A Λ^0 particle at rest decays into a proton and a π^- meson....

A $\Lambda^{0}$ particle at rest decays into a proton and a $\pi^{-}$ meson. (a) Use the data in Table 44.2 to find the $Q$ value for this decay in MeV. (b) What is the total kinetic energy shared by the proton and the $\pi^{-}$ meson after the decay? (c) What is the total momentum shared by the proton and the $\pi^{-}$ meson? (d) The proton and the $\pi^{-}$ meson have momenta with the same magnitude after the decay. Do they have equal kinetic energies? Explain.

A $\Lambda^{0}$ particle at rest decays into a proton and a $\pi^{-}$ meson.
(a) Use the data in Table 44.2 to find the $Q$ value for this decay in MeV. (b) What is the total kinetic energy shared by the proton and the $\pi^{-}$ meson after the decay? (c) What is the total momentum shared by the proton and the $\pi^{-}$ meson?
(d) The proton and the $\pi^{-}$ meson have momenta with the same magnitude after the decay. Do they have equal kinetic energies? Explain.

Key Concepts

Q value

The Q value in a decay process is defined as the difference in the total rest mass energy of the initial particle and the sum of the rest mass energies of the decay products. It represents the energy released in the decay, which is then available as kinetic energy for the decay products. Calculating the Q value involves understanding mass-energy equivalence and is crucial in nuclear and particle physics processes.

Conservation of Energy

In any particle decay process, the total energy (including both rest mass energy and kinetic energy) is conserved. For a decay occurring at rest, the Q value translates into the total kinetic energy of the resulting products. This conservation principle is essential for determining how the energy released by the decay is partitioned among the products.

Conservation of Momentum

Momentum conservation is a fundamental principle that remains valid in all decay processes, including those in the rest frame of the decaying particle. In a two-body decay, momentum conservation dictates that the decay products have equal and opposite momenta. This concept is used to analyze the resulting motion of the products even when their kinetic energies differ.

Two-Body Decay Kinematics

In two-body decays, the kinematics are simplified by the condition that the two products have equal magnitude of momentum in the rest frame of the parent particle. However, because kinetic energy depends inversely on mass (K = p²/2m for non-relativistic cases or derived via the relativistic formula), the particle with lower mass will generally acquire a greater share of the kinetic energy. This understanding is key when analyzing how energy is distributed among decay products with different masses.

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