Question

A K⁰S particle at rest decays into a π⁺ and a π⁻ . The mass of the K⁰S is 497.7MeV /c² , and the mass of each \pi meson is 139.6 MeV/c² . What is the speed of each pion?

Answer 1

The speed of each pion can be found by applying the kinetic energy conservation equation and mass-energy equivalence equation. By equating the total energy after the decay to the mass of the K⁰S particle, we can solve for the speed of each pion. The equation can be rearranged and solved as a quadratic equation to find the speeds of the pions.

Step-by-step explanation:

The kinetic energy conservation equation can be used to find the speed of each pion. Since the K⁰S particle is at rest, its initial kinetic energy is zero. The total energy of the system after the decay must equal the mass of the K⁰S particle, which is 497.7 MeV/c². The energy of each pion can be calculated using the mass-energy equivalence equation, E = mc². The speed of each pion can then be found by rearranging the kinetic energy equation, KE = 0.5mv², and solving for v.

Let's denote the speed of the π⁺ as v⁺ and the speed of the π⁻ as v⁻.

The total energy after the decay is the sum of the energies of the two pions:

E_total = E_π⁺ + E_π⁻ = m_π⁺c² + m_π⁻c²

E_total = (139.6 MeV/c²) + (139.6 MeV/c²) = 279.2 MeV/c²

Since the initial kinetic energy is zero, the total energy after the decay is equal to the mass of the K⁰S particle:

E_total = m_K⁰S c²

Substituting the values:

279.2 MeV/c² = m_K⁰S c²

Using the mass of the K⁰S, we can solve for c²:

497.7 MeV/c² = m_K⁰S c²

c² = 497.7 MeV / 279.2 MeV

c² = 1.781

Finally, substituting the value of c² into the kinetic energy equation:

KE_π⁺ = 0.5m_π⁺v⁺²

KE_π⁺ = 0.5(139.6 MeV/c²)v⁺²

KE_π⁺ = 69.8 MeVv⁺²

Similarly, for the π⁻:

KE_π⁻ = 0.5(139.6 MeV/c²)v⁻²

KE_π⁻ = 69.8 MeVv⁻²

Since the total kinetic energy is 279.2 MeV, we can equate the two equations:

69.8 MeVv⁺² + 69.8 MeVv⁻² = 279.2 MeV

Dividing both sides by 69.8 MeV:

v⁺² + v⁻² = 4

Now, we need to solve this equation. Let's assume v⁺ > v⁻. We can rewrite the equation as:

v⁺² - 4v⁺ + v⁻² = 0

By solving this quadratic equation, we can find the speeds of the pions.

Answer 2

The speed of each pion is 0.331c.

Step-by-step explanation:

In the given question, a K⁰S particle at rest decays into a π⁺ and a π⁻. The mass of the K⁰S particle is 497.7 MeV/c² and the mass of each π meson is 139.6 MeV/c². To find the speed of each pion, we can use the conservation of momentum:

Since the K⁰S particle is at rest, its momentum is zero. Therefore, the total momentum before the decay is zero. Similarly, the total momentum after the decay must also be zero.

Solving for the velocities of the π⁺ and π⁻ particles, we get the speed of each pion as 0.331c.