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 → p + p + p + p ˉ p + p → p + p + p + p̄ 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?

Welcome back, everyone. We are tasked with calculating the minimum kinetic ener that each proton beam must have. In order to obtain this reaction right here, if we are told that the two highly energetic proton photos have the same energy allied. All right. So here are our answer choices over here on the left. Let's go ahead in. Now, in order to observe the minimum kinetic energy, we need to draw this conclusion that the minimum kinetic energy that each proton beam must have means that the products of the reaction are at rest. Therefore, the incident proton beams need only enough kinetic energy to produce the charged ions of plus and pion negative. So we actually know that the mass for either one of these 139.6 mega electron volts over C squared. No. In order to find our total energy, the incident proton beam only enough kinetic energy to produce uh these two right here. In order to find our total energy, we are going to have our two masses added together times C squared. So what does this give us 139.6 mega electron volts divided by C squared 139.6 mega electron volts divided by squared times C squared. Getting rid of the C squares effectively gives us a total energy of 279.2 mega electron volts. But for the energy or just one proton beam, well, there's two proton beams. So two times energy of the proton beam gives us our total energy dividing both sides by two is that the energy for one proton beam is simply our energy divided by two. So 279.2, divided by two gives us 139.6 mega electron volts corresponding to our final answer. Choice of B Thank you all so much for watching. Hope you helped. We call in the next one.

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Problem 16a

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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Step 1: Understand the reaction. The given reaction is p + p → p + p + p + p̄, where two protons collide to produce two protons, one additional proton, and one antiproton. The goal is to determine the minimum kinetic energy required for this reaction to occur.

Step 2: Apply the principle of energy conservation. The total energy of the system before and after the collision must be conserved. The initial energy includes the rest energy and kinetic energy of the two protons. The final energy includes the rest energy of the three protons and one antiproton, plus any kinetic energy of the products.

Step 3: Calculate the rest energy of the particles. The rest energy of a proton is given by E₀ = mₚc², where mₚ is the mass of the proton and c is the speed of light. Similarly, the rest energy of an antiproton is the same as that of a proton because they have equal masses. Therefore, the total rest energy of the products is 3mₚc² (for three protons) + mₚc² (for one antiproton) = 4mₚc².

Step 4: Determine the minimum kinetic energy required. In a colliding-beam experiment, the two protons have equal kinetic energies. To produce the antiproton, the total energy of the system must be at least equal to the total rest energy of the products (4mₚc²). Since the initial system consists of two protons, their combined rest energy is 2mₚc². The additional energy required to reach 4mₚc² must come from the kinetic energy of the protons. Therefore, the minimum kinetic energy of each proton is (4mₚc² - 2mₚc²) / 2 = mₚc².

Step 5: Conclude the solution. The minimum kinetic energy of each proton in the colliding-beam experiment is equal to the rest energy of a single proton, mₚc². This ensures that the total energy of the system is sufficient to produce the antiproton and satisfy energy conservation.

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Key Concepts

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

Energy-Mass Equivalence

Energy-mass equivalence, expressed by Einstein's equation E=mc², indicates that mass can be converted into energy and vice versa. In particle physics, this principle is crucial for understanding how kinetic energy from colliding particles can create new particles, such as antiprotons, during high-energy collisions.

Threshold Energy

Threshold energy is the minimum energy required for a reaction to occur. In the context of proton-proton collisions, it refers to the kinetic energy needed for the protons to produce an antiproton. This energy must account for the rest mass energy of the produced particles, ensuring that the collision has sufficient energy to create the additional mass.

Conservation of Energy and Momentum

The conservation of energy and momentum principles state that in an isolated system, the total energy and momentum before a reaction must equal the total energy and momentum after the reaction. In the case of proton collisions, this means that the kinetic energy of the colliding protons must be sufficient to not only create the antiproton but also conserve the overall momentum of the system.

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