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Understanding the total energy of an electron moving at speeds close to the speed of light involves a relativistic approach. According to special relativity, the total energy (\(E_{total}\)) of an object is expressed by the famous Einstein equation: \[ E = \frac{mc^2}{\sqrt{1 - (\frac{v^2}{c^2})}} \] where \(m\) is the rest mass of the electron, \(c\) is the speed of light (approximately \(3.0 \times 10^8\,\text{m/s}\)), and \(v\) is the velocity of the electron. As the velocity of an electron approaches the speed of light, relativistic effects become significant and the classical formula \(E = mc^2\) for the rest energy is no longer sufficient. Instead, the Lorentz factor, \[ \gamma = \frac{1}{\sqrt{1 - (\frac{v^2}{c^2})}} \] comes into play, accounting for time dilation and length contraction effects.

When plugging in the values, you'd get the total energy for an electron with speed of \(0.97c\), which is a substantial fraction of the speed of light, thus necessitating the relativistic treatment for accurate results. It's critical to remember that the rest mass energy (\(mc^2\)) is included in the total energy, representing the intrinsic energy the electron possesses even when at rest.

Kinetic energy in the realm of relativistic physics diverges from the classical \(\frac{1}{2}mv^2\) formula. The relativistic kinetic energy (KE) of an electron is the difference between its total energy and rest mass energy. The equation for calculating the kinetic energy is: \[ KE = E_{total} - mc^2 \] Where \(E_{total}\) is derived from the total energy equation we used in the context of relativistic speed. Subtracting the rest mass energy (\(mc^2\)) from the total energy gives us the additional energy attributed to the motion of the electron.

To calculate the kinetic energy of our electron travelling at \(0.97c\), you'd use the calculated total energy and subtract the rest mass energy. The resulting value reflects how much energy is due to the electron's high-speed movement. Since the velocities are so high, the kinetic energy will be a huge number, showcasing just how much additional energy an electron needs to be propelled to such extreme speeds.

When we talk about relativistic electron velocity, we are referring to electrons traveling at a speed that's a significant percentage of the speed of light. At these velocities, classical physics no longer gives us an accurate picture of the electron's behavior, and we must turn to Einstein's theory of special relativity to understand the implications.

In the example of an electron moving at \(0.97c\), the velocity is extremely high, causing time dilation and length contraction as described by Lorentz transformations. This velocity not only influences the total and kinetic energy calculations but also plays a pivotal role in other relativistic phenomena, such as mass increase (also known as relativistic mass), where the effective mass of the electron seems to increase as its velocity approaches the speed of light.

This kind of velocity has practical implications in particle accelerators and other high-energy physics domains, where understanding the true capabilities and consequences of such speeds is crucial for experimentation and innovation.