Ist-order binomial approximation: (1+x)"≈1+nx. The second-order approximation keeps the second-order term, 2nd-order binomial approximation: (1+x)" ≈ 1+nx+ n(n − 1) x², - 2!

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Say that the mean lifetime of a π+ meson in its own rest frame is τ = 2.6 x 10-8 s. A pion of this lifetime is created at an altitude 100 km in the atmosphere by the collision of an incoming cosmic·ray proton with an atmospheric nucleus, and it has lifetime τ.

Use the binomial approximation (see attached image) to determine how fast would this πmeson need to move in order to reach the ground before decaying. Express the velocity in the form v/c = 1 - ϵ, where ϵ << 1. 

Ist-order binomial approximation: (1+x)"≈1+nx.
The second-order approximation keeps the second-order term,
2nd-order binomial approximation: (1+x)" ≈ 1+nx+
n(n − 1) x²,
2!
Transcribed Image Text:Ist-order binomial approximation: (1+x)"≈1+nx. The second-order approximation keeps the second-order term, 2nd-order binomial approximation: (1+x)" ≈ 1+nx+ n(n − 1) x², 2!
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