Final answer:
The fraction of the total energy released that goes into the kinetic energy of the proton during neutron decay is approximately 0.054% of the total kinetic energy.
Step-by-step explanation:
To determine what fraction of the total energy released goes into the kinetic energy of the proton when a neutron decays, we must apply the principles of conservation of momentum and energy, considering that a neutron decays into a proton and an electron.
Since the neutron is initially at rest, the momentum of the system is zero before decay. After decay, the momentum of the proton and the electron must also cancel each other out because of the conservation of momentum. Let's denote the kinetic energy of the proton as Kp and the kinetic energy of the electron as Ke. The mass of the proton is given as 1836 times the mass of an electron, so:
1836 × me × vp = me × ve
This states that the momentum of the proton is 1836 times less than that of the electron assuming they have the same velocity, but since their velocities are different, it also means that:
vp = (1/1836) × ve
Using the relation for kinetic energy, K = (1/2) × m × v^2, the kinetic energy of the proton can be expressed in terms of the electron's kinetic energy:
Kp = (1/2) × (1836 × me) × ((1/1836) × ve)^2
This simplifies to:
Kp = (1/2) × me × ve^2 × (1/1836)
which indicates that:
Kp = Ke / 1836
Therefore, the fraction of total kinetic energy KEtot that goes into the proton's kinetic energy is:
Kp / Ke = 1 / (1 + 1836) = 1 / 1837
This means that Kp is approximately 0.054% of the total kinetic energy, which is the correct answer when expressed as a percentage.