Question

Can you replace c with velocity in `E = mc^2`?

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.

Answer 1

see below

Explanation:

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") :}`

• With rest energy: `qquad E_o = m_o c^2`

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`

• From `square`: `qquad v^2 = p^2/(gamma^2 m_o^2)`

`= 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