Calculate the maximum kinetic energy of the electron when a mu...

Question

Calculate the maximum kinetic energy of the electron when a muon decays from rest via μ⁻ ⟶ e⁻ + vₑ + vμ. [Hint: In what direction do the two neutrinos move relative to the electron in order to give the electron the maximum kinetic energy? Both energy and momentum are conserved; use relativistic formulas.]

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with special relativity. This problem says during a cosmic event, a stationary pylon decays into a muon and a muon neutrino, calculate the maximum kinetic energy of the muon. And we're given the rest mass for both pyon and the muon. For the pylon. It is 135.6 mega electron volts per C squared. And for the muon, it is 105.7 mega electron volts per C squared. We're also given four possible choices as our answers. For choice A we have 50 mega electron volts. For choice B, we have 34 mega electron volts for choice C, we have 21 mega electron volts. And for choice D we have 4.1 mega electron volts. Now, the question asks us to calculate the maximum kinetic energy of the muon. And to do this, we're gonna use our concept of energy conservation. So for energy conservation, we're gonna need the total energy of the pion which we'll call E pi and that's gonna be equal to the total energy of the muon, which will be E mu. Thus the total energy of the muon neutrino, which we'll call e new. Now, since a neutrino is basically a massless particle, we can write its total energy in terms of its momentum. To recall for a massless particle, we have e new is equal to P new multiplied by C where P new is the momentum of the neutrino. However, since we have a stationary pion, that means the initial momentum is zero. So we can also use conservation momentum to say that the magnitude of the momentum or the neutrino has to be equal to the magnitude of the momentum of the U one. So they'll have the same value for the momentum, they'll just be going in opposite directions. And so we can replace RPU in this formula with pu. And so that means that he knew is going to be equal to P mu, which is the momentum of the muon multiplied by C. We can also write the total energy of the muon in terms of its momentum. And to recall from our energy momentum relationship or our relativistic version of that anyways, we'll have emu is equal to the square root of the quantity of P mu multiplied by C in quantity squared plus the quantity of M mu multiplied by C squared in quantity squared. Where pu here is our momentum for the muon and mu is the mass of the muon and C is the speed of light vacuum. So what we can do here is use this expression and plug the these back in and actually solve with the momentum of the muon and then use that momentum to calculate the energy. So coming back to our energy conservation equation, we have for the total energy of the pion since is at rest, it has no connect energy. So it just has its rest energy which is gonna be its mass, which we'll call M pi multiplied by C squared. And that's gonna be equal to or emu will play in our expression from the uh energy momentum relationship. That's gonna be the square root of the quantity of P mu multiplied by C in quantity squared plus the quantity of M mu multiplied by C squared in quantity squared. And then we'll add to that the energy of the neutrino, which is just gonna be plus mu sorry PM multiplied by C. So at this point, I need to um actually solve this for our momentum of the muon. And so the first step I'm going to do to solve that is I'm gonna subtract P mu multiplied by C from both sides and then square both sides. And so when I do that, I'll get the quantity of MPI multiplied by C squared in quantity squared minus two pu multiplied by M pi multiplied by C cubed. They'll have plus the quantity of P mu multiplied by C in quantity squared. And that's gonna be equal to the quantity of P mu multiplied by C in quantity squared plus the quantity of M mu multiplied by C squared in quantity squared. So the quantity of P mu multiplied by C and quantity squared is going to subtract out from both sides since I have that on both sides of the equation. So it'll cancel out. And then I just need to solve this equation for pu. And so to do that, I'm gonna subtract the quantity of M pi multiplied by C squared and quantity squared from both sides and then divide both sides by the quantity of minus two multiplied by M pi multiplied by C cubed. And so when I do that, I'll get team U is equal to the quantity of M pi multiplied by C squared in quantity squared minus the quantity of mu multiplied by squared in quantity squared divided by the quantity of two MPI multiplied by C cubed. Now we were given the masses for both the pion and the muon. So we can actually plug those in and calculate a value for our momentum. So we'll have Pu is equal to the quantity for M pi the mass of the pion that is going to be the 139.6 mega electron volts per C squared. This could be multiplied by C squared and that whole quantity is going to be squared. Don't have minus the quantity for the mass of the muon. That is the 105.7 mega electron volts per C squared. And that quantity is multiplied by C squared. And then that whole quantity is going to be squared. And that's gonna be divided by the quantity of two multiplied by the mass of the pion again, which is 139.6 mega electron volts or C squared. And then that's gonna be multiplied by C cubed. So the reason why we didn't plug into value for C is that they're going to cancel in the numerator, we'll have the C squares canceling. And in the denominator, I'll have the C squared canceling out with two powers of the C cubed, leaving just a single power of C. So when I plug those values into my calculator, I'll get P me is equal to 29.78 mega electron volts per C. So now what I can do is use this to calculate my kinetic energy. And so recall that your kinetic energy which we call kmu is gonna be equal to our total energy, which is emu minus the rest energy, which is mu multiplied by C squared. Now, we have an expression for the emu from our energy and uh momentum relationship. And so we can actually plug that in and we have values for everything else. So we'll have KU is equal to emu. That's gonna be the square root of the quantity of puc. So for pu, we just calculated that that's gonna be the 29.78 mega electron volts per C and then that is gonna be multiplied by C and then that whole quantity is gonna be squared, then we'll have plus the quantity of the mass of the muon, which is 105.7 mega electron volts or C squared. That's could be multiplied by C squared. And that whole quantity is going to be squared. And then I'm gonna subtract from that square root term, the rest energy for the muon, which is just going to be it's mass or its rest mass, which is 105.7 mega electron volts per C squared. And that's gonna be multiplied by C squared. So we can plug all of those values into our calculator and we'll get our connect energy ku it's going to be equal to 4.1 mega electron volts. So that's the answer that I'm looking for. And if I look at my answer choices, this corresponds to answer choice D. So just to recap of what we've done in this problem, we started off by using our concept of conservation of energy where we set the total energy of the Pyon equal to the total energy of the muon plus the total energy of the neutrino. We then wrote the energy total energy of the muon in terms of its momentum. And we also wrote the total energy of the neutrino in terms of its momentum. We then used the concept of conservation momentum since our pyon initially had zero momentum, that meant that the magnitude of the momentum for the neutrino and the muon had to be equal. And so that meant that we could write an equation that had just um the momentum of the muon in it. We could then solve that equation for a value for the momentum for the muon and then use that value for the momentum to calculate the kinetic energy. So I hope that this has been useful and I'll see you in the next video.

Key Concepts

Conservation of Energy

Conservation of Momentum

Relativistic Kinetic Energy

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