Question

35.91 Two identical nuclei, each with rest mass \(50.30 \mathrm{GeV} / c^{2},\) are accelerated in a collider to a kinetic energy of \(503.01 \mathrm{GeV}\) and made to collide head on. If one of the two nuclei were instead kept at rest, what would the kinetic energy of the other nucleus have to be for the collision to achieve the same center-of-mass energy?

Short answer

Answer: The kinetic energy required for the other nucleus is 731.94 GeV.

Step-by-step solution

Calculate the relativistic total energy in the collider scenario

In the collider scenario, both nuclei have equal kinetic energies. To find the total energy, we need to add their rest mass energies and kinetic energy of each nucleus. Rest mass energy = rest mass × (\(c^2\)) = (50.30 GeV/c^2) × (c^2) = 50.30 GeV Kinetic energy of one nucleus in collider scenario = 503.01 GeV Total energy of one nucleus in collider scenario = Rest mass energy + Kinetic energy = 50.30 GeV + 503.01 GeV = 553.31 GeV Since the nuclei are identical, the other nucleus will also have the same total energy of 553.31 GeV.

Determine the center-of-mass energy

Center-of-mass energy in collider scenario is represented by the variable 𝑠. Using the energy-momentum conservation, we can represent 𝑠=(2𝐸𝑃)², where 𝐸 is the total energy and 𝑃 is the total momentum. Since both nuclei have equal and opposite momenta, their momenta in the center-of-mass frame cancel each other, as: 𝑃 = 0 Thus, the variable 𝑠 is equal to: 𝑠 = (2𝐸 × 0)² = 0 Now, we can calculate the center-of-mass energy: Center-of-mass energy \(\sqrt{𝑠}\) = \(\sqrt{2 × 553.31^2}\) = 783.31 GeV

Calculate the required kinetic energy

In the scenario where one nucleus is at rest, let's find the kinetic energy required for the other nucleus to achieve the same center-of-mass energy as the collider scenario (783.31 GeV). Let E1 be the total energy of the nucleus at rest and E2 be the total energy of the other nucleus. We have: E1 = Rest mass energy = 50.30 GeV Center-of-mass energy = \(\sqrt{E1^2 + E2^2}\) Now, substituting given values: 783.31 GeV = \(\sqrt{50.30^2 + E2^2}\) Squaring both sides: 783.31^2 = 50.30^2 + E2^2 Solving for E2: E2^2 = 783.31^2 - 50.30^2 E2 = \(\sqrt{783.31^2 - 50.30^2}\) = 782.24 GeV Now, to find the required kinetic energy for the other nucleus: Required kinetic energy = E2 - Rest mass energy = 782.24 GeV - 50.30 GeV = 731.94 GeV The kinetic energy required for the other nucleus to achieve the same center-of-mass energy with one of the nuclei at rest is 731.94 GeV.