Electron Boost (Homework 8-323)

The energy of an electron is twice its rest mass (rest energy). a. Find the electron’s boost factor $γ$ and the velocity factor $β$. b. Find the electron’s velocity factor β for the case where its momentum is its rest mass (times c).


$E^2=p^2 c^2+m_0^2 c^4$    with $E=2m_0 c^2$
$4m_0^2 c^4=p^2 c^2+m_0^2 c^4$  

or $3m_0^2 c^4=p^2 c^2$     with $p=γm_0 v$

$3m_0^2 c^4=(m_0^2 v^2)/(1-β^2 )*c^2$    
or   $3c^2=v^2/(1-β^2 )$    
or $3/β^2 =1/(1-β^2 )$   or $3-3β^2=β^2$

$β^2=3/4=0.75$    and $γ=1/\sqrt{1-β^2 }=1/\sqrt{(1-0.75)}=2$

b) If
$p=m_0 c$    this is $γm_0 v=m_0 c$    or  $1/\sqrt{(1-β^2 )}=1/β$     or $β^2=1-β^2$    or $β^2=1/2$ 
$β=1/√2=0.707$  and $γ=1/\sqrt{(1-1/2)}=1/√0.5=√2=1.41$