In this problem, you will calculate the difference in the time of arrival of two neutrinos of different energy from a supernova that is 170,000 light-years away. Let the energies of the neutrinos be E₁ = 20 MeV and E₂ = 5 MeV, and assume that the rest mass of a neutrino is 20 eV/c². Because their total energy is so much greater than their rest energy, the neutrinos have speeds that are very nearly equal to c and energies that are approximately E ≈ pc.

(a) If t₁ and t₂ are the times it takes for neutrinos of speeds u₁ and u₂ to travel a distance x, show that

(b) The speed of a neutrino of rest mass m₀ and total energy E can be found from Equation 39-25. Show that when E >> m₀c², the speed u is given approximately by

(c) Use the results for (b) to calculate u₁ – u₂ for the energies and rest mass given, and calculate Δt from the result for (a) for x = 170,000c·y.

(d) Repeat the calculation in (c) using m₀c² = 40 eV for the rest energy of a neutrino.

Zn - 'n En'n C2 At = t, -, = x-