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In relativistic physics, the Lorentz factor (\(\gamma\)) plays a crucial role when dealing with objects moving at speeds comparable to the speed of light. This factor can be thought of as a measure of how much time, length, and relativistic mass change for an object moving close to light speed.
It is defined through the equation:\[\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}\]

• \(v\) represents the speed of the object.
• \(c\) is the speed of light, approximately \(3.00 \times 10^8 \, \text{m/s}\).

As an object's speed approaches the speed of light, the denominator of the equation above approaches zero, causing \(\gamma\) to increase drastically. This effectively means that the relativistic effects become more pronounced. In the context of this exercise, by calculating the Lorentz factor, we can understand how the proton behaves when it possesses a large amount of kinetic energy that's significant compared to its rest energy.

When dealing with particles in relativistic speeds, like a proton accelerated to high energies, we need to take extra care in converting kinetic energy from one unit to another and using it appropriately in formulas. Initially, the kinetic energy is given in mega-electronvolts (MeV), which is common in particle physics. Conversion to joules is done using:\[1 \, \mathrm{MeV} = 1.60218 \times 10^{-13} \, \mathrm{J}\]So for 200 MeV, this becomes:\[E_k = 200 \, \mathrm{MeV} \times 1.60218 \times 10^{-13} \, \mathrm{J/MeV} = 3.20436 \times 10^{-11} \, \mathrm{J}\]
It’s critical to convert energies into consistent units (like joules here) to correctly compute subsequent quantities and factors, such as the Lorentz factor, which uses mass in kilograms and speed in meters per second. This ensures that our calculations for speed, energy, and other related values remain precise and reliable.

The speed of light, denoted by \(c\), is a core constant in physics with a value of \(3.00 \times 10^8 \, \text{m/s}\). It represents the ultimate speed limit in the universe, as no object with mass can reach or exceed this speed. In relativistic kinematics, which handles objects moving at or near this speed, \(c\) is critical as a conversion factor.
This speed comes up frequently, such as when calculating the Lorentz factor or when determining relativistic kinetic energy and momentum. Particularly in the exercise at hand, we use \(c\) to express the proton's speed as a fraction of the speed of light, helping us understand how relativity affects the particle:\[v = c \sqrt{1 - \frac{1}{\gamma^2}}\]Here, \(v\) is the speed as obtained after computing the Lorentz factor, indicating how close the speed of the proton is to \(c\), thus revealing the significant relativistic effects exerted on the proton when accelerating near the speed of light.