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The theory of special relativity, put forth by Albert Einstein in 1905, revolutionized the way we understand space, time, and energy. Special relativity is based on two postulates: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is constant and independent of the motion of the light source or observer.

One of the most famous equations arising from special relativity is the equation for relativistic mass-energy equivalence, which shows that energy and mass are interchangeable. This equation, often expressed as \( E = mc^2 \), indicates that an object has a greater energy content as its velocity approaches the speed of light. When solving physics problems, such as calculating the total energy of a particle, we use the principles of special relativity to ensure our calculations are accurate at high velocities.

Every object with mass possesses an intrinsic energy, even when it is at rest. This energy is known as 'rest mass energy', symbolized by \( E_0 \) and given by the equation \( E_0 = m_0c^2 \), where \( m_0 \) is the rest mass of the object and \( c \) is the speed of light.

For subatomic particles like protons and electrons, the rest mass energy is a crucial aspect to consider when calculating their total energy. A proton’s rest mass energy is approximately \(938.27 \text{MeV}\), and an electron's is about \(0.511 \text{MeV}\). To acquire a complete understanding of a particle's energy, one must consider its kinetic energy in addition to its rest mass energy, especially at relativistic speeds.

The speed of light, denoted by \( c \), is a fundamental physical constant important in many areas of physics. Its value is approximately \(3.00 \times 10^8 \text{m/s}\) in a vacuum. Under special relativity, the speed of light is the upper limit for the speed at which conventional matter, energy, or any signal carrying information can travel.

When dealing with high-velocity entities approaching the speed of light, ordinary Newtonian mechanics no longer suffice, and one must use relativistic mechanics to accurately describe their behavior. For instance, in our exercise, we consider particles traveling at \(99\%\) of the speed of light to understand the vast increase in energy due to such high velocities.

The total energy of an object in the relativistic context is the sum of its rest mass energy and the kinetic energy it gains due to motion. It is given by the formula \( E = \gamma m_0c^2 \), where \( \gamma \) (Gamma) is the Lorentz factor, defined as \( \gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \). Here \( v \) represents the velocity of the object, and \( c \) is the speed of light.

This equation shows that as an object moves faster, its total energy increases dramatically. For velocities approaching the speed of light, the denominator of the Lorentz factor becomes very small, leading to a huge increase in the object's total energy. In solving our exercise, we calculate the total energy for a proton and an electron at \(99\%\) of the speed of light, demonstrating the significant impact of relativistic speeds on the total energy of particles.