# How do I determine how far a muon neutrino of 1 GeV will travel in the atmosphere if it's a product of the decay of a muon?

I have a question that is asking how far a muon neutrino of 1 GeV will travel in the atmosphere is the muon lifetime is $$2.2\mu$$ sec. This is in the context of muons being created by cosmic rays hitting the atmosphere. I know that when cosmic rays hit the atmosphere they create pions which mostly decay into muons and muon neutrinos, so I'm assuming that since it's giving the lifetime of the muon itself it only wants me to consider the distance traveled by a daughter muon neutrino from the decay of the muon.

How do I calculate this?

Edit: I have a theory that I need to calculate the lifetime of the muon neutrino, then calculate the distance it travels using relativity (time dilation), but I'm not sure how I factor in the lifetime of the muon into this.

• The muon neutrino - although it's mass is very small - will most likely pass through the other side of the Earth and keep on going. – Cinaed Simson Apr 15 at 22:36
• I believe since you know the energy and lifetime of the muon, the problem is about the decay of the muon as observed from Earth. The lifetime of muon is in the rest frame of the muon. – Cinaed Simson Apr 15 at 22:47
• My reaction was similar to @Cinaed's: I think you have misunderstood or misscommunicated the problem. The range of the neutrino is ... well, long. The range of the associated muons is of an interesting scale: the same order of magnitude as the scale height of the atmosphere and takes on that value in part due to relativistic effects. – dmckee Apr 16 at 2:33
• " I need to calculate the lifetime of the muon neutrino" neutrnos do not decay, they only interact very very weakly – anna v Apr 16 at 4:30