The highest-energy cosmic ray ever detected had an energy of...

The highest-energy cosmic ray ever detected had an energy of about $3.0 \times 10^{20} \mathrm{eV}$. Assume that this cosmic ray was a proton. a. What was the proton's speed as a fraction of $c ?$ b. If this proton started at the same time and place as a photon traveling at the speed of light, how far behind the photon would it be after traveling for 1 ly?

The highest-energy cosmic ray ever detected had an energy of about $3.0 \times 10^{20} \mathrm{eV}$. Assume that this cosmic ray was a proton.
a. What was the proton's speed as a fraction of $c ?$
b. If this proton started at the same time and place as a photon traveling at the speed of light, how far behind the photon would it be after traveling for 1 ly?

Key Concepts

Ultra-High Energy Cosmic Rays

Ultra-high energy cosmic rays are particles, such as protons, that are accelerated to extremely high energies, far exceeding those produced in terrestrial accelerators. Studying these particles involves applying relativistic kinematics to understand their behavior as they traverse space at speeds very close to the speed of light. Their analysis provides insights into extreme astrophysical processes and the limitations imposed by the speed of light over cosmic distances.

Relativistic Energy-Momentum Relation

In the realm of high speeds, the classical equations of energy and momentum are replaced by the relativistic energy-momentum relation, E² = (pc)² + (m?c²)². For particles traveling at speeds close to that of light, this relation provides a way to link an object's total energy to its mass and velocity. It is essential for calculating the speed of particles when given extraordinary energy values, as it captures the nonlinear increase of kinetic energy as an object’s speed approaches c.

Special Relativity

Special relativity is the framework that governs the behavior of objects moving at speeds comparable to the speed of light. It introduces concepts such as time dilation, length contraction, and the invariance of the speed of light in all inertial reference frames. This theory provides the necessary tools to analyze how energy, momentum, and velocity relate at ultra?high speeds, which is critical when dealing with particles accelerated to nearly light speed.

Lorentz Factor

The Lorentz factor, often denoted by ?, quantifies the relativistic effects experienced by a moving object. It is defined as ? = 1/?(1 - ?²), where ? is the ratio of the object’s speed to the speed of light. This factor is used to relate the energy and momentum of a particle to its mass and speed, and it becomes extremely large as the speed approaches the speed of light, significantly affecting measurements of time and distance.

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