Question

A K⁺-meson (with a rest mass of 493.7 MeV/c2) initially at rest decays into a μ⁺ (positive muon with rest mass of 105.7 MeV/c2), Two photons and a neutrino (assume exactly zero rest mass - not quite true). The Muon is observed to move along the x-axis with momentum 100MeV/c. The two photons are observed to have identical energies but opposite directions along the positive and negative y-axes. The neutrino (not shown) is unseen. Find the energies of the photons.Hint: momentum is a vector quantity!

Answer 1

given
K+ meson, mo = 493.7 MeV/c^2
initially at rest
hence initial momentum = 0

decays into
mu+ muon, m'o = 105.7 MeV/c^2
two photons, and neutrino are released too, assume 0 rest mass

momentum of muon along +x axis, Pm = 100 MeV/c
energy of each photon = E
momentum of each photon = -P, P (opposite direction)
energy of neutrino = E'
momentum on neutrino = P'

now, from consevation of energy

(mo*c^2) = sqrt((Pm*c)^2 + (m'o*c^2)^2) + 2E
=> 493.7 = sqrt(100^2 + 105.7^2) + 2E + E'
E + E'/2 = 174.0961513 MeV

from conservation of momentum
0 = Pm + P'
P' = -Pm = 100 MeV/c along -x axis

now, E' = Pm*c
hence E' = 100 MeV
hence
E = 124.0961513 MeV