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Problem 47 Hard Difficulty

A $\pi$ -meson at rest decays according to
$$
\pi^{-} \rightarrow \mu^{-}+\overline{\nu}_{\mu}
$$
What is the energy carried off by the neutrino? Assume the neutrino has no mass and moves off with the speed of light. Take $m_{\pi} c^{2}=139.6 \mathrm{MeV}$ and $m_{\mu} c^{2}=105.7 \mathrm{MeV.}$ Note: Use relativity; see Equation 26.13 .

Answer

the energy of the neutrino is 29.8 $\mathrm{MeV}$ .

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Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

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University of Washington

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Hope College

Video Transcript

this question, given that the high on it's at rest initially, anything case into emu on rivets. Anti neutrino. Not a key, would. Here is that the pilot is at rest my ready to case. So that's no kinetic energy whatsoever. And the momentum of the system is zero. And so, by conservation meant of the final momentum or the system must be zero as well, right? So first off, you're gonna use conservation of energy, tells us that the initial energy goes to the final energy system. Initial energy is just e rest energy off pion. So that is given it a question. The final energy NOPD and itchy off you one first the energy off. Oh, a neutrino. Right? And the energy involves both kinetic energy escrow as the rest mass energy. All right, So, Dennis, first keep this my then we move on to using the momentum Conservation of momentum. Right, because our initial momentum is zero. So final momentum spc Race well, and this implies that Mm magnitude off mentum off the mu one must be equal to the minute your momentum from the neutrino. Right? And they must be equal and opposite so that they will cancel each other out to gift the momentum of the system to be zero. Now I know any momentum off our neutrino with you is given s. Yeah, We'll see front. Since neutrino is assumed to be mess, Lis writes for the relativistic forming right astute relativistic equation. Yes, great. Goes to disgraces great plus scripts before. Since it is a mess lists the second term goes to zero. So you only have equals two PC and so p even went to Mexico's to be overseas. What? You're in for the momentum off our Mulan. This? Ah, it's a bit more complicated. Right? So we start off from this equation over here. So the energy or the new one square discourse to be we went toe to me one c square this I am square. See, for from here, we can actually substitute in. Yeah, momentum from the momentum equation into over here into the movement of Hue. Right? No way we can do is to 50 equation around a little bit. So I'm gonna bring the energy off the neutrino to the left in sight, kids. All right. And now we're gonna do a little bit off manipulation Oh, actually, we can simply substitute in, right? Since we no, from equation one that the energy off, the everyone can be written in terms off energy off the neutrino, right? No, We have to open up the briquettes then can self for this. So every right now we want to find the energy off the neutrino. So rearranging to make sure that the energy neutrino Christie subject. Right? So we are given also from the question that the restoration out everyone this year Yes, ones were 5.7 MTV. So you just have to square this and we can substitute that in. Sure. Have 40 failures. Step you need I'll find an answer. Should be straight night point. This is the energy that was carried away by the news, you know?

National University of Singapore
Top Physics 103 Educators
Andy C.

University of Michigan - Ann Arbor

LB
Liev B.

Numerade Educator

Marshall S.

University of Washington

Zachary M.

Hope College