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Chapter 30: Problem 11

The \(K^{0}\) meson can decay to two pions: \(K^{0} \rightarrow \pi^{+}+\pi^{-}\) The rest energies of the particles are: \(K^{0}=497.7 \mathrm{MeV}\) \(\pi^{+}=\pi^{-}=139.6 \mathrm{MeV} .\) If the \(K^{0}\) is at rest before it decays, what are the kinetic energies of the \(\pi^{+}\) and the \(\pi^{-}\) after the decay?

Short Answer

Expert verified
Answer: The kinetic energies of the π+ and π- pions after the decay of a K0 meson at rest are each 109.25 MeV.

Step by step solution

01

Calculate the total energy of the pion system after the decay

Since the \(K^{0}\) is at rest before it decays, its total energy is equal to its rest energy: \(E_{K^{0}}=mc^2=497.7\,\mathrm{MeV}\) After the decay, the total energy of the pions is conserved, so: \(E_{\text{total pions}} = E_{K^{0}} = 497.7\,\mathrm{MeV}\)
02

Calculate the rest energies of the pions

The rest energies of the \(\pi^{+}\) and \(\pi^{-}\) pions are given as: \(E_{\pi^{+}}=E_{\pi^{-}}=mc^2=139.6\,\mathrm{MeV}\)
03

Calculate the total kinetic energy of the pions

Before calculating the kinetic energy, we must determine the total energy of the pions after the decay. Since there are two pions, we can add their rest energies together: \(E_{\text{total rest pions}} = E_{\pi^{+}} + E_{\pi^{-}} = 2 \times 139.6\,\mathrm{MeV} = 279.2\,\mathrm{MeV}\) Now, we can find the total kinetic energy by subtracting the total rest energy from the total energy: \(E_{\text{total kinetic pions}} = E_{\text{total pions}} - E_{\text{total rest pions}} = 497.7\,\mathrm{MeV} - 279.2\,\mathrm{MeV} = 218.5\,\mathrm{MeV}\)
04

Determine the individual kinetic energies of the pions

Since the decay is symmetric, the two pions will have the same kinetic energy. Therefore, we can divide the total kinetic energy by 2 to find the kinetic energy of each pion: \(E_{\mathrm{kinetic}\,\pi^{+}}=E_{\mathrm{kinetic}\,\pi^{-}}=\frac{E_{\text{total kinetic pions}}}{2}=\frac{218.5\,\mathrm{MeV}}{2}=109.25\,\mathrm{MeV}\) So, the kinetic energies of the \(\pi^{+}\) and \(\pi^{-}\) after the decay are each \(109.25\,\mathrm{MeV}\).

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