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Question
Which of the following quantities related to an electron has a finite upper limit?
Options
Mass
Momentum
Speed
Kinetic energy
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Solution
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.
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