(a)
The kinetic energy of the muon.
(a)
Answer to Problem 61P
The kinetic energy of the muon is
Explanation of Solution
A pion at rest decays to a muon and an antineutrino. The reaction for the decay is shown below.
Here,
The decay reaction obeys both energy conservation law and momentum conservation law. That is energy and momentum of the particles before and after the decay remains the same.
Write the expression for the energy of pion at rest.
Here,
After decay of pion, muon and antineutrino are formed.
Write the expression for the relativistic energy of muon.
Here,
Write the expression of the energy of antineutrino.
Here,
Pion is at rest. So its momentum is zero. Thus the final momentum also should be zero. Thus sum of momentum of muon and antineutrino is equal to zero.
Here,
Write the expression for the momentum of muon.
Here,
Use expression (V) in (IV).
Use expression (VI) in (III) to find
By conservation of energy, energy of pion is equal to the sum of energy of muon and antineutrino.
Use expressions (VII), (II), and (I) in expression (VIII).
Solve expression (IX) for
Write the expression for Lorentz factor.
Use expression (XI) in (X).
Reduce expression (XII).
Solve the right hand side of expression (XIII).
Use expression (XIV) in (XIII).
Write the expression for kinetic energy of muon.
Here,
Conclusion:
Substitute
Substitute
Substitute
Therefore, the kinetic energy of the muon is
(b)
The energy of the antineutrino in electron volts.
(b)
Answer to Problem 61P
The energy of the antineutrino is
Explanation of Solution
Write the expression for the kinetic energy of antineutrino.
Here,
Use expression (I), and (II) in expression (XVII) to find
Substitute
Conclusion:
Substitute
Therefore, the energy of the antineutrino is
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Chapter 39 Solutions
Physics for Scientists and Engineers with Modern Physics, Technology Update
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