Question

Which of the following quantities related to an electron has a finite upper limit?

Options

• Mass

• Momentum

• Speed

• Kinetic energy

Answer 1

Speed

 

If an electron is given a very high speed v, its mass

\[m = \gamma m_o  = \frac{m_o}{\sqrt{1 - \frac{v^2}{c^2}}}\]

\[\text{momentum, }p = mv = \gamma m_o v = \frac{m_o v}{\sqrt{1 - \frac{v^2}{c^2}}}\]

\[\text{kinetic energy, }k = \frac{1}{2}m v^2  = \frac{1}{2}\gamma m_o  v^2  = \frac{1}{2}\frac{m_o v^2}{\sqrt{1 - \frac{v^2}{c^2}}}\]

\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]

at \[v = c, \gamma = \frac{1}{\sqrt{1 - \frac{c^2}{c^2}}} =  \infty \]

\[ \Rightarrow\text{ at }v = c,   m = p = k =  \infty\]

Therefore, there's an upper bound for v to be always less than c but no upper limits for mass, momentum and kinetic energy of the electron.