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Rest energy is a key concept in physics that describes the energy contained within a particle at rest, relative to an observer. It's represented by the ubiquitous equation:

• \( E_0 = mc^2 \)

Here, \( m \) represents the rest mass of the particle, and \( c \) symbolizes the speed of light. The rest energy is an inherent property of matter, implying that even a particle not in motion has energy due to its mass.
This principle stems from Einstein's theory of relativity, which suggests that mass and energy are interchangeable. This energy can be seen as the intrinsic energy that an object possesses simply by "existing". It is important in understanding how energy is conserved in various systems.

The speed of light, denoted by \( c \), is a universal constant that plays a crucial role in many areas of physics, particularly in relativity. Its approximate value is about 299,792,458 meters per second (m/s).
Light speed acts as a cosmic speed limit for how fast information and matter can travel through the universe.
In the equation \( E_0 = mc^2 \), the speed of light serves not only as a conversion factor between mass and energy but also reinforces the idea that the closer a particle's speed is to \( c \), the more significant relativistic effects become. This influences how we perceive and calculate energies and velocities in high-speed scenarios.

The Lorentz factor, \( \gamma \), is a component used in calculating relativistic effects when velocities reach a significant fraction of the speed of light. It is given by the equation:

• \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

Within this formula, \( v \) is the velocity of the object, and \( c \) is the speed of light. The Lorentz factor approaches infinity as \( v \) nears \( c \), indicating that reaching or exceeding light speed requires infinite energy, which is impossible for massive particles.
The Lorentz factor helps adjust for time dilation and length contraction, two relativistic effects observed at high speeds. In terms of energy, it helps to determine the total energy of moving particles by modifying their rest energy.

Calculating the speed of a particle whose kinetic energy is a multiple of its rest energy involves solving equations derived from relativistic principles.
For example, if the kinetic energy equals the rest energy (as described in part (a) of our exercise), the equation becomes:

• \( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} = 2mc^2 \)

Solving this, we find that \( v = \frac{\sqrt{3}}{2}c \), meaning the particle moves at about 86.6% of the speed of light.
Similarly, when kinetic energy is five times the rest energy (part (b)), the equation becomes:

• \( \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} = 6mc^2 \)

This yields \( v = \frac{\sqrt{35}}{6}c \), or about 98.2% of the speed of light. Calculations like these highlight how energy requirements rapidly increase as velocities approach light speed. This illustrates why, in reality, massive objects can never reach the speed of light.