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 of the newly formed (daughter) nucleus, we can use the principle of conservation of momentum. According to this principle, the total momentum before the decay should be equal to the total momentum after the decay.
Before the decay, the radioactive nucleus is at rest, so its momentum is zero. The momentum of the emitted electron is given as 8.11 x 10^-19 kg·m/s and it is directed along the positive x-axis. The momentum of the neutrino is given as 5.08 x 10^-19 kg·m/s.
Since the electron momentum is only in the x-direction and the neutrino's momentum is perpendicular to it, we can treat their momenta as two-dimensional vectors in the x-y plane.
Let's denote the momentum of the daughter nucleus as P_nucleus and its direction angle (measured counterclockwise from the positive x-axis) as θ.
Using vector addition, the total momentum after the decay can be calculated as:
P_total = P_electron + P_neutrino + P_nucleus
Since the neutrino's momentum is perpendicular to the electron's momentum, we can represent it as:
P_neutrino = (0, P_neutrino)
The momentum of the electron can be represented as:
P_electron = (P_electron, 0)
The total momentum after the decay in vector form is:
P_total = (P_electron, 0) + (0, P_neutrino) + (P_nucleus * cosθ, P_nucleus * sinθ)
According to the conservation of momentum, the total momentum before the decay is zero, which means the total momentum after the decay should also be zero:
P_total = (0, 0)
Since the x-components of the momenta cancel out, we can equate the y-components to zero:
0 = P_neutrino + P_nucleus * sinθ
We are given the magnitudes of P_electron and P_neutrino. Now, we need to solve for the magnitude of P_nucleus.
Using the Pythagorean theorem, we can write:
(P_nucleus * cosθ)^2 + (P_nucleus * sinθ)^2 = P_nucleus^2
Simplifying, we get:
P_nucleus^2 (cos^2θ + sin^2θ) = P_nucleus^2
Therefore, cos^2θ + sin^2θ = 1.
Substituting this into the equation 0 = P_neutrino + P_nucleus * sinθ, we get:
0 = P_neutrino + P_nucleus
Now we can solve for the magnitude of P_nucleus:
P_nucleus = -P_neutrino
Since the neutrino's momentum is directed opposite to the x-axis (negative x-direction), the direction angle θ for the nucleus recoil is:
θ = 180 degrees or π radians
Therefore, the newly formed nucleus recoils in the direction opposite to the x-axis.
To summarize:
Momentum of the newly formed nucleus (P_nucleus) = -5.08 x 10^-19 kg·m/s
Direction of nucleus recoil = Opposite to the positive x-axis (180 degrees or π radians)