(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given \(0.750\space \mathrm{MeV}\) of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.

Beta decay is a nuclear process in which an unstable nucleus transitions to a more stable state by emitting particles. In the specific type of beta decay mentioned in the exercise, an electron, often designated as a beta particle, is ejected from the nucleus when a neutron is transformed into a proton. This occurs due to the weak nuclear force, one of the four fundamental forces in physics.

During beta decay, the electron is not just emitted randomly; it carries away excess energy from the nucleus in the form of kinetic energy. This is important, as conservation of energy must be upheld in all physical processes. In the given exercise, an electron is emitted with a substantial amount of kinetic energy, specifically 0.750 megaelectronvolts (MeV), which is a common unit of energy in nuclear physics due to the small energy scales involved.

Kinetic energy, in the context of relativity, becomes more complex than the classical formula of \(\frac{1}{2}mv^2\). The relativistic kinetic energy accounts for the fact that as an object moves faster and approaches the speed of light, its effective mass increases, and thus, more energy is required to continue accelerating it. This situation is precisely what the electron in the beta decay process experiences.

The formula for relativistic kinetic energy is given by \(K = (\gamma - 1)m_0c^2\). Here, \(m_0\) is the rest mass of the electron, \(c\) is the speed of light, and \(\gamma\) is the Lorentz factor, which accounts for time dilation and length contraction effects as the electron's velocity approaches \(c\). Calculating the relativistic kinetic energy allows us to understand the relationship between energy and velocity when dealing with particles like the electron that are moving at significant fractions of the speed of light.

The Lorentz factor, denoted by \(\gamma\), is a fundamental quantity in special relativity, which becomes relevant as objects approach significant fractions of the speed of light. For velocities much lower than \(c\), the speed of light, \(\gamma\) is approximately equal to 1 and classical mechanics holds true. However, as velocity increases, \(\gamma\) also increases, which indicates the growing influence of relativistic effects.

To find the Lorentz factor, we use the equation \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the object's velocity. In beta decay, with the electron reaching extremely high velocities, \(\gamma\) notably departs from 1, thus requiring a relativistic approach to calculate kinetic energy and predict the behavior of the electron accurately.

Rest mass energy is the energy contained within an object simply by virtue of its mass, according to Einstein’s famous equation \(E = mc^2\). For an electron, the rest mass energy is an essential reference point since it provides a baseline for comparing other forms of energy, such as kinetic energy acquired through acceleration.

The rest mass of an electron is \(9.109 \times 10^{-31} \text{kg}\), and its rest mass energy, derived from the famous equation, calculates to approximately 0.511 MeV. This rest mass energy is critical in understanding beta decay since the kinetic energy given to the emitted electron will be large compared to its rest mass energy, highlighting the fundamental importance of relativistic physics in accurately describing the energy and velocity of particles moving at high velocities.