37.52•• Quarks and gluons are fundamental particles that will bediscussed in Chapter 44. A proton, which is a bound state of two upquarks and a down quark, has a rest mass of m_p=1.67×10^−27 kg.This is significantly greater than the sum of the rest mass of theup quarks, which is m_u=4.12×10^−30 kg each, and the rest mass ofthe down quark, which is m_d=8.59×10^−30 kg. Suppose we(incorrectly) model the rest energy of the proton m_pc^2 as derivedfrom the kinetic energy of the three quarks, and we split thatenergy equally among them. (a) Estimate the Lorentz factorγ=(1−v^2/c^2)^-1/2 for each of the up quarks usingKE=(mc^2)/sqrt(1-v^2/c^2) - mc^2 = (γ-1)mc^2. (b) Similarlyestimate the Lorentz factor γ for the down quark. (c) Are thecorresponding speeds V_u and V_d greater than 99% of the speed oflight? (d) More realistically, the quarks are held together bymassless gluons, which mediate the strong nuclear interaction.Suppose we model the proton as the three quarks, each with a speedof 0.90c, with the remainder of the proton rest energy supplied bygluons. In this case, estimate the percentage of the proton restenergy associated with gluons. (e) Model a quark as oscillatingwith an average speed of 0.90c across the diameter of a proton,1.7×10^−15 m. Estimate the frequency of that motion.