A particle of rest energy m c^2 is moving with speed v in...

A particle of rest energy $m c^{2}$ is moving with speed $v$ in the positive $x$ direction. The particle decays into two particles, each of rest energy $140 \mathrm{MeV}$. One particle, with kinetic energy $282 \mathrm{MeV},$ moves in the positive $x$ direction, and the other particle, with kinetic energy $25 \mathrm{MeV}$, moves in the negative $x$ direction. Find the rest energy of the original particle and its speed.

Key Concepts

Conservation of Momentum in Relativistic Processes

Momentum must be conserved in any closed system, regardless of whether the components are moving at relativistic speeds. In the context of relativistic collisions or decays, momentum conservation must account for both magnitude and direction. This principle is applied to ensure that the vector sum of the momenta before and after an event remains constant, thereby providing additional equations necessary to solve for unknown variables such as the speed or energy of the particles involved.

Relativistic Energy–Momentum Relation

This concept expresses the fundamental relation between energy, momentum, and mass in special relativity. It is given by E² = (m c²)² + (p c)², where E is the total energy, m is the rest mass, p is the momentum, and c is the speed of light. This relation is essential when analyzing processes involving high-speed particles, such as decays or collisions, and allows for connecting energy and momentum in a consistent framework.

Lorentz Factor

The Lorentz factor, denoted as ? (gamma), quantifies the effect of time dilation, length contraction, and the increase in relativistic mass as an object approaches the speed of light. It is defined as ? = 1/?(1 ? v²/c²), where v is the velocity of the particle. This factor is fundamental in relating rest energy and kinetic energy in relativistic scenarios since it scales the rest mass energy by ?.

Conservation of Energy in Relativistic Processes

In any isolated system, the total energy (including both rest energy and kinetic energy) must be conserved, even when particles are moving at relativistic speeds. This principle is used to equate the total initial energy with the sum of the energies of the resultant particles after a decay or collision, which allows one to solve for unknown quantities like the original rest energy in decay problems.

Relativistic Kinetic Energy

Unlike classical kinetic energy, the relativistic kinetic energy is defined as the total energy minus the rest energy of a particle. It is given by (? ? 1)mc², where m is the rest mass and ? is the Lorentz factor. This concept is crucial for analyzing events where particles move at speeds comparable to the speed of light, as it accurately describes the energy difference due to motion when classical approximations fail.

Particle Decay in Special Relativity

This concept involves the transformation of a single particle into multiple particles along with the application of conservation laws. In relativistic particle decay, both energy and momentum must be conserved, and the decay products typically share the original particle's energy in a way that respects the relativistic energy–momentum relation. This framework is essential for understanding high-energy physics experiments and scenarios where particles decay into various products with both kinetic and rest energies.

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