Question

37.52•• Quarks and gluons are fundamental particles that will be discussed in Chapter 44. A proton, which is a bound state of two up quarks 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 the up quarks, which is m_u=4.12×10^−30 kg each, and the rest mass of the 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 derived from the kinetic energy of the three quarks, and we split that energy equally among them. (a) Estimate the Lorentz factor γ=(1−v^2/c^2)^-1/2 for each of the up quarks using KE=(mc^2)/sqrt(1-v^2/c^2) - mc^2 = (γ-1)mc^2. (b) Similarly estimate the Lorentz factor γ for the down quark. (c) Are the corresponding speeds V_u and V_d greater than 99% of the speed of light? (d) More realistically, the quarks are held together by massless gluons, which mediate the strong nuclear interaction. Suppose we model the proton as the three quarks, each with a speed of 0.90c, with the remainder of the proton rest energy supplied by gluons. In this case, estimate the percentage of the proton rest energy associated with gluons. (e) Model a quark as oscillating with an average speed of 0.90c across the diameter of a proton, 1.7×10^−15 m. Estimate the frequency of that motion.

Answer 1

a) The rest energy of proton is

Kinetic energy of up quark is 1/3 of this

Substitue the values and solve to obtain

or

velocity of up quark is

b) Kinetic energy of down qaurk is

this gives

or

this gives

c) these speeds are clearly above 99% of speed of light, c.

d) The gamma factor of up quark moving with 0.9 c is

the kinetic energy of up quark is

the kinetic energy of down quark is

The total kinetic energy is

This as a Percentage of proton rest mass energy is

e) Frequency of oscillation of quark is