Explanation The momentum before the decay is zero as the pion decays from the rest. Therefore, the magnitudes of momentum of the positron, and neutrons are equal because the momentum after the decay is also zero. The neutrinos are also treated as massless. Write the relation for the energy and momentum conservation, and use the relativistic relationship between energy, and momentum. m π + c 2 = E e + + E ν m π + c 2 − E e + = E ν Here, m π + is the mass of the pion, E ν is the energy of neutrino and E e + is the energy of the positron. Momentum conservation says that the magnitudes of the momentum of the two products are equal. p e + = p ν ( p e + 2 c 2 ) = ( p ν 2 c 2 ) (I) Here, p e + is the momentum of the positron, and p ν is the momentum of the neutrino. Write the expression for the energy of the particle. E e + 2 = p e + 2 c 2 + m e + 2 c 4 p e + 2 c 2 = E e + 2 − m e + 2 c 4 Neutrino is considered as mass less. Substitute E e + 2 − m e + 2 c 4 for p e + 2 c 2 , and E ν 2 for p ν 2 c 2 in equation (I). E e + 2 − m e + 2 c 4 = E ν 2 (II) Substitute ( m π + c 2 − E e + ) for E ν in equation (II). E e + 2 − m e + 2 c 4 = ( m π + c 2 − E e + ) 2 E e + 2 − m e + 2 c 4 = m π + 2 c 4 − 2 E e + m π + c 2 + E e + 2 2 E e + m π + c 2 = m π + 2 c 4 + m e + 2 c 4 E e + = 1 2 m π + c 2 + m e + 2 c 2 2 m π + (III) Write the expression to calculate the kinetic energy of e + ion

Physics for Scientists and Engineers with Modern Physics

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ISBN: 9780136139225

Author: Douglas C. Giancoli

Publisher: Prentice Hall

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Chapter 43, Problem 27P

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The kinetic energy of the positron in the rare decay π+→e++νe.

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Physics for Scientists and Engineers with Modern Physics

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