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Ch 36: Special Relativity
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 36: Special Relativity문제 75
36장, 문제 75

Some particle accelerators allow protons (p⁺) and antiprotons (p⁻) to circulate at equal speeds in opposite directions in a device called a storage ring. The particle beams cross each other at various points to cause p⁺ + p⁻ collisions. In one collision, the outcome is p⁺ + p⁻ → e⁺ + e⁻ + γ + γ, where γ represents a high-energy gamma-ray photon. The electron and positron are ejected from the collision at 0.9999995c and the gamma-ray photon wavelengths are found to be 1.0 x 10-6 nm. What were the proton and antiproton speeds, as a fraction of c, prior to the collision?

검증된 단계별 안내
1
Understand the problem: The collision involves protons and antiprotons moving at equal speeds in opposite directions. The goal is to determine their speeds as a fraction of the speed of light (c) before the collision. The key concepts here are relativistic energy and momentum conservation.
Step 1: Write down the total energy of the system before the collision. The total energy of each proton or antiproton is given by the relativistic energy formula: E = γmc², where γ = 1 / √(1 - v²/c²) is the Lorentz factor, m is the rest mass of the proton, and v is the speed of the proton as a fraction of c. Since the protons and antiprotons are moving at equal speeds, their total energy is 2γmc².
Step 2: Write down the total energy of the system after the collision. The products of the collision are an electron (e⁻), a positron (e⁺), and two gamma-ray photons (γ). The total energy after the collision is the sum of the relativistic energies of the electron and positron, plus the energy of the two gamma-ray photons. The energy of each photon is given by E_γ = hc/λ, where h is Planck's constant, c is the speed of light, and λ is the wavelength of the photon.
Step 3: Apply conservation of energy. The total energy before the collision (2γmc²) must equal the total energy after the collision. Set up the equation: 2γmc² = 2γ_e m_e c² + 2E_γ, where γ_e is the Lorentz factor for the electron and positron, m_e is the rest mass of the electron, and E_γ is the energy of each gamma-ray photon. Substitute the given values for the photon wavelength and the electron speed (0.9999995c) to calculate γ_e and E_γ.
Step 4: Solve for the proton speed. Rearrange the energy conservation equation to isolate γ for the proton. Use the relationship γ = 1 / √(1 - v²/c²) to solve for v, the speed of the proton as a fraction of c. This will give you the proton and antiproton speeds prior to the collision.

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주요 개념

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Relativistic Speeds

In physics, relativistic speeds refer to velocities that are a significant fraction of the speed of light (c). At these speeds, classical mechanics no longer accurately describes motion, and relativistic effects such as time dilation and length contraction become significant. Understanding how objects behave at relativistic speeds is crucial for analyzing high-energy particle collisions, as it affects the energy and momentum calculations involved.
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Conservation of Energy and Momentum

The conservation of energy and momentum is a fundamental principle in physics stating that in an isolated system, the total energy and total momentum remain constant before and after a collision. In particle physics, this principle is used to analyze the outcomes of collisions, such as the creation of new particles. For the given collision, one must account for the initial energies and momenta of the protons and antiprotons to determine their speeds.
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Photon Wavelength and Energy Relation

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. In high-energy collisions, the emitted gamma-ray photons can provide insights into the energy released during the interaction. Understanding this relationship is essential for calculating the energy involved in the collision and the resulting particle speeds.
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