Question

The kinetic energy of an electron is equal to its rest mass energy. Determine the

magnitude of its velocity and momentum.

Answer 1

$E_0=m_0c^2=9.1\cdot10^{-31}\cdot(3\cdot10^{8})^2=819\cdot10^{-16}\ (J)\approx512\ (keV)$

$KE=m_0c^2\bigg(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\bigg)$ , $KE=E_0$

$m_0c^2=m_0c^2\bigg(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\bigg)\to 1=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1$

$\sqrt{1-\frac{v^2}{c^2}}=\frac{1}{2}\to1-\frac{v^2}{c^2}=\frac{1}{4}\to v=\frac{\sqrt3}{2}c$ . Answer

Total energy

$E=m_0c^2/\sqrt{1-\frac{v^2}{c^2}}=512/\sqrt{1-\frac{(\frac{\sqrt3}{2}c)^2}{c^2}}=1024\ (keV)$

Momentum

$E^2=p^2c^2+(m_0c^2)^2\to p=\sqrt{(E^2-(m_0c^2)^2)/c^2}=$

$=\sqrt{(1024^2-(512)^2)/c^2}=887\ (keV/c)$ . Answer