Question

An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy $10 \,keV$, and the second with $100 \,keV$. The ratio of their speeds is
(where $m_e$ and $m_p$ are masses of electron and proton respectively)

• A. $\sqrt{\frac{1}{10} \frac{m_{e}}{m_{p}}}$
• B. $\sqrt{\frac{1}{10} \frac{m_{p}}{m_{e}}}$
• C. $\\frac{1}{10} \frac{m_{e}}{m_{p}}$
• D. $\frac{1}{10} \frac{m_{e}}{m_{p}}$

Answer 1

Answer: (B)

Kinetic energy of the electron,
$K_{e} = \frac{1}{2}m_{e}v^{2}_{e} = 10\,eV\quad\cdots\left(i\right)$
Kinetic energy of the proton,
$K_{p} = \frac{1}{2}m_{p}v^{2}_{p} = 100eV\quad\cdots\left(ii\right)$
Divide $\left(i\right)$ by $\left(ii\right)$, we get
$\frac{m_{e}v^{2}_{e}}{m_{p}v^{2}_{p}} = \frac{1}{10}$
or $\frac{v^{2}_{e}}{v^{2}_{p}} = \frac{1}{10} \frac{m_{p}}{m_{e}}$
or $\frac{v_{e}}{v_{p}} = \sqrt{\frac{1}{10} \frac{m_{p}}{m_{e}}}$