A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of 8.55×10−23 and 5.33×10−23 , respectively. What is the magnitude of the momentum of the second (recoiling) nucleus?
To solve this problem, we can use the conservation of momentum. Since the initial nucleus is at rest, the total initial momentum is zero. Therefore, the total momentum after the decay must also be zero.
Let's assume the mass of the second (recoiling) nucleus is M and its momentum is p. The momentum of the electron is given as 8.55×10^−23, and the momentum of the neutrino is given as 5.33×10^−23.
According to conservation of momentum, we have:
0 = p + 8.55×10^−23 + 5.33×10^−23.
To find the magnitude of the momentum of the second nucleus, we need to find the value of p.
Adding the momenta on the right-hand side, we get:
0 = p + (8.55+5.33)×10^−23
0 = p + 13.88×10^−23
Simplifying, we have:
p = -13.88×10^−23.
The magnitude of the momentum of the second nucleus can be found by taking the absolute value of p:
|p| = |-13.88×10^−23|
|p| = 13.88×10^−23.
Therefore, the magnitude of the momentum of the second (recoiling) nucleus is 13.88×10^−23.
To find the magnitude of the momentum of the second (recoiling) nucleus, we can use the principle of conservation of momentum. According to this principle, the total momentum before the decay must be equal to the total momentum after the decay.
In this case, we have three particles with known momenta: the electron, the neutrino, and the recoiling nucleus.
Let's denote the momentum of the recoiling nucleus as P_nucleus, the momentum of the electron as P_electron, and the momentum of the neutrino as P_neutrino.
Since the electron and neutrino are emitted at right angles, we can use the Pythagorean theorem to find the magnitude of the total momentum before the decay:
P_total(before) = √(P_electron^2 + P_neutrino^2)
P_total(before) = √((8.55×10^-23)^2 + (5.33×10^-23)^2)
P_total(before) = √(7.3125×10^-46 + 2.8489×10^-46)
P_total(before) = √(10.1614×10^-46)
P_total(before) = √(1.01614×10^-45)
P_total(before) ≈ 3.186×10^-23 kg·m/s
According to the conservation of momentum, the total momentum after the decay must also be equal to P_total(before). Since the recoiling nucleus is the only unknown momentum, we can write:
P_total(after) = P_nucleus
P_total(after) = P_total(before) = 3.186×10^-23 kg·m/s
Therefore, the magnitude of the momentum of the second (recoiling) nucleus is approximately 3.186×10^-23 kg·m/s.