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The concept of total energy in the realm of relativity is an extension of classical physics, designed to include the effects that arise at high speeds, close to the speed of light. In relativistic physics, the total energy (\(E\)) of a particle is not just the sum of its kinetic and potential energy. The total energy includes both the kinetic energy it possesses due to its motion and its rest mass energy.
Using Einstein's famous equation, the rest energy (\(E_0\)) is given by \(E_0 = mc^2\), where \(m\) is the rest mass and \(c\) is the speed of light. Relativistic total energy is expressed as \(E = \sqrt{(pc)^2 + (mc^2)^2}\), where \(p\) stands for momentum.
This representation can be rearranged and squared to look like \(E^2 = p^2c^2 + (mc^2)^2\). This equation reveals how rest mass energy and momentum contribute to the total energy, offering a deeper understanding of the unity of energy and mass at relativistic speeds.

In relativistic physics, momentum is intertwined with energy in ways that aren't evident in everyday experiences. It's crucial to calculate momentum with these extended ideas to capture all phenomena at high speeds.
The relativistic momentum (\(p\)) can be derived from the total energy equation \(E^2 = p^2c^2 + (mc^2)^2\). When calculating for momentum, one first rearranges to find:

• \(p^2c^2 = E^2 - (mc^2)^2\)
• \(p^2 = \frac{E^2 - (mc^2)^2}{c^2}\)
• \(p = \sqrt{\frac{E^2 - (mc^2)^2}{c^2}}\)

For example, for a proton with a total energy of 1938 MeV and rest energy (mass energy) of 938 MeV, substituting these into the equation helps find the momentum with great precision. The result gives us an understanding of how energy is distributed in moving particles beyond classical expectations.

Kinetic energy represents the extra energy a proton (or any other particle) has due to its motion. In relativistic terms, kinetic energy (\(K\)) is the energy a particle has when moving, minus its rest energy (\(E_0\)).
For a proton, the kinetic energy can be integrated into its total energy using the relation \(E = K + E_0\). In our scenario, a proton has a kinetic energy of 1000 MeV. The full total energy of this proton will then be the sum of 1000 MeV kinetic energy and its rest energy of 938 MeV, bringing it to 1938 MeV.
This energy plays a vital role in determining how the proton behaves, showing its increased dynamical properties due to the relativistic speeds. Adding to the overall energy helps illustrate how energy in particles at these scales is a blend of kinetic and intrinsic mass energies.