Ch 37: The Foundations of Modern Physics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 37: The Foundations of Modern Physics

Problema 31

Capitolo 37, Problema 31

The factor γ appears in many relativistic expressions. A value γ = 1.01 implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is γ = 1.01 for (a) an electron, (b) a proton, and (c) an alpha particle?

Guida verificata passo dopo passo

Step 1: Recall the definition of the Lorentz factor γ, which is given by the equation:

γ = 1 / √(1 - v²/c²)

, where v is the velocity of the particle and c is the speed of light. Rearrange this equation to solve for v in terms of γ:

v = c √(1 - 1/γ²)

Step 2: The relativistic kinetic energy (K) is given by the equation:

K = (γ - 1)mc²

, where m is the rest mass of the particle. For each particle (electron, proton, and alpha particle), substitute the given γ = 1.01 into this equation.

Step 3: Use the known rest masses of the particles: for an electron,

m_e = 0.511 \, \(\text{MeV}\)/c²

; for a proton,

m_p = 938.27 \, \(\text{MeV}\)/c²

; and for an alpha particle,

m_\(\alpha\) = 3727.38 \, \(\text{MeV}\)/c²

. Substitute these values into the kinetic energy equation for each particle.

Step 4: Simplify the kinetic energy equation for each particle. For example, for the electron:

K_e = (1.01 - 1)(0.511 \, \(\text{MeV}\)/c²)c²

. Repeat this process for the proton and the alpha particle.

Step 5: After substituting and simplifying, the resulting kinetic energy values for each particle will be in MeV. These values represent the kinetic energy at which relativistic effects become significant for γ = 1.01.

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Lorentz Factor (γ)

The Lorentz factor, denoted as γ (gamma), is a crucial component in the theory of relativity that describes how time, length, and relativistic mass change for an object moving relative to an observer. It is defined as γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. A γ value greater than 1 indicates that relativistic effects become significant, affecting measurements of time and space.

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Kinetic Energy in Relativity

In classical mechanics, kinetic energy (KE) is given by the formula KE = 1/2 mv². However, in relativistic physics, the kinetic energy of an object is expressed as KE = (γ - 1)mc², where m is the rest mass and c is the speed of light. This formula accounts for the increase in energy as an object's speed approaches the speed of light, highlighting the differences between classical and relativistic kinetic energy.

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Rest Mass vs. Relativistic Mass

Rest mass is the mass of an object measured when it is at rest, while relativistic mass increases with velocity and is given by m' = γm, where m' is the relativistic mass. This distinction is important in relativistic physics, as it affects how we calculate energy and momentum. Understanding the difference helps clarify how objects behave at high speeds and the implications for energy calculations in relativistic scenarios.

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