Assume that when a free neutron decays, it transforms into a proton and an electron. Calculate the kinetic energy of the electron.
OpenStax College Physics for AP® Courses, Chapter 33, Problem 14 (Test Prep for AP® Courses)
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This is College Physics Answers with Shaun Dychko. Suppose a neutron at rest decays into a proton and an electron, the question is what is the kinetic energy of the electron? The total kinetic energy of the two particles combined will be the mass defect, which is the difference in mass between the reactant and the products times c squared. So the mass of the neutron is 939.57 megaelectron volts per c squared and we subtract from that the total mass of the electron and the proton and all this is times c squared and we have 0.789 megaelectron volts is the total kinetic energy of the two particles. Now the initial momentum is 0 of this single neutron that's at rest so that means the total momentum of these two particles must also be zero which is to say that the momentum of one of them is going to be an equal magnitude to the momentum of the other and they will go in opposite directions so that's what this says here. And because this total energy is small compared to the rest energy of the proton and you know, it's only 50 percent more than the energy of the electron we can use non-relativistic formulas here. So we have mass of the proton times speed of the proton equals mass of the electron times speed of the electron in other words and that's the momentum conservation. Now the total kinetic energy that we calculated is going to be the energy of the proton plus the kinetic energy of the electron. And from the momentum formula, we can solve for the speed of the proton by dividing both sides by mass of the proton and we get speed of the proton then is mass of the electron times speed of the electron divided by mass of the proton and we can substitute that in for v p here and so we have one-half mass of the proton times this substitution and then squaring this bracket and we end up with this line here. Now what we want to know is the kinetic energy of the electron and so we are going to factor out one-half mass of an electron times speed of the electron squared from both of these terms and we end up with this line here. So this is one-half mass of an electron speed of an electron squared times mass of an electron divided by mass of the proton plus 1 so each of these terms was just divided by one-half m ev e squared and we end up with this. And then we divide both sides by this bracket and we end up with this line here and so this is the kinetic energy of the electron equals the total kinetic energy divided by the mass of the electron divided by mass of the proton plus 1 and this works out to 0.789 megaelectron volts and so it's pretty much the total kinetic energy is the kinetic energy of the electron and that wasn't a surprise because given that the momenta needs to be the same, the mass of the electron is so much less than the mass of the proton there needs to be a compensating much greater speed for the electron and therefore given that kinetic energy depends on speed squared, the kinetic energy of the electron will be much greater than that of the proton.