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To begin with, Einstein's special relativity is a branch of physics that was introduced by Albert Einstein in 1905. It consists of two postulates: firstly, the laws of physics are invariant (identical) in all inertial frames of reference (i.e., frames of reference with no acceleration), and secondly, the speed of light in a vacuum is the same for all observers, regardless of their relative motion or the motion of the light source.

Understanding special relativity is crucial when dealing with objects moving at significant fractions of the speed of light, referred to as 'relativistic speeds'. Traditional Newtonian mechanics fails to accurately describe motions at these high velocities since time, length, and even mass are no longer constant but vary with speed. In our exercise, we apply these principles to calculate the potential difference needed for an electron to achieve such high speeds.

Kinetic energy is the energy that an object possesses due to its motion. Traditionally, its calculation for an object with mass 'm' and velocity 'v' is provided by the equation \( \frac{1}{2}mv^2 \). Nevertheless, this formula does not hold true at relativistic speeds, like those close to the speed of light (c).

Einstein's theory modifies the kinetic energy calculation for these situations. The correct formula, which takes into account relativistic effects, is given by \( K = m_0c^2(1 - \sqrt{1 - v^2/c^2}) \), where \( m_0 \) is the rest mass of the object and \( v \) is its velocity. The term \( m_0c^2 \) represents the rest energy of the object, and the complete equation accounts for the additional energy required to move the object from rest up to the velocity 'v'.

In our context, electron acceleration refers to the process of increasing an electron's velocity. It is commonly achieved by applying a potential difference, which creates an electric field that impels the electron to accelerate. Since electrons have charge, they naturally respond to such electric fields.

When electrons are accelerated to high speeds approaching the speed of light, as in particle accelerators or certain electronic devices, relativistic effects become significant. The mass of an electron appears to increase as its speed increases, demanding more and more energy to continue accelerating. This concept is vital for the exercise at hand, as we calculate the potential difference needed to attain a given fraction of the speed of light.

Finally, the notion of energy conversion is pivotal in moving from one unit of energy to another. In the realm of atomic and subatomic particles, it is often convenient to express energies in electron volts (eV), rather than the standard Joules (J) used in classical mechanics. One electron volt is defined as the amount of kinetic energy gained by an electron when it's accelerated through an electric potential difference of one volt.

The conversion factor between Joules and electron volts is a key piece of information, with \( 1 eV = 1.602 \times 10^{-19} J \). Using this conversion factor, we can translate the kinetic energy determined through special relativity calculations in Joules to a more conveniently small and manageable figure in electron volts, which is done in the final step of our exercise.