A neutron is an electrically neutral subatomic particle with a mass just slightly greater than that of a proton. A free neutron is radioactive and decays after a few minutes into other subatomic particles. In one experiment, a neutron at rest was observed to decay into a proton (mass 1.67×10−²⁷ kg) and an electron (mass 9.11×10−³¹ kg) . The proton and electron were shot out back-to-back. The proton speed was measured to be 1.0 ×10⁵ m/s and the electron speed was 3.0×10⁷ m/s . No other decay products were detected. c. How much momentum did this neutrino 'carry away' with it?

Video transcript

Hey everyone. So this problem is dealing with conservation of momentum. Let's see what it's asking us. The disintegration of a carbon 14 into nitrogen 14 is an example of beta minus decay. Consider the carbon 14 decay where carbon decays into nitrogen plus an electron plus an anti neutrino. Initially, carbon 14 is at rest the nitrogen 14. And electron speeds in the laboratory frame are measured to be four times 10 to the second meters per second and one times 10 to the seven m per second respectively. Nitrogen 14 and the electron move in opposite directions. The mass of nitrogen 14 is 2.32 times to the negative 26 kg. And the mass of the electron is 9.11 times 10 to the negative kg. Calculate the momentum of the produced anti neutrino. Our multiple choice answers here are a 1.7 times to the negative 25 kg meters per second squared in the same direction as the electron b 1.7 times 10 to the negative 25 kg multiplied by meters per second squared in the same direction as the nitrogen nucleus. See 9.3 times 10 to the negative 24 kg meters per second squared in the same direction as the electron or D 9.3 times 10 to the negative kg meters per second squared in the same direction as the nitrogen nucleus. Mm OK. So there is a lot of information in this problem, but it actually comes down to a pretty simple conservation of momentum question. So we can recall that the conservation of momentum states that the initial momentum or P I is equal to the final momentum. And that has in the initial momentum of all of the objects in the system is equal to the final momentum of all of the objects in the system. And we can also recall that momentum is given by mass multiplied by velocity. So we can write out this P I equals P F equation in terms of the various objects or in this case, particles, our initial, we're working with our carbon. Um So we have M CV C is equal to. And then after the decay, we have our nitrogen atom. So M nitrogen V nitrogen, we have our electron so massively electron multiplied by velocity electron. And then we have our anti neutrino mass of the anti neutrino and speed of the anti neutrino. Now, they're a actually asking for the momentum of the anti nerina. So we can simplify this last term back into just P sub A for momentum of the anti treatment, the initial speed of the carbon atom is zero. It's at rest prior to the decay. So this full term goes to zero. And then we are left with the momentum of the anti neutrino is equal to negative mass of the neutrino or sorry mass of the nitrogen multiplied by velocity of the nitrogen minus mass of the electron multiplied by the state of the electron. We can look at from our problem all of the values here and ensure that we have everything we need to solve for our momentum. The mass of the nitrogen at nitrogen 14 is given as 2.32 times times the negative 26 kg. Our speed of our nitrogen Adam is given as four times 10 to the two m per second. Our mass of our electron is given as 9.11 times 10, the negative 31 kg. And then the speed of the electron is given as one times 10 to the seven m per second. We need to recall that that is in the opposite direction as the nitrogen atom. So that's going to be negative one times 10 to 7 meters per second. And so now we have, we do have everything we need to solve for our. So we have momentum that's equal to negative 2.32 times 10 to the negative kg multiplied by four times 10 to the second meters per second. And then this um velocity piece of E term, it's negative. So negative and this minus term makes a makes a positive or a plus. So I say plus mass of the electron 9.11 times 10 to the negative 31 kg multiplied by one times 10 to the seven m per second. And so when we plug all of that into our calculator, we get negative 1.7 times 10 to the negative kilogram meters per second squared. And because it is negative, that means that it is moving in the same direction as the electron as we can recall that that electron uh velocity was negative. And so when we look at our multiple choice of answers that aligns with answer choice A, so a is the correct answer for this problem. And that's all we have for this one. We'll see you in the next video.