A cosmic-ray proton is traveling at nearly the speed of light \(\left(3 \times 10^{8} \mathrm{m} / \mathrm{s}\right)\) a. Using Einstein's familiar relationship between mass and energy \(\left(E=m c^{2}\right),\) show how much energy (in joules) the cosmic-ray proton would have if \(m\) were based only on the proton's rest mass \(\left(1.7 \times 10^{-27} \mathrm{kg}\right)\) b. The actual measured energy of the cosmic-ray proton is 100 J. What, then, is the relativistic mass of the cosmic-ray proton? c. How much greater is the relativistic mass of this cosmicray proton than the mass of a proton at rest?

Einstein's mass-energy equivalence principle is encapsulated by the famous equation \(E = mc^2\). This equation tells us that energy (E) is equal to mass (m) times the speed of light (c) squared. It reveals a profound connection between mass and energy, showing that mass can be converted into energy and vice versa. This principle is crucial in various aspects of physics, including nuclear reactions and particle physics.
For example, in the context of a cosmic-ray proton, the rest mass of the proton can be used to calculate how much energy it has when considered at rest. In this scenario, the rest mass is given, and by plugging it into Einstein’s equation along with the speed of light, we determine the energy associated with the proton's rest mass.

Relativistic mass is the concept that as an object moves faster and its velocity approaches the speed of light, its mass increases. This happens because the energy needed to continue accelerating the object increases.
The relativistic mass \(m_{rel}\) can be found using the formula \(E = m_{rel} c^2\). If we have a cosmic-ray proton with an actual measured energy, for instance, 100 joules, we can calculate its relativistic mass by rearranging this equation to \(m_{rel} = \frac{E}{c^2}\). This concept explains why particles traveling at high velocities have much greater mass than their rest mass.

Rest mass, or invariant mass, is the mass of an object when it is at rest relative to an observer. It is a fundamental property of matter and does not change regardless of the object's motion. For a proton, the rest mass is given as \(1.7 \times 10^{-27} \text{kg}\).
Using the rest mass in Einstein's mass-energy equivalence equation (\(E = mc^2\)), we can calculate the energy of the proton when it's not moving. This is essential in understanding the cosmic-ray proton's energy profile and comparing it to its relativistic mass when it is moving at speeds close to the speed of light.

Cosmic-ray physics studies high-energy particles originating outside the Earth's atmosphere. These particles include protons, which travel at speeds close to the speed of light. The study of these energetic particles helps scientists understand the universe's most violent processes.
High-energy protons can be observed and measured, and their energy calculated using principles like Einstein's equations. By analyzing these high-speed protons, scientists track their sources, such as supernovae, and study fundamental physics under extreme conditions that cannot be replicated on Earth.

The speed of light \(c\) is a universal constant approximately equal to \(3 \times 10^8 \text{m/s}\). It is a critical factor in many physical equations, including Einstein's mass-energy equivalence equation.
Knowing the speed of light allows us to calculate the energy equivalent of a mass. For a cosmic-ray proton, traveling nearly at the speed of light, understanding this relationship is essential. As per special relativity, no object with mass can reach the speed of light, but particles like cosmic-ray protons can get significantly close, thus gaining tremendous relativistic mass and energy.