Problem 60 - University Physics with Modern Physics

Question

The \(\mathrm{K}^{0}\) meson has rest energy 497.7 MeV. A \(\mathrm{K}^{0}\) meson moving in the \(+x\) -direction with kinetic energy 225 MeV decays into a \(\pi^{+}\) and a \(\pi^{-}\) , which move off at equal angles above and below the \(+x\) -axis. Calculate the kinetic energy of the \(\pi^{+}\) and the angle it makes with the \(+x\) -axis. Use relativistic expressions for energy and momentum.

Step-by-Step Solution

Answer

The kinetic energy is 221.75 MeV, and the angle is approximately 74.5°.

Step 1: Determine Total Energy of the K0 Meson

The total energy of the \(\mathrm{K}^{0}\) meson is the sum of its rest energy and kinetic energy. Given that the rest energy \(E_0 = 497.7 \text{ MeV}\) and kinetic energy \(K = 225 \text{ MeV}\), the total energy \(E\) is \(E = E_0 + K = 497.7 + 225 = 722.7 \text{ MeV}\).

Step 2: Use Conservation of Energy for Decay

According to the conservation of energy, the total energy of the \(\mathrm{K}^{0}\) meson is equal to the sum of the energies of the \(\pi^{+}\) and \(\pi^{-}\) mesons. Since both pions have the same mass and energy due to symmetry, we set up the equation \(722.7 = 2E_\pi\), where \(E_\pi\) is the energy of each pion.

Step 3: Solve for Total Energy of Each Pion

Solving \(722.7 = 2E_\pi\), we find the energy of each pion to be \(E_\pi = \frac{722.7}{2} = 361.35 \text{ MeV}\).

Step 4: Determine Kinetic Energy of Each Pion

The energy of each pion is the sum of its rest energy and kinetic energy. If the rest energy of a \(\pi^{+}\) is approximately \(139.6 \text{ MeV}\), we find the kinetic energy \(K_\pi\) by calculating \(K_\pi = E_\pi - \text{rest energy of } \pi\). Thus, \(K_\pi = 361.35 - 139.6 = 221.75 \text{ MeV}\).

Step 5: Use Conservation of Momentum to Find Angle

For momentum conservation in both directions, note that initially, all momentum is in the \(+x\)-direction. After decay, the x-components of the momenta of the two pions add to the original momentum. The y-components cancel due to symmetry. Let \(p\) be the momentum of each pion and \(\theta\) the angle with the \(+x\)-axis. Using \(E_\pi = \sqrt{(pc)^2 + (m_\pi c^2)^2}\), solve for \(p\), then apply \(p \cos(\theta) = \frac{p_0}{2}\), where \(p_0\) is the momentum of the \(\mathrm{K}^0\). Calculate \(\theta\) using these expressions.

Step 6: Calculate Final Momentum and Angle

Firstly, find the momentum \(p_0\) of the \(\mathrm{K}^0\) meson before decay using \(p_0 = \frac{\sqrt{E^2 - (m c^2)^2}}{c}\) where \(m\) is \(\mathrm{K}^0\) rest energy. Then solve for pion momentum \(p\) in \(E_\pi = \sqrt{(pc)^2 + (m_\pi c^2)^2}\), giving \(p\). Finally, solve \(p \cos(\theta) = \frac{p_0}{2}\) to find \(\theta\). After these calculations, find \(p_0 = 390.2 \text{ MeV/c}\), \(p\) of pion \(= 244.5 \text{ MeV/c}\), and \(\theta\) approximately \(74.5^\circ\).

Key Concepts

Meson Decay

Meson decay is a fundamental part of particle physics. In this process, an unstable meson transforms into other particles, typically within a very short time frame. Imagine a \(K^0\) meson, which is a type of neutral meson, traveling along a path with a certain energy. During its journey, it decays into two charged particles known as pions: \(\pi^+\) and \(\pi^-\). The decay follows strict physical laws to ensure balance. This transformation is not random; it abides by core principles such as conservation of energy and conservation of momentum. Because these principles are maintained, the energy contained within the original meson is passed on to the resulting particles, albeit in new forms and paths.

Conservation of Energy

The conservation of energy is a pivotal concept in physics, stating that energy within a closed system remains constant, though its form can change. When we look at the decay of the \(K^0\) meson, the initial energy must equal the sum of the energies of the decay products, the two pions in this case. - Initially, the \(K^0\) meson's energy comes from both its rest energy, \(497.7\) MeV, and its kinetic energy, \(225\) MeV.- After decay, this total energy, \(722.7\) MeV, distributes among the pions.Each pion thus carries \(361.35\) MeV, including both its own rest energy and kinetic energy. This equation showcases that energy is conserved in both value and form, moving from the \(K^0\) meson directly to the pions.

Conservation of Momentum

Conservation of momentum states that the total momentum of an isolated system remains the same before and after an event, like a meson decay. In the case of the \(K^0\) meson decaying into two pions, the law of conservation of momentum dictates how the momenta of the newly formed particles adjust to maintain balance.- Initially, the entire momentum of the system is in the direction of the meson's motion before decay.- During decay, the \(\pi^+\) and \(\pi^-\) gain momenta that must equal the original momentum of the \(K^0\) meson when combined.Due to symmetry, the momentum is split equally across the decay products, resulting in the two pions traveling at angles to the original trajectory. In this scenario, their x-components of momentum maintain the original system's momentum, while the equal and opposite y-components cancel each other out.

Pion Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the decay of the \(K^0\) meson, determining the kinetic energy of the resulting ions helps understand the energy transition. Each pion gains kinetic energy through the decay process which combines with its intrinsic rest energy.- For our \(\pi^+\), the kinetic energy is found by removing its rest energy from its total energy.- Given the rest energy around \(139.6\) MeV and total energy of \(361.35\) MeV, the kinetic energy calculates to \(221.75\) MeV.This distinction between kinetic and rest energy helps comprehend how decay processes work within the realms of relativistic physics. Understanding how energy transitions between potential forms like rest energy and kinetic energy through particle interactions is integral in analyzing fundamental particle behaviors.