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In the realm of physics, specifically within the framework of Einstein's special relativity, the rest mass energy is a fundamental concept. It is the energy that is inherently contained within a particle merely by virtue of it having mass. The famous equation that represents this relationship is Einstein’s mass-energy equivalence formula,
\( E_0 = m_0c^2 \)
, where \( E_0 \) is the rest mass energy, \( m_0 \) is the rest mass of the particle, and \( c \) is the speed of light in a vacuum (~299,792,458 meters per second). This equation shows that even when a particle is at rest, it possesses a tremendous amount of energy due to the incredibly large value of \( c^2 \). The concept of rest mass energy is crucial as it provides an understanding of the minimum energy an object contains before any external forces act upon it. In the context of the exercise, the electron’s rest mass energy is the baseline to which the additional kinetic energy - should the electron move - is compared, in order to find the required velocity for a specific total energy.

Special relativity is a theory proposed by Albert Einstein in 1905, transforming our understanding of space, time, and energy. One of the pivotal insights of special relativity is the relativity of simultaneity, which explains how two events that might be simultaneous for one observer may not be for another, if they are in relative motion. Another key aspect is the notion that nothing can travel faster than the speed of light in a vacuum.

Within this framework, the equation for relativistic kinetic energy becomes important when an object moves at a significant fraction of the speed of light. An object’s total energy \( E \) in special relativity is given by:
\( E = \frac{m_0 c^2}{\sqrt{1-\frac{v^2}{c^2}}} \)
where \( v \) is the object's velocity. As the velocity of an object increases and approaches the speed of light, its energy increases dramatically due to the relativistic effects encapsulated by this equation. The challenge in the given exercise is to apply these principles from special relativity to determine an electron's velocity so that its total energy exceeds its rest mass energy by 10%.

Solving for velocity in the context of special relativity often involves working with equations that contain velocities approaching the speed of light. This requires an understanding of both algebraic manipulation and the physical implications of relativistic effects on mass and energy.

For the particular problem at hand, we follow the steps outlined: starting with equating 110% of the rest mass energy to the total energy when the electron is moving. After simplification, the equation relates the square of the electron's velocity to a fraction of the speed of light squared. Algebraically, this requires careful squaring, isolation of terms, and taking square roots, ensuring to maintain the relationship between the velocity \( v \) and the speed of light \( c \).

The final solution is obtained by calculating the square root of the fraction obtained after simplification, multiplied by \( c \) to find the actual velocity:
\( v = c \times \sqrt{0.174} \)
This computed velocity reveals how fast the electron must move so its total energy is 10% more than its rest mass energy, exemplifying the process of solving for velocity within the framework of special relativity.