## Conceptual Physics (12th Edition)

The neutron and the alpha particle fly apart with equal and opposite momenta. You may write kinetic energy in terms of the momentum. $$KE = \frac{1}{2}mv^{2} = \frac{(mv)^{2}}{2m} = \frac{p^{2}}{2m}$$ For particles with the same momentum, KE is inversely proportional to mass. The alpha particle has four times the mass of the neutron, and thus has one-fourth the kinetic energy. Alternate solution: the neutron has $\frac{1}{4}$ the mass of the alpha particle, and equal momentum, so it has four times the speed. Compare the KE directly. For the neutron: $$KE = \frac{1}{2}m(4v)^{2} = 8 mv^{2}$$ For the alpha particle: $$KE = \frac{1}{2}4m(v)^{2} = 2 mv^{2}$$ The kinetic energies are in the ratio of 4:1, or 80:20. The neutron gets about 80 percent of the kinetic energy, and the alpha particle gets 20 percent.