Math  /  Calculus

QuestionProblem 1: When a cosmic ray (e.g. a proton) hits a molecule in the atmosphere, muons are created during the particle shower. Muons decay with a mean lifetime τ=\langle\tau\rangle= 2.20μ s2.20 \mu \mathrm{~s}. (a) What is the decay rate, Γ\Gamma, for the muon?
Let us now model our cosmic ray as a particle of mass M. Assume that the particle of mass MM decays into two particles of mass mM:Mmmm \neq M: M \rightarrow m m. (a) Write down the conservation of the quadrimomentum in the reference frame of the particle of mass MM (hint: the particle MM is at rest in its reference frame) (b) Using the conservation equation above, compute the energies and momentum of the two particles of mass mm after the decay. (c) Is there a maximum value for mm over which the decay is no longer admissible? If the answer is yes, write such value.

Studdy Solution
Determine the maximum value for mm for which the decay is admissible:
For the decay to be admissible, the expression under the square root must be non-negative:
M24m20 M^2 - 4m^2 \geq 0
Solving for mm:
M24m2 M^2 \geq 4m^2
mM2 m \leq \frac{M}{2}
Thus, the maximum value for mm is:
M2 \boxed{\frac{M}{2}}

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