How do I prove e=moc2 (o=naught)

Question

• Hello, I am Ahsan. I am trying to derive the e=mc2 equation from the relalative mass equation. But I have stumbled upon where it says rest motion = moc2(o=naught), how do I prove that?

Answer 1

A good approach is to begin with the calculation of work, which gives kinetic energy.  Use relativistic mass.

 

K = W = ∫Fdx

 

F = d(mv)/dt = mdv/dt + vdm/dt

 

K = ∫(mdv/dt + vdm/dt)dx

 

m = m₀/√(1 - v²/c²)

 

m²(1 - v²/c²) = m₀²

 

m²c² - m²v² = m₀²c²

 

Take d/dt of the above.

 

2c²mdm/dt - 2mv²dm/dt - 2m²vdv/dt = 0

 

Divide through by 2mv.

 

c²(1/v)dm/dt - vdm/dt - mdv/dt = 0

 

Rearrange.

 

mdv/dt + vdm/dt = c²(1/v)dm/dt

 

v = dx/dt ⇒ 1/v = dt/dx

 

mdv/dt + vdm/dt = c²(dt/dx)(dm/dt) = c²dm/dx

 

mdv/dt + vdm/dt = c²dm/dx

 

Now put this in the K integral.

 

K = ∫(mdv/dt + vdm/dt)dx = ∫(c²dm/dx)dx =  c²∫dm = c²(m_f - m_i)

 

If the particle starts from rest, m_i = m₀.  But when it reaches speed v, 

 

m_f =  m₀/√(1 - v²/c²)

 

Therefore,

 

K = c²[m₀/√(1 - v²/c²) - m₀] = m₀c²/√(1 - v²/c²) - m₀c²

 

Rearrange.

 

K + m₀c² = m₀c²/√(1 - v²/c²)

 

The left hand side is some kind of total energy E.

 

E = K + m₀c²

 

Since K = kinetic energy, K ≠ 0 if the particle is moving, and K = 0 if the particle is at rest.  Then when the particle is at rest,

 

E = m₀c²