energy in the rest frame of a relativistic free particle due to its motion can be explored as:

$E^2 - E_o^2 = (gamma^2 - 1) (m_o c^2)^2$

$= ( 1/ (1-v^2/c^2) - 1) (m_o c^2)^2$

$= ( ( v^2/c^2)/ (1-v^2/c^2 ) ) (m_o c^2)^2$

$= gamma^2 v^2/c^2 m_o^2 c^4$

$= gamma^2 (p^2/(gamma^2 m_o^2))/c^2 \ m_o^2 c^4$

$= p^2 c^2 qquad [= E^2 - E_o^2]$

Therefore:

$E^2 = color(blue)( p^2 c^2) + (m_o c^2)^2$

In the real world, we work with the blue term