Textbook Question
A proton (rest mass kg) has total energy that is 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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Ch 37: Special Relativity
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610
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Table of contents
- Ch 01: Units, Physical Quantities & Vectors41
- Ch 02: Motion Along a Straight Line67
- Ch 03: Motion in Two or Three Dimensions47
- Ch 04: Newton's Laws of Motion23
- Ch 05: Applying Newton's Laws59
- Ch 06: Work & Kinetic Energy14
- Ch 07: Potential Energy & Conservation8
- Ch 08: Momentum, Impulse, and Collisions18
- Ch 09: Rotation of Rigid Bodies42
- Ch 10: Dynamics of Rotational Motion48
- Ch 11: Equilibrium & Elasticity27
- Ch 12: Fluid Mechanics31
- Ch 13: Gravitation35
- Ch 14: Periodic Motion32
- Ch 15: Mechanical Waves25
- Ch 16: Sound & Hearing32
- Ch 17: Temperature and Heat36
- Ch 18: Thermal Properties of Matter38
- Ch 19: The First Law of Thermodynamics33
- Ch 20: The Second Law of Thermodynamics22
- Ch 21: Electric Charge and Electric Field36
- Ch 22: Gauss' Law24
- Ch 23: Electric Potential31
- Ch 24: Capacitance and Dielectrics18
- Ch 25: Current, Resistance, and EMF26
- Ch 26: Direct-Current Circuits30
- Ch 27: Magnetic Field and Magnetic Forces18
- Ch 28: Sources of Magnetic Field10
- Ch 29: Electromagnetic Induction28
- Ch 30: Inductance17
- Ch 31: Alternating Current21
- Ch 32: Electromagnetic Waves1
- Ch 33: The Nature and Propagation of Light20
- Ch 34: Geometric Optics34
- Ch 35: Interference19
- Ch 36: Diffraction25
- Ch 37: Special Relativity20
- Ch 38: Photons: Light Waves Behaving as Particles15
- Ch 39: Particles Behaving as Waves26
- Ch 40: Quantum Mechanics I: Wave Functions21
- Ch 41: Quantum Mechanics II: Atomic Structure25
- Ch 42: Molecules and Condensed Matter18
- Ch 43: Nuclear Physics16
- Ch 44: Particle Physics and Cosmology23
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
Chapter 37, Problem 37a
A particle has rest mass kg and momentum 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 guidance1
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.
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05:17
Intro to Momentum
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.
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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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04:10Intro to Energy & Types of Energy
Related Practice
Textbook Question
Electrons are accelerated through a potential difference of kV, so that their kinetic energy is eV.
(a) What is the ratio of the speed of an electron having this energy to the speed of light, ?
(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 kg) using both the nonrelativistic and relativistic expressions, and compute the ratio of the two results (relativistic divided by nonrelativistic) for speeds of (a) m/s and (b) m/s.
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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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