Question

With what velocity should an electron move so that its kinetic energy equals its rest mass energy?

• A. \( \frac{2}{3} c \)
• B. \( \frac{\sqrt{3}}{4} c \)
• C. \( \frac{\sqrt{3}}{2} c \)
• D. \( \frac{c}{2} \)

Hint:

When kinetic energy equals rest energy, the velocity can be found using \( \gamma = 2 \).

Answer 1

The Correct Option is C

Relativistic total energy is:
\[ E = \gamma mc^2 \] Given that the kinetic energy is equal to the rest mass energy:
\[ KE = (\gamma - 1) mc^2 = mc^2 \] \[ \gamma - 1 = 1 \Rightarrow \gamma = 2 \] Since \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \), solving for \( v \):
\[ \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = 2 \] \[ 1 - \frac{v^2}{c^2} = \frac{1}{4} \] \[ \frac{v^2}{c^2} = \frac{3}{4} \] \[ v = \frac{\sqrt{3}}{2} c \]