91Ó°ÊÓ

The velocity of an electron can be derived from its kinetic energy using the equation for kinetic energy \( E_k = \frac{1}{2}mv^2 \), where \( E_k \) is kinetic energy, \( m \) is mass, and \( v \) is velocity. Given that the electron in this exercise has a kinetic energy of \( 10 \mathrm{keV} \), equivalent to \( 1.60 \times 10^{-15} \, \text{J} \) after conversion, we can calculate its velocity.
By rearranging the formula to solve for velocity, we get \( v = \sqrt{\frac{2E_k}{m}} \). Substituting the values:\( v_e = \sqrt{\frac{2 \times 1.60 \times 10^{-15} \, \text{J}}{9.11 \times 10^{-31} \, \text{kg}}} \approx 5.93 \times 10^7 \, \text{m/s} \).
This calculation shows that the electron moves extremely fast due to its light mass relative to its energy. Such a high velocity characterizes electrons in high-energy conditions like cosmic rays.

• The mass of the electron: \( 9.11 \times 10^{-31} \text{kg} \)
• The kinetic energy: \( 1.60 \times 10^{-15} \, \text{J} \)

To calculate the velocity of a proton, we use the same kinetic energy equation, \( v = \sqrt{\frac{2E_k}{m}} \). Protons are much heavier than electrons, with a mass of \( 1.67 \times 10^{-27} \, \text{kg} \). This significantly impacts their calculated speed.
The given kinetic energy of the proton is \( 100 \mathrm{keV} \), converted to \( 1.60 \times 10^{-14} \, \text{J} \). Plugging these values into the kinetic energy formula gives us:
\( v_p = \sqrt{\frac{2 \times 1.60 \times 10^{-14} \, \text{J}}{1.67 \times 10^{-27} \, \text{kg}}} \approx 1.38 \times 10^6 \, \text{m/s} \).
Although the proton has more kinetic energy than the electron, its much greater mass results in a substantially slower speed than that of the electron. This is a typical result due to the mass dependency in the velocity computation.

• Proton mass: \( 1.67 \times 10^{-27} \, \text{kg} \)
• Kinetic energy: \( 1.60 \times 10^{-14} \, \text{J} \)

Knowing that electron and proton velocities can differ greatly, calculating the ratio of their speeds gives useful insights. It allows us to compare their velocities directly in a meaningful way. The ratio is defined as \( \frac{v_e}{v_p} \), where \( v_e \) and \( v_p \) are the velocities of the electron and proton, respectively.
Substituting the calculated values for our example:
\[ \frac{v_e}{v_p} = \frac{5.93 \times 10^7}{1.38 \times 10^6} \approx 43.0 \] This means that the electron is approximately 43 times faster than the proton under the specified experimental conditions.
Such a speed difference is significant and is primarily due to the mass difference between the electron and the proton. While both particles carry energy, the lesser mass of the electron enables it to reach much higher speeds for the same or even lesser energy levels.

• Velocity of Electron: \( 5.93 \times 10^7 \, \text{m/s} \)
• Velocity of Proton: \( 1.38 \times 10^6 \, \text{m/s} \)
• Speed Ratio: \( \approx 43 \)