Answer
(a) In beta decay, the daughter nucleus is usually not stationary, but if it is, the conservation of momentum requires that the beta particle and the neutrino are emitted in opposite directions. This is because the total momentum of the system before and after the decay must be conserved. Since the daughter nucleus is stationary, it has zero momentum before the decay, so the beta particle and the neutrino must have equal but opposite momenta to conserve momentum.
(b) In a decay with a Q value of 1.5 MeV, if the beta particle is emitted with very little kinetic energy, most of the energy must be carried away by the neutrino. The reason for this is the conservation of energy and momentum in the decay process. The daughter nucleus recoils in the opposite direction to the beta particle, carrying away some of the energy and momentum. However, since the beta particle has very little kinetic energy, its momentum is small, and most of the momentum of the system is carried away by the neutrino.
To understand this, consider the momentum conservation equation: p_daughter = p_beta + p_neutrino, where p_daughter, p_beta, and p_neutrino are the momenta of the daughter nucleus, the beta particle, and the neutrino, respectively. Since the daughter nucleus is initially at rest, its momentum is zero, so p_beta + p_neutrino must also be zero. However, the neutrino has a very small mass compared to the beta particle, so it carries most of the momentum. As a result, the daughter nucleus recoils with a very small momentum, while the neutrino carries away most of the energy and momentum of the system.
In summary, if the beta particle is emitted with very little kinetic energy in a decay with a Q value of 1.5 MeV, most of the energy is carried away by the neutrino, and the daughter nucleus recoils with a very small momentum.
Work Step by Step
(a) In beta decay, the daughter nucleus is usually not stationary, but if it is, the conservation of momentum requires that the beta particle and the neutrino are emitted in opposite directions. This is because the total momentum of the system before and after the decay must be conserved. Since the daughter nucleus is stationary, it has zero momentum before the decay, so the beta particle and the neutrino must have equal but opposite momenta to conserve momentum.
(b) In a decay with a Q value of 1.5 MeV, if the beta particle is emitted with very little kinetic energy, most of the energy must be carried away by the neutrino. The reason for this is the conservation of energy and momentum in the decay process. The daughter nucleus recoils in the opposite direction to the beta particle, carrying away some of the energy and momentum. However, since the beta particle has very little kinetic energy, its momentum is small, and most of the momentum of the system is carried away by the neutrino.
To understand this, consider the momentum conservation equation: p_daughter = p_beta + p_neutrino, where p_daughter, p_beta, and p_neutrino are the momenta of the daughter nucleus, the beta particle, and the neutrino, respectively. Since the daughter nucleus is initially at rest, its momentum is zero, so p_beta + p_neutrino must also be zero. However, the neutrino has a very small mass compared to the beta particle, so it carries most of the momentum. As a result, the daughter nucleus recoils with a very small momentum, while the neutrino carries away most of the energy and momentum of the system.
In summary, if the beta particle is emitted with very little kinetic energy in a decay with a Q value of 1.5 MeV, most of the energy is carried away by the neutrino, and the daughter nucleus recoils with a very small momentum.
