Ch 37: Special Relativity

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 37: Special Relativity

Problem 35

Chapter 36, Problem 35

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?
(c) What is the ratio of the kinetic energy to the rest energy of the particle?

Verified step by step guidance

Step 1: Understand the concept of total energy in relativistic physics. The total energy $ E $ of a particle is given by the equation $ E = \sqrt{(pc)^2 + (m_0c^2)^2} $, where $ p $ is the momentum, $ m_0 $ is the rest mass, and $ c $ is the speed of light.

Step 2: Calculate the rest energy of the particle using the formula $ E_0 = m_0c^2 $. Substitute $ m_0 = 6.64 \times 10^{-27} $ kg and $ c = 3 \times 10^8 $ m/s into the equation.

Step 3: Use the total energy formula to find the total energy $ E $. Substitute the given momentum $ p = 2.10 \times 10^{-18} $ kg•m/s and the calculated rest energy into the equation $ E = \sqrt{(pc)^2 + (m_0c^2)^2} $.

Step 4: Determine the kinetic energy $ K $ of the particle. The kinetic energy is given by $ K = E - E_0 $, where $ E $ is the total energy and $ E_0 $ is the rest energy.

Step 5: Calculate the ratio of the kinetic energy to the rest energy using the formula $ \text{Ratio} = \frac{K}{E_0} $. Substitute the values of $ K $ and $ E_0 $ obtained from previous steps.

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Key Concepts

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

Relativistic Energy-Momentum Relation

The relativistic energy-momentum relation is a fundamental concept in physics that connects a particle's total energy (E), momentum (p), and rest mass (m₀) through the equation E² = (pc)² + (m₀c²)², where c is the speed of light. This equation allows us to calculate the total energy of a particle when its momentum and rest mass are known, accounting for relativistic effects at high speeds.

Kinetic Energy in Relativity

In relativistic physics, kinetic energy (K) is defined as the difference between the total energy (E) and the rest energy (m₀c²) of a particle. It is given by K = E - m₀c². This definition extends the classical concept of kinetic energy to account for the effects of relativity, which become significant at velocities close to the speed of light.

Rest Energy

Rest energy is the energy inherent to a particle due to its rest mass, given by the equation E₀ = m₀c². This concept, introduced by Einstein's theory of relativity, highlights that mass itself is a form of energy. Rest energy is a crucial component in calculating the total energy of a particle, especially when considering relativistic effects.

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Related Practice

Textbook Question

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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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

Compute the kinetic energy of a proton (mass $1.67 \times 10^{- 27}$ kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) $8.00 \times 10^{7}$ m/s and (b) $2.85 \times 10^{8}$ m/s.

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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

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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