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When dealing with particles moving at speeds close to the speed of light, it is essential to consider relativistic effects. This is because classical physics does not apply to particles moving at such high velocities. Relativistic energy includes not just the energy from motion but also the intrinsic energy contained in the mass of the particles. The total relativistic energy of a particle is given by the equation:\[ E = \gamma mc^2 \]where \( E \) is the total energy, \( m \) is the mass, \( c \) is the speed of light, and \( \gamma \) is the Lorentz factor. This equation shows how energy increases significantly as an object's speed approaches the speed of light. Relativistic energy is particularly important when calculating energies involved in high-speed particle interactions, such as electron-positron annihilation.

Every particle possesses rest energy, which is the energy inherent to the mass of the particle when it's not in motion. This energy is given by Einstein’s famous equation:\[ E_0 = mc^2 \]Here, \( E_0 \) represents the rest energy, \( m \) is the rest mass of the particle, and \( c \) is the speed of light. Rest energy is the most basic form of energy a particle can have, and for the electron and the positron, this is a substantial portion of their total energy. Understanding rest energy is crucial in particle physics because it sets a baseline for conversion during particle reactions, such as in annihilation, where rest mass contributes to radiation energy.

Kinetic energy in the realm of relativity is a bit different from the classical perspective because it includes the Lorentz factor. At high velocities, kinetic energy is no longer simply \(1/2 mv^2\). Instead, we use:\[ KE = (\gamma - 1)mc^2 \]This formula shows how, as particles move faster and closer to the speed of light, their kinetic energy increases disproportionately compared to what Newtonian physics would predict. For an electron and positron moving at \(0.20c\), this results in additional energy being available for conversion during their annihilation, which adds to the overall energy seen in the resulting electromagnetic radiation.

The Lorentz factor, denoted as \( \gamma \), is a crucial component when discussing relativistic dynamics. It accounts for the effects of time dilation and length contraction at relativistic speeds, given by:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]Where \( v \) is the velocity of the particle and \( c \) is the speed of light. As the speed of a particle approaches the speed of light, \( \gamma \) increases rapidly. This factor modifies both kinetic energy and momentum in relativistic equations. For example, an electron moving at \(0.20c\) has a \( \gamma \) of approximately 1.0204, introducing corrections to the energy calculations that are necessary to accurately describe the behavior of particles at relativistic speeds.

In physics, one of the most fundamental principles is the conservation of energy, which states that energy in an isolated system remains constant. During electron-positron annihilation, this principle is perfectly illustrated. Initially, the system's energy consists of both the rest energy and the kinetic energy of the particles. When they collide and annihilate, all of this energy is transformed into electromagnetic radiation.The equation that represents this conservation in an annihilation scenario is:\[ E_{\text{initial}} = E_{\text{final}} \]Here, \( E_{\text{initial}} \) is the total energy before collision, including both rest and kinetic energy, and \( E_{\text{final}} \) is the energy of the resulting radiation. For the electron and positron in the exercise, their combined energy of \(1.6712 \times 10^{-13} \text{ J} \) validates the principle of conservation, as the same amount is converted into radiant energy post-annihilation.