How much energy is released when a muon at rest decays into an electron and two neutrinos? Neglect the small masses of the neutrinos.
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1. Intro to Physics Units
Introduction to Units
Problem 28
Textbook Question
What is the total kinetic energy of the decay products when an upsilon particle at rest decays to ?

1
Step 1: Understand the problem. The upsilon particle (Υ) is initially at rest and decays into two tau particles (t⁺ and t⁻). Since the upsilon particle is at rest, its initial momentum is zero. The total kinetic energy of the decay products can be determined using conservation of energy and momentum principles.
Step 2: Apply the conservation of energy. The total energy of the upsilon particle before decay is equal to its rest energy, which is given by E = m_Υ * c², where m_Υ is the mass of the upsilon particle and c is the speed of light. After decay, this energy is distributed between the rest energy and kinetic energy of the tau particles.
Step 3: Write the energy equation for the decay products. The total energy of the system after decay is the sum of the rest energy and kinetic energy of the tau particles: E_total = 2 * (m_τ * c²) + KE_total, where m_τ is the mass of a tau particle and KE_total is the total kinetic energy of the tau particles.
Step 4: Use conservation of momentum. Since the upsilon particle is at rest, the momentum of the decay products must cancel out. This means the tau particles move in opposite directions with equal magnitudes of momentum. Use the relativistic energy-momentum relation: E² = (pc)² + (m * c²)², where p is the momentum, m is the mass, and E is the total energy of each tau particle.
Step 5: Solve for the total kinetic energy. Subtract the rest energy of the tau particles from their total energy to find their kinetic energy. Add the kinetic energy of both tau particles to find KE_total. The final expression for KE_total will depend on the mass of the upsilon particle and tau particles, as well as the speed of light.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity of the object. In particle physics, understanding kinetic energy is crucial for analyzing the energy distribution among decay products after a particle decay.
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Conservation of Energy
The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In the context of particle decay, the total energy before the decay (rest energy of the upsilon particle) must equal the total energy after the decay, which includes the kinetic energy of the decay products and their rest mass energy.
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Particle Decay
Particle decay refers to the process by which an unstable particle transforms into other particles, releasing energy in the form of kinetic energy and radiation. In this case, the upsilon particle decays into a pair of tau particles (t^+ and t^-), and understanding the dynamics of this decay is essential for calculating the total kinetic energy of the resulting particles.
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