A radioactive nucleus is at rest when it spontaneously decays by emitting an electron and neutrino (as shown in the figure below ). The momentum of the electron is 8.11 x 10−19 kg·m/s and it is directed at right angles to that of the neutrino. The neutrino's momentum has magnitude 5.08 x 10−19 kg·m/s. In what direction does the newly formed (daughter) nucleus recoil? Let the electron direction be along the positive x-axis and find the direction of nucleus recoil with respect to the electron's direction.
What is its momentum?
To find the momentum and direction of the newly formed (daughter) nucleus, we need to use the principle of conservation of momentum.
According to this principle, the total momentum before the decay is equal to the total momentum after the decay.
In this case, we know the momentum of the electron and the neutrino before the decay. The momentum of the electron is given as 8.11 x 10^(-19) kg·m/s along the positive x-axis, and the momentum of the neutrino is given as 5.08 x 10^(-19) kg·m/s.
Since momentum is a vector quantity, we can use vector addition to find the total momentum before the decay. Since the electron and the neutrino are traveling at right angles to each other, the total momentum before the decay is equal to the vector sum of their individual momenta. This can be calculated using the Pythagorean theorem:
Total momentum before decay = √(momentum of electron)^2 + (momentum of neutrino)^2
Total momentum before decay = √((8.11 x 10^(-19))^2 + (5.08 x 10^(-19))^2) kg·m/s
Total momentum before decay ≈ 9.60 x 10^(-19) kg·m/s
Now that we know the total momentum before the decay, according to the conservation of momentum principle, the total momentum after the decay must also be equal to 9.60 x 10^(-19) kg·m/s.
Since the electron has been emitted, the recoil momentum of the daughter nucleus must be equal in magnitude and opposite in direction to the total momentum before the decay. Therefore, the momentum of the daughter nucleus is also 9.60 x 10^(-19) kg·m/s, but in the opposite direction to that of the electron.
To find the direction of the daughter nucleus recoil with respect to the electron's direction, we can consider the x-axis as a reference. Since the electron is traveling along the positive x-axis, the daughter nucleus will be traveling in the opposite direction, i.e., along the negative x-axis.
So, the momentum of the newly formed (daughter) nucleus is 9.60 x 10^(-19) kg·m/s in the negative direction of the x-axis.