Answer
(a) Since the kinetic energy is a significant fraction of the rest energy, the electrons are relativistic.
(b) $v = 0.63~c$
Work Step by Step
(a) The rest energy of an electron is $511~keV$ and the kinetic energy is $150~keV$. Since the kinetic energy is a significant fraction of the rest energy, the electrons are relativistic.
(b) We can find the Lorentz factor $\gamma$:
$K = mc^2~(\gamma-1)$
$\gamma = \frac{K}{mc^2}+1$
$\gamma = \frac{150~keV}{511~keV}+1$
$\gamma = 1.2935$
We can find the speed:
$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
$\sqrt{1-\frac{v^2}{c^2}} = \frac{1}{\gamma}$
$1-\frac{v^2}{c^2} = (\frac{1}{\gamma})^2$
$\frac{v^2}{c^2} = 1-(\frac{1}{\gamma})^2$
$v = \sqrt{1-(\frac{1}{\gamma})^2}~c$
$v = \sqrt{1-(\frac{1}{1.2935})^2}~c$
$v = 0.63~c$
