Question

A kaon (denoted \(\left.\mathrm{K}^{\bullet}\right)\) is an unstable particle of mass \(8.87 \times 10^{-28} \mathrm{~kg}\). One of the means by which it decays is by spontaneous creation of two pions. a \(\pi^{+}\) and a \(\pi^{-}\). The decay process may be represented as $$ \mathrm{K}^{0} \rightarrow \pi^{+}+\pi^{-} $$ Both the \(\pi^{+}\) and \(\pi^{-}\) have mass \(2.49 \times 10^{-28} \mathrm{~kg}\). Suppose that a kaon moving in the \(+x\) direction decays by this process, with the \(\pi^{+}\) moving off at speed \(0.9 \mathrm{c}\) and the \(\pi^{-}\) at \(0.8 c\). (a) What was the speed of the kaon before decay? (b) In what directions do the pions move af ter the decay?

Short answer

a) The speed of the kaon before the decay can be found by solving the rearranged formula for relativistic momentum. b) The directions of the pions after the decay are symmetric to the x-axis with one moving counterclockwise and the other moving clockwise from the x-axis. The actual angle can be resolved by the ratio of the magnitudes of their momentums.

Step-by-step solution

Identify the known parameters.

The mass of the kaon \(m_k = 8.87 \times 10^{-28}\) kg and the mass of each pion \(m_p = 2.69 \times 10^{-28}\) kg. The velocity of pion+\(v_{p+}=0.9c\) and the velocity of pion-\(v_{p-}=0.8c\).

Calculate the momentum of each pion.

Using the formula for relativistic momentum \(p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}}\), calculate the momentum for each pion. Let's denote the momentum of pion+ as \(p_{p+}\) and pion- as \(p_{p-}\). So, \(p_{p+} = \frac{m_pv_{p+}}{\sqrt{1 - \frac{v_{p+}^2}{c^2}}}\) and \(p_{p-} = \frac{m_pv_{p-}}{\sqrt{1 - \frac{v_{p-}^2}{c^2}}}\).

Calculate the total momentum after the decay.

After the decay, the kaon turned into two pions moving in different directions. We assume that the total momentum remains on the x-axis, in the sanity of no external forces. So, the total momentum after the decay is the vector sum of the two pions' momentums, \(p_{total} = p_{p+} + p_{p-}\).

Infer the magnitude and direction of the initial kaon's momentum.

The magnitude of the kaon's momentum is exactly the same as the total momentum after the decay because of the conservation of momentum. The direction must also be the same, which is in the +x direction.

Calculate the speed of the kaon before the decay.

We now have the magnitude of the kaon momentum and its mass. We can find its speed by rearranging the formula for relativistic momentum: \(v_k = \frac{pc}{\sqrt{m_k^2c^2 + p^2}}\), where \(p = p_{total}\).

Find out the directions of the pions after the decay.

In order to have their vector sum along the +x-axis, they must have moved in directions symmetric to the x-axis. One moves in a certain angle counterclockwise from the +x-axis, while the other moves in the same angle clockwise from the +x-axis. The actual angle can be resolved by the ratio of the magnitudes of their momentums.

Key concepts

Conservation of Momentum

In particle physics, the law of conservation of momentum is pivotal, stating that the total momentum of an isolated system remains constant if it is not influenced by external forces. This principle anchors the analysis of particle interactions and decays, such as that of a kaon into pions. When a kaon decays, producing a \(\pi^{+}\) and a \(\pi^{-}\), the vector sum of the pions' momenta must equal the kaon's initial momentum.

To illustrate, if you have a moving cart that breaks apart into two pieces, regardless of how those pieces move apart, their combined momenta will match that of the whole cart before separation. This is because momentum is conserved in both magnitude and direction. In our kaon decay example, even though the pions fly off in different directions, their total momentum will align with the original motion of the kaon, preserving the system's overall momentum along the +x direction.

Relativistic Momentum

The concept of relativistic momentum is crucial when dealing with particles moving at speeds close to the speed of light, denoted as \(c\). This is a modification from classical mechanics, where momentum is simply the product of mass and velocity. In the realm of the fast-moving particles, we use the formula \(p = \frac{mv}{\sqrt{1 - \frac{v^2}{c^2}}}\) to account for the increase in mass with velocity as per Einstein’s theory of relativity.

For example, when a sprinter runs faster, their momentum straightforwardly increases with their speed. However, if we could watch a particle like a pion sprinting at 90% of the speed of light, its effective mass for calculating momentum would seem to increase, requiring the relativistic formula to correctly determine its momentum. This formula plays a critical role in accurately describing the behavior of particles in high-energy environments.

Particle Physics

Particle physics delves into the study of the smallest known components of matter and the interactions that govern their behavior. It seeks to understand and explain the fundamental forces and particles in the universe, such as quarks, leptons, bosons, and the ever-fascinating pions and kaons. Particle accelerators like the Large Hadron Collider (LHC) enable physicists to investigate the properties of these particles by colliding them at high energies and observing the resulting decay processes and particle creation.

When analyzing events like kaon decay, particle physicists must consider a variety of conservation laws, including momentum, energy, and other quantum numbers. These laws help to predict and verify the outcomes of particle interactions. In educational terms, think of particle physics as the scientific detective work that explores the microscopic world, piecing together the puzzle of our universe's smallest building blocks.

Pions

Pions, or pi mesons, are subatomic particles that play a significant role as the mediators of the strong nuclear force within atomic nuclei. They come in three charge states: \(\pi^{+}\), \(\pi^{0}\), and \(\pi^{-}\). With a mass about one-seventh that of the proton, pions are relatively light by particle standards, and they have a short lifetime, decaying to lighter particles like muons or electrons and neutrinos.

To make it relatable, consider pions as the glue sticks of the atomic world; just like glue holds craft materials together, pions help bind the nucleus’ protons and neutrons. Their role in the universe is subtle yet indispensable, and when they are produced in particle collisions or decays - like from a kaon - they provide key insights into the fundamental forces at play. The kaon decay we’re discussing involves the creation of a \(\pi^{+}\) and a \(\pi^{-}\), each carrying away a part of the kaon's original momentum.