Meson Decay in Relativistic Mechanics

[pcandrepair]

Homework Statement

The K^o meson is a subatomic particle of rest mass M_K = 498 MeV/c² that decays into two charged pions, K^o $$\rightarrow$$ $$\pi$$positive+ $$\pi$$negative. Both pions have the same mass, m_$$\pi$$ = 140 Mev/c². A K^o at rest decays into two pions. Use conservation of energy and momentum to find the energies, momenta and speeds of the two pions. (Give algebraic answers, in terms of the symbols M_K and m_$$\pi$$. Then plug in numbers.)

Homework Equations

Total E = Mc²

Mc² = (K₁ + K₂) + (m₁ + m₂)*c²

$$\Delta$$M = M - (m₁ + m₂)

$$\Delta$$Mc² = K₁ + K₂

The Attempt at a Solution

I don't know how to find the energies of the two pions. The back of the book says 249 MeV. M_K/2 = 249 MeV but I don't know why. I'm not sure how they got that answer. Any help would be appreciated. Thanks!

 

[Dick]

Because the sum of the total energy of the two pions is equal to the total energy of the K (total energy includes rest mass). To conserve momentum the sum of the momenta of the two pions must be zero, since the initial momentum of the K was zero. That means the momenta are equal and opposite. So they have equal energy.

 

[pcandrepair]

I got it. Thank you!