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California State Polytechnic University, Pomona

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Problem 93 Easy Difficulty

A neutron at rest decays (breaks up) to a proton and an electron. Energy is released in the decay and appears as kinetic energy of the proton and electron. The mass of a proton is 1836 times the mass of an electron. What fraction of the total energy released goes into the kinetic energy of the proton?

Answer

$0.0544 \%$

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Video Transcript

{'transcript': "once again welcome to a new problem This time we're given, it's almost like chemist, too, where we have a neutron that breaks up into a proton and an electron. And it's just so happens that the muss off the pro tem eyes equivalent to one thousand eight hundred and thirty six times the mass ofthe the election. So this is the mass of the electron. And so you know, there's these energy released in the process because so these energy released. So we want to know. Um, you know what? What fraction? What fraction? A fraction off the total kinetic energy is the kinetic energy off the pro tem. So in relative times, obviously you're seeing that. You know, the mass of the proton is one thousand eight hundred thirty six times the mass of the neutral, and that affects thie kinetic energy. So, you know, the fast then we're going to do is to use the law of conservation of momentum. We'Ll use the love conservation Ah, off momentum. That says the initial momentum off the system equals to the final momentum off the system. Initially, you only have the neutron like that. And then this is like before and then after you have the proton and the electron this is what happens after. So after these motion before there's no motion And so the initial momentum is zero The initial momentum is zero In the final momentum is going to be the you know, the velocity a CZ they're moving You could see that the proton is gonna have its own velocity and then the electron will also have its own velocity. The proton has it's sorry that he had the project will have its velocity the election who have its velocity Remember, the election is negative so we'LL have zero equals to, you know, the mass of the election, the momentum The momentum of the electron is negative but the momentum off the proton iss positive Remember, the electron has negative charge and the part in its positive charge And so this will help us solve for the will of the velocity of the electrons are empty and V, This is pure algebra divide both sides by the mass of the electron. So we see that the velocity of the electron is the ratio off their masses. That's why this numbers very helpful. The racial off their mass is times the velocity of the project. You know, it kind of like makes sense. If one is huge than it's probably going to move slower. And then if one is small, it's gonna move faster. Relative the last is in the next page member. Our target is to get what fraction of the total energy total Kinetic energy off the project is what fraction of the kinetic energy of the Parton is is is ah is you know what fraction of the total energy Sorry it goes to the conn. The pardon. So on this side, we kind of have the total energy. And then on this side, we have the total kinetic energy we have the kinetic energy off the potent plus the kinetic energy off the electron. So that's what's gonna happen in the next page. We have the total kinetic energy off the system is split. This is kind of like what's in the neutron, and it split into the kinetic energy off the Pro Tem. This is the product, plus the kinetic energy off Electra. Um, this is the electric. And so the the total kinetic energy becomes the same as one half Marcel's Proton times velocity ofthe proton squared plus one half Marcel's electron times velocity of Election Square. That's what we have remember previously were able to get the velocity of the electron, and we can do the replacement and say this is equal to one half mast ofthe proton velocity of Porton squared plus one half the mass of the electron is the ratio's off the mass of the proton and the electron times the velocity off the pro tem. And this helps because now everything is in terms ofthe pretty much in terms ofthe problems with with the exception off the mass of the election. But that's good, because on a relative terms, we know what the mass of the proton is relative to the election. Okay, so you know, it means that we could still solve our problem. So going back, we simplify this and say, Ah, one Huff, Marcel's Porton. Remember this one? This velocity now this, um, velocity of the electrons. So let's go back. Velocity of the election is this one. So we have to square that and then we have the mass of the election there. So we have one must velocity of Parton squared plus one one half muscled electron times Ah, muss off mass off the muscle the prata mb squared with that over the mass of the electron squared. Okay. And then velocity of Porton squared like that. This one also is squared. So this will cancel one of these. And we're left with one half Marcel's Porton velocity of Porton squared. And then one Los Marce off Rotten off the mass of the electron. So this is just pure algebra on DH. So in the next page, we notice something special. When you see this, this one is the kinetic energy off the project. So we're gonna use that in the next page. And so the total kinetic energy off the system equals to the kinetic energy off the Pro Tem. You know, which is this part? And then times one plus months off Parton of a muscle electron. Um and then we can We can divide both sides by the kinetic energy of the project. Look, a united Kennedy off the Porton and what we wanted to find out in relative terms, You know, we wanted to find out in relative times. What's thie? No. What's the kinetic energy off the off the proton relative to the total. So we have total over kinetic energy off the Parton like that equals to these to cancel out Ah, we can put a common denominator. Their mass of electrons are multiplying both sides by me of m e. So this is mass of electron plus mass off the Parton, and then we're gonna flip it because we want the fractions in Connecticut. Ledge off the Parton over the total Connecticut energy becomes the mass of the electron over the muscle, the election plus the moss off the porton, which is the same as if you go back here. We're interested in the ER in the totals. So the mass of the proton is eighteen thirty six the mass of the election. So we'll have mass of the electron in off the muscles. The election plus eighteen thirty six months of the election simplify that we get muscles the electron over. If you this is like having a one there so eighteen, thirty seven mass off the election, these to cancel out and then it means that finally we can see that the kinetic energy off the porton relative to the total kinetic energy of the system is one out off, one thousand eight hundred and thirty seven. If you multiply that by hundred percent, we see that the fraction zero point zero five four percent hope you enjoy the problem. You know, if you have any questions, send them my way. The deal here is to remember the fractional relationship off the masses and then to use the law of conservation of momentum before nothing happened. Afterwards, there was motion that helped us to solve for the final velocity of the electric on DH. Then we plugged in those values to look at the fractional relationship so used the kinetic energy. The total kinetic energy is the kinetic energy of the port on and the electron. So using that in algebra were able to get the the fractional relationships. OK, eso if you have any questions, send them my way and have a wonderful day"}