## The $β$-decay process, discovered around 1900, is basically the decay of a neutron $(n)$. In the laboratory, a proton $(p)$ and an electron $(e_{−})$ are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process, i.e. $n→p+e_{−}+v_{e}$, around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino $(v_{e})$ to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is $0.8×10_{6}eV$. The kinetic energy carried by the proton is only the recoil energy.

If the anti-neutrino had a mass of 3 $eV/c_{2}$ (where $c$ is the speed of light) instead of zero mass, what should be the range of the kinetic energy, $K$, of the electron?