Ch. 36 - The Special Theory of Relativity

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 36 - The Special Theory of Relativity

Problem 85

Chapter 35, Problem 85

A pi meson of mass m_π decays at rest into a muon (mass m_μ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is K_μ = (m_π - m_μ)² c² / (2m_π).

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Start by applying the principle of conservation of energy. Since the pi meson is initially at rest, its total energy is equal to its rest energy, which is given by E_π = m_π c². After the decay, the total energy is shared between the muon and the neutrino.

Next, apply the principle of conservation of momentum. Since the pi meson is initially at rest, the total momentum of the system after the decay must also be zero. This means the muon and the neutrino must have equal and opposite momenta.

Express the total energy of the muon and neutrino. The muon's total energy is E_μ = √(p²c² + m_μ²c⁴), where p is the magnitude of the muon's momentum. The neutrino's energy is E_ν = pc, since its mass is negligible.

Use conservation of energy to write E_π = E_μ + E_ν. Substituting the expressions for E_μ and E_ν, you get m_π c² = √(p²c² + m_μ²c⁴) + pc.

Square both sides of the equation to eliminate the square root, simplify, and solve for the muon's kinetic energy K_μ = E_μ - m_μ c². After algebraic manipulation, you will find K_μ = (m_π - m_μ)² c² / (2m_π).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Energy

In any physical process, the total energy before the event must equal the total energy after the event. In the case of the decay of a pi meson, the initial rest energy of the meson is converted into the kinetic energy of the resulting particles, the muon and the neutrino. This principle allows us to relate the masses of the particles involved to their kinetic energies.

Rest Mass Energy

The rest mass energy of a particle is given by Einstein's equation E = mc², where m is the rest mass and c is the speed of light. For the pi meson at rest, its total energy is simply its rest mass energy. This energy is crucial for calculating the kinetic energy of the decay products, as it provides the initial energy available for conversion into kinetic energy.

Kinetic Energy in Relativistic Physics

In relativistic physics, the kinetic energy of a particle is not simply given by the classical formula (1/2)mv², especially when dealing with particles moving at speeds close to the speed of light. Instead, the kinetic energy can be derived from the difference in rest mass energy before and after a decay process, which is essential for deriving the expression for the muon's kinetic energy in this decay scenario.

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A pi meson of mass mπ decays at rest into a muon (mass mμ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is Kμ = (mπ - mμ)² c² / (2mπ).

Verified video answer for a similar problem:

Key Concepts

Conservation of Energy

Rest Mass Energy

Kinetic Energy in Relativistic Physics

A pi meson of mass m_π decays at rest into a muon (mass m_μ) and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is K_μ = (m_π - m_μ)² c² / (2m_π).