A neutron initially at rest, decays into a proton, an electron, and an antineutrino. The ejected electron has a momentum of 1.4 10⁻²⁶ kg-m/s while the antineutrino has 6.4 10⁻²⁷ kg-m/s . Determine the recoil speed of the proton if the electron and the antineutrino are ejected along perpendicular directions. The mass of the proton is 1.67 10⁻²⁷ kg .

Step-by-step solution

By the principle of conservation of linear momentum, the total initial momentum of the neutron is zero. Thus, the vector sum of the momenta of the decay products must be zero. p _p + p _e + p _ = 0 p _p = -( p _e + p _ ) Since the electron and antineutrino are ejected along perpendicular directions, the magnitude of the proton's momentum is given by: p_p = p_e^2 + p_ ^2 Given: p_e = 1.4 10⁻²⁶ kg-m/s = 14 10⁻²⁷ kg-m/s p_ = 6.4 10⁻²⁷ kg-m/s Substituting the values: p_p = (14 10⁻²⁷)^2 + (6.4 10⁻²⁷)^2 p_p = 196 + 40.96