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Electron-positron annihilation is a fascinating process that illustrates the mass-energy equivalence in a vivid way. When an electron, a negatively charged particle, meets its antimatter counterpart, the positron, which has a positive charge, they collide and effectively cancel each other out. This collision results in their mass being converted into energy. This energy is released in the form of electromagnetic radiation, typically gamma rays.

The incredible aspect of this process is that the entire rest mass of these particles disappears, with only energy left as a product. According to the mass-energy equivalence principle, articulated by Einstein, the rest mass energy of a particle is defined by the equation: \[ E = mc^2 \]where \( E \) is the energy, \( m \) is the rest mass, and \( c \) is the speed of light. This formula demonstrates that mass can be converted into a large amount of energy, even if the mass is quite small, due to the enormous value of \( c^2 \).

In electron-positron annihilation, the conservation of energy dictates that the energy available before the annihilation, including kinetic energy and rest mass energy, must equal the energy of the photons emitted, which is the essence of mass-energy equivalence.

Kinetic energy is the energy an object possesses due to its motion. For everyday speeds, it's often calculated using the classical formula:\[ KE = \frac{1}{2}mv^2 \] where \( KE \) is kinetic energy, \( m \) is mass, and \( v \) is velocity. However, when particles like electrons and positrons approach relativistic speeds, meaning speeds close to the speed of light, their kinetic energy calculations require modifications to account for relativistic effects.

Under these circumstances, we use the relativistic expression for kinetic energy:\[ KE = (\gamma - 1)mc^2 \] where \( \gamma \) (gamma) is the Lorentz factor, given by:\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]This factor accounts for the relativistic changes experienced due to high velocities.

It ensures that as an object's velocity increases, particularly as it nears the speed of light, its mass effectively becomes infinitely heavy, and thus, infinite energy would be required to reach the speed of light, which is why it's impossible.

In the context of the electron-positron annihilation exercise, both particles have significant kinetic energy, contributing to the total energy that will be converted into electromagnetic radiation upon annihilation.

Relativistic effects become significant when particles travel at speeds approaching the speed of light. These effects include changes in mass, time, and energy as perceived by an observer.

One of the key differences within relativistic physics, as opposed to classical physics, is how velocity affects an object's mass and energy. As an object's speed increases, its relativistic mass increases, which is an indicator of how much energy it has. Consequently, it takes more force to accelerate the object than would be predicted by Newtonian physics alone.

This is explained by the previously mentioned Lorentz factor, \( \gamma \), which becomes significantly larger as the speed of an object nears the speed of light, leading to notable relativistic effects. These adjustments ensure that fundamental laws of physics, like conservation of energy and momentum, hold true in all frames of reference.

In scenarios involving particles like electrons and positrons moving at relativistic speeds before annihilation, these effects ensure accurate calculations of kinetic and total energies. They factor importantly into the total energy, predicted accurately using the equation: \[ E = \gamma mc^2 \]. This encompasses both rest mass energy and kinetic energy in systems where relativity cannot be ignored. Thus, understanding relativistic effects is imperative when dealing with high-speed particles in both theoretical physics and practical applications.