What is the speed of a proton that has total energy 10001000 GeV?

Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 44: Particle Physics and Cosmology

Problem 13a

Chapter 44, Problem 13a

What is the speed of a proton that has total energy $1000$ GeV?

Verified step by step guidance

Understand the problem: The total energy of the proton is given as 1000 GeV. The total energy (E) of a particle is related to its rest energy and kinetic energy. The relationship between total energy, rest energy, and momentum is given by the relativistic energy-momentum relation.

Write the relativistic energy-momentum relation: $ E^2 = (pc)^2 + (m_0c^2)^2 $, where $ E $ is the total energy, $ p $ is the momentum, $ c $ is the speed of light, and $ m_0 $ is the rest mass of the proton.

Rearrange the equation to solve for the momentum $ p $: $ p = \sqrt{\frac{E^2 - (m_0c^2)^2}{c^2}} $. Substitute the given total energy $ E = 1000 \text{ GeV} $ and the rest energy of the proton $ m_0c^2 = 0.938 \text{ GeV} $.

Relate the momentum $ p $ to the speed $ v $ of the proton using the relativistic momentum formula: $ p = \frac{m_0v}{\sqrt{1 - \frac{v^2}{c^2}}} $. Solve for $ v $ by substituting the value of $ p $ obtained in the previous step.

Simplify the expression for $ v $ and solve numerically if needed. The speed $ v $ will be very close to the speed of light $ c $, as the total energy is much larger than the rest energy.

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Key Concepts

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

Total Energy in Relativity

In the context of special relativity, the total energy of a particle is the sum of its rest mass energy and its kinetic energy. For a proton, this is given by the equation E = mc² + K.E., where E is the total energy, m is the rest mass, and c is the speed of light. Understanding this relationship is crucial for calculating the speed of a proton when its total energy is known.

Relativistic Momentum

Relativistic momentum is defined as p = γmv, where γ (gamma) is the Lorentz factor, m is the rest mass, and v is the velocity of the particle. As the speed of a particle approaches the speed of light, its momentum increases significantly, which is essential for understanding how energy and speed relate in high-energy physics scenarios, such as with a proton at 1000 GeV.

Lorentz Factor

The Lorentz factor, denoted as γ, is a crucial component in relativity that accounts for time dilation and length contraction at high speeds. It is defined as γ = 1 / √(1 - v²/c²). This factor becomes significant when calculating the speed of particles moving close to the speed of light, as it affects both energy and momentum, making it essential for solving the given problem.

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Related Practice

Textbook Question

Calculate the minimum beam energy in a proton-proton collider to initiate the $p + p \to p + p + η^{0}$ reaction. The rest energy of the $eta to the 0 power$ is $547.3$ MeV (see Table $44.3$ ).

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Textbook Question

In Example $44.3$ , it was shown that a proton beam with an $800$ -GeV beam energy gives an available energy of $38.7$ GeV for collisions with a stationary proton target. In a colliding-beam experiment, what total energy of each beam is needed to give an available energy of $2 open paren 38.7$ GeV $close paren equals 77.4$ GeV?

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Textbook Question

Deuterons in a cyclotron travel in a circle with radius $32.0$ cm just before emerging from the dees. The frequency of the applied alternating voltage is $9.00$ MHz. Find the magnetic field.

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Textbook Question

You work for a start-up company that is planning to use antiproton annihilation to produce radioactive isotopes for medical applications. One way to produce antiprotons is by the reaction $p + p \to p + p + p + \overset{ˉ}{p}$ in proton-proton collisions. You first consider a colliding-beam experiment in which the two proton beams have equal kinetic energies. To produce an antiproton via this reaction, what is the required minimum kinetic energy of the protons in each beam?

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Textbook Question

A high-energy beam of alpha particles collides with a stationary helium gas target. What must the total energy of a beam particle be if the available energy in the collision is $16.0$ GeV?

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Textbook Question

The magnetic field in a cyclotron that accelerates protons is $1.70$ T. How many times per second should the potential across the dees reverse? (This is twice the frequency of the circulating protons.)

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