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 and its direction, we can use the principle of conservation of momentum. According to this principle, the total momentum before the decay (which is zero since the nucleus is at rest) should be equal to the total momentum after the decay.
Let's break down the given information:
- The momentum of the electron is 8.11 x 10^(-19) kg·m/s in the positive x-axis direction.
- The momentum of the neutrino has a magnitude of 5.08 x 10^(-19) kg·m/s.
Since the electron and the neutrino are emitted at right angles to each other, we can use vector addition to find the total momentum. The magnitude of the total momentum will be the sum of the magnitudes of individual momenta, and the direction will be the direction of the vector sum.
Given:
Magnitude of electron momentum, P(e) = 8.11 x 10^(-19) kg·m/s
Magnitude of neutrino momentum, P(ν) = 5.08 x 10^(-19) kg·m/s
To find the magnitude of the total momentum:
P(total) = √(P(e)^2 + P(ν)^2)
P(total) = √((8.11 x 10^(-19))^2 + (5.08 x 10^(-19))^2)
P(total) = √(65.92 x 10^(-38) + 25.81 x 10^(-38))
P(total) = √(91.73 x 10^(-38))
P(total) ≈ 9.58 x 10^(-19) kg·m/s
Now, to determine the direction of the newly formed nucleus recoil with respect to the electron's direction, we can use trigonometry. Since the electron is along the positive x-axis, the direction of the recoil will be the angle with respect to the positive x-axis.
tan(θ) = (P(ν) / P(e))
tan(θ) = (5.08 x 10^(-19) / 8.11 x 10^(-19))
tan(θ) ≈ 0.626
To find the angle, θ:
θ ≈ tan^(-1)(0.626)
θ ≈ 31.30 degrees
Therefore, the momentum of the newly formed nucleus is approximately 9.58 x 10^(-19) kg·m/s, and its direction is at an angle of about 31.30 degrees with respect to the positive x-axis.