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When it comes to calculating the velocity of an electron, understanding how kinetic energy plays a role is key. Electrons are small particles with a negative charge and are found in atoms orbiting the nucleus. To find out how fast an electron is moving with a given kinetic energy, we use the formula that emerges from the kinetic energy expression:

• Kinetic energy (KE) is given by \( KE = \frac{1}{2} mv^2 \), where \( m \) is mass and \( v \) is velocity.
• We can rearrange this to solve for velocity \( v \) as: \( v = \sqrt{\frac{2 \times KE}{m}} \).

The mass of an electron is approximately \(9.109 \times 10^{-31} \text{ kg}\). Given a kinetic energy of \(2.403285 \times 10^{-16} \text{ Joules}\) (as converted from 1.5 keV), plug the values into the formula to find the velocity:

• Substitute the mass and kinetic energy into the equation.
• Solve: \( v \approx 2.282 \times 10^7 \text{ m/s} \).

This solution gives us an idea of how swiftly electrons can travel when they possess a small amount of kinetic energy, highlighting the dynamic nature of these subatomic particles.

While both electrons and protons are subatomic particles, the key difference in velocity calculations lies in their mass. Protons are significantly heavier than electrons, impacting their velocity when endowed with the same kinetic energy. Here’s how you calculate the proton’s velocity:

• Protons possess a mass of approximately \(1.673 \times 10^{-27} \text{ kg}\), which is much heavier than electrons.
• Using the kinetic energy of \(2.403285 \times 10^{-16} \text{ Joules}\) (converted from 1.5 keV), apply the velocity formula: \( v = \sqrt{\frac{2 \times KE}{m}} \).

Insert the proton’s mass and the kinetic energy into this equation:

• Calculate to find: \( v \approx 1.384 \times 10^5 \text{ m/s} \).

As observed, even with the same amount of kinetic energy, protons move significantly slower than electrons due to their greater mass. This disparity in mass and movement illustrates why protons behave differently in various physical and chemical contexts.

In physics, energy is a crucial concept, represented in various units depending on the context. Kilo-electron volts (keV) are commonly used in particle physics, while joules are preferred in many other fields. Understanding how to convert between the two is essential for applications like these velocity calculations.

• 1 electron volt (eV) equals \(1.60219 \times 10^{-19} \text{ Joules}\).
• To convert keV to eV, remember: 1 keV = 1000 eV.

In our exercise, we had 1.5 keV:

• Multiply by 1000 to get 1500 eV.
• Convert eV to Joules: \(1500 \times 1.60219 \times 10^{-19} = 2.403285 \times 10^{-16} \text{ J}\).

This conversion allows us to use the energy in the standard equation for kinetic energy, ensuring consistency and accuracy in our calculations. It’s a simple but important step that ensures that all calculations align correctly with the metric system.