A proton (rest mass $1.67 \times 10^{- 27}$ kg) has total energy that is $4.00$ times its rest energy. What are (a) the kinetic energy of the proton; (b) the magnitude of the momentum of the proton; (c) the speed of the proton?

검증된 단계별 안내

Step 1: Understand the concept of total energy in relativistic physics. The total energy $ E $ of a particle is given by $ E = \gamma m_0 c^2 $, where $ \gamma $ is the Lorentz factor, $ m_0 $ is the rest mass, and $ c $ is the speed of light. In this problem, the total energy is 4 times the rest energy, so $ E = 4 m_0 c^2 $.

Step 2: Calculate the kinetic energy $ K $ of the proton. The kinetic energy in relativistic physics is given by $ K = E - m_0 c^2 $. Substitute the expression for total energy from Step 1 to find $ K = 4 m_0 c^2 - m_0 c^2 = 3 m_0 c^2 $.

Step 3: Determine the magnitude of the momentum $ p $ of the proton. The relativistic momentum is given by $ p = \gamma m_0 v $, where $ v $ is the velocity of the particle. Alternatively, you can use the relation $ E^2 = (pc)^2 + (m_0 c^2)^2 $ to solve for $ p $. Substitute $ E = 4 m_0 c^2 $ into the equation to find $ p $.

Step 4: Calculate the speed $ v $ of the proton. Use the Lorentz factor $ \gamma = \frac{E}{m_0 c^2} = 4 $ to find $ v $ using the relation $ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} $. Solve for $ v $ in terms of $ c $.

Step 5: Review the steps and ensure understanding of the relativistic relationships used. The key concepts include the Lorentz factor, relativistic energy, momentum, and the relationship between speed and energy. These are crucial for solving problems involving particles moving at speeds close to the speed of light.

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12m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Relativistic Energy

Relativistic energy accounts for the total energy of a particle, including its rest energy and kinetic energy. The total energy is given by E = γmc², where γ is the Lorentz factor, m is the rest mass, and c is the speed of light. In this problem, the total energy is four times the rest energy, indicating significant relativistic effects.

Lorentz Factor

The Lorentz factor, γ, is crucial in relativistic physics, defined as γ = 1/√(1-v²/c²), where v is the velocity of the particle and c is the speed of light. It describes how time, length, and relativistic mass change for an object moving at a significant fraction of the speed of light. It helps determine the kinetic energy and momentum of the proton in this scenario.

Relativistic Momentum

Relativistic momentum extends the classical concept of momentum to high-speed particles, calculated as p = γmv, where m is the rest mass, v is the velocity, and γ is the Lorentz factor. This concept is essential for finding the momentum of the proton, as its speed is a significant fraction of the speed of light, requiring adjustments from classical mechanics.