The mass of a proton is 1,850 times the mass of an electron. If a proton and an electron have the same kinetic energy \(\left(E_{K}=1 / 2 m v^{2}\right),\) how many times greater is the velocity of the electron than that of the proton?

The mass of a proton is significantly larger than the mass of an electron. Specifically, the mass of a proton is about 1,850 times that of an electron. This large difference in mass has profound effects on their behavior in physics. For instance, protons are found in the nuclei of atoms and contribute greatly to the atomic mass, while electrons are much lighter and orbit around the nucleus. The symbol for proton mass is often denoted as \(m_p\). Understanding the mass difference is crucial for calculating other properties, such as kinetic energy and velocity in various physics problems.

Electrons are tiny particles that have much less mass compared to protons. The mass of an electron is denoted by the symbol \(m_e\), and as mentioned before, it is 1,850 times lighter than a proton. This smaller mass allows electrons to move much faster than protons when they have the same kinetic energy. Because of their light mass, electrons play key roles in electricity, magnetism, and chemical reactions. They are also essential in explaining phenomena such as electron clouds in atoms and the conduction of electricity in materials.

When two particles have the same kinetic energy, the ratio of their masses directly influences their velocities. The kinetic energy formula is \( E_K = \frac{1}{2}mv^2 \). This means if both a proton and an electron have the same kinetic energy, we can express their kinetic energies as:

\( \frac{1}{2}m_p v_p^2 = \frac{1}{2}m_e v_e^2 \).

Here, \(v_p\) and \(v_e\) are the velocities of the proton and electron, respectively. Since \(m_p = 1850 m_e\), we substitute this into the equation to get:

\( 1850 m_e v_p^2 = m_e v_e^2 \).

Cancelling out \(m_e\) from both sides, we get:

\( 1850 v_p^2 = v_e^2 \).

Taking the square root of both sides, we find that:

\( v_e = \sqrt{1850} v_p \). Hence, the velocity of the electron is approximately 43 times greater than that of the proton. This dramatic difference arises because the electron’s much smaller mass allows it to achieve a much higher velocity for the same amount of kinetic energy.