Question

If a ${\mathit{\Sigma }}^{\mathit{+}}$at rest decays into a proton and a $\mathit{\pi }\mathbf{°}$, what is the total kinetic energy of the decay products?

Short answer

The total kinetic energy of the decay products is $115.7\mathrm{MeV}$.

Step-by-step solution

State Principle of conservation of energy in the decay process

Energy in the decay process is always conserved. Energy can not be destroyed. It can change forms, but the total amount always remains the same.

Calculate the kinetic energy of decay products

The ${\Sigma }^{+}$particle decays into meson ${\pi }^0$and proton.

The $\Sigma °$is initially at rest, so all of its energy is in the form of its rest energy.

• Rest energy of ${\Sigma }^a$is, localid="1667801562408" ${E_{\Sigma }}^{+}=1189\mathrm{MeV}$
• Rest energy of $\pi °$is, localid="1667801582728" $E_{{\mathrm{\pi }}^o}-135\mathrm{MeV}$
• Rest energy of $\rho$is $E_{\rho }=938.3\mathrm{MeV}$

From the conservation of energy,

localid="1667801599078" $$\begin{gathered} E_{{\Sigma }^{+}}=E_{{\alpha }^v}+E_{\rho }+K \\ K=1189\mathrm{MeV}-135\mathrm{MeV}-938.3\mathrm{MeV} \\ K=115.7\mathrm{MeV} \end{gathered}$$

Therefore, The total kinetic energy of the decay products is$115.7\mathrm{MeV}$.