The charged pion $\pi^{-}$ usually decays into a muon and a neutrino,
$$
\pi^{-} \rightarrow \mu^{-}+\bar{\nu}_{\mu}
$$
but occasionally into an electron and a neutrino,
$$
\pi^{-} \rightarrow \mathrm{e}^{-}+\bar{\nu}_{\mathrm{e}}
$$
The relative frequency of the electron decay (compared to the muon decay) is on the order of 1 in $10^{4}$. The large difference between the probabilities for these two decays can be explained in terms of the $\mu-\mathrm{e}$ mass difference as follows: The theory of weak interactions predicts that the probability for either decay is proportional to $(1-v / c)$, where $v$ is the speed of the outgoing $\mu^{-}$ or $\mathrm{e}^{-} .$ Calculate the quantity $(1-v / c)$ for each decay, and compute their ratio. Note that this ratio has the same order of magnitude as the observed relative frequency. (Use the result of Problem 2.40; $m_{\pi} \approx 140, m_{\mu} \approx 106$, and $\left.m_{\mathrm{e}} \approx 0.51 \mathrm{MeV} / c^{2} .\right)$