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Ch 37: The Foundations of Modern Physics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Capitolo 37, Problema 27b

What is the velocity, as a fraction of c, of an electron with 2.0 GeV total energy? Hint: This problem uses relativity.

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Start by recalling the total energy equation in special relativity: E = γmc², where γ is the Lorentz factor, m is the rest mass of the electron, and c is the speed of light.
Rearrange the equation to solve for the Lorentz factor γ: γ = E / (mc²). Here, E is the total energy (2.0 GeV), and mc² is the rest energy of the electron (approximately 0.511 MeV). Convert units so they are consistent (e.g., GeV to MeV).
The Lorentz factor is also related to velocity by the equation: γ = 1 / √(1 - v²/c²). Substitute the value of γ obtained in the previous step into this equation.
Rearrange the equation to solve for v/c, the velocity as a fraction of the speed of light: v/c = √(1 - 1/γ²). Plug in the value of γ to compute v/c.
Simplify the expression to find the numerical value of v/c. This will give the velocity of the electron as a fraction of the speed of light.

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Relativistic Energy

In the context of special relativity, the total energy of an object is given by the equation E = γmc², where γ (gamma) is the Lorentz factor. This factor accounts for the effects of relativistic speeds, which become significant as an object's velocity approaches the speed of light (c). The total energy includes both rest mass energy and kinetic energy, and understanding this concept is crucial for solving problems involving high-energy particles like electrons.
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Intro to Energy & Types of Energy

Lorentz Factor

The Lorentz factor, denoted as γ, is defined as γ = 1 / √(1 - v²/c²), where v is the velocity of the object and c is the speed of light. This factor increases as the velocity of the object approaches the speed of light, leading to significant relativistic effects such as time dilation and length contraction. It is essential for calculating the relativistic momentum and energy of particles moving at high speeds.
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Lorentz Transformations of Position

Velocity as a Fraction of c

In relativistic physics, expressing velocity as a fraction of the speed of light (c) allows for a clearer understanding of how fast an object is moving relative to the ultimate speed limit in the universe. For example, if an electron's velocity is 0.9c, it is traveling at 90% of the speed of light. This concept is vital for analyzing the behavior of particles in high-energy physics, particularly when calculating their relativistic effects.
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Static & Kinetic Friction
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