Ch 37: Special Relativity

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 37: Special Relativity

Problem 33

Chapter 37, Problem 33

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?

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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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Key Concepts

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

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.

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Related Practice

Textbook Question

A particle has rest mass $6.64 \times 10^{- 27}$ kg and momentum $2.10 \times 10^{- 18}$ kgm/s.

(a) What is the total energy (kinetic plus rest energy) of the particle?

(b) What is the kinetic energy of the particle?

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Textbook Question

A proton has momentum with magnitude p₀ when its speed is 0.400c. In terms of p₀, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

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Textbook Question

Electrons are accelerated through a potential difference of $750$ kV, so that their kinetic energy is $7.50 \times 10^{5}$ eV.

(a) What is the ratio of the speed $v$ of an electron having this energy to the speed of light, $c$ ?

(b) What would the speed be if it were computed from the principles of classical mechanics?

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Textbook Question

A force is applied to a particle along its direction of motion. At what speed is the magnitude of force required to produce a given acceleration twice as great as the force required to produce the same acceleration when the particle is at rest? Express your answer in terms of the speed of light.

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Textbook Question

Relativistic Baseball. Calculate the magnitude of the force required to give a 0.145 kg baseball an acceleration a = 1.00 m/s² in the direction of the baseball's initial velocity when this velocity has a magnitude of (a) 10.0 m/s; (c) 0.990c.

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Textbook Question

An observer in frame S′ is moving to the right (+x-direction) at speed u = 0.600c away from a stationary observer in frame S. The observer in S′ measures the speed v′ of a particle moving to the right away from her. What speed v does the observer in S measure for the particle if (a) v′ = 0.400c; (b) v′ = 0.900c; (c) v′ = 0.990c?

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A proton (rest mass 1.67×10−271.67\(\times\)10^{-27} kg) has total energy that is 4.004.00 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?

Verified video answer for a similar problem:

Key Concepts

Relativistic Energy

Lorentz Factor

Relativistic Momentum

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?