10.5 Problems

Example 10.1: 

Suppose a pion decays into a muon and a neutrino,

$${\pi }^{+}={\mu }^{+}+{\nu }_{\mu }.$$

(10.20)

Express the momentum of the muon and the neutrino in terms of the mass of pion and muon. Assume that the neutrino mass is zero, and that the pion is at rest. Calculate the momentum using $m_{{\pi }^{+}}=139.6\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$, $m_{\mu }=105.7\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$.

Example 10.2: 

Calculate the lowest energy at which a $\Lambda \left(1115\right)$ can be produced in a collision of (negative) pions with protons at rest, throught the reaction ${\pi }^{-}+p\to K^0+\Lambda .$ $m_{{\pi }^{-}}=139.6\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$, $m_p=938.3\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$, $m_{K^0}=497.7\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$. (Hint: the mass of the $\Lambda$ is $1115\mathrm{\text{MeV}}∕{\mathrm{\text{c}}}^2$.)

Example 10.3: 

a) Find the maximum value for $v$ such that the relativisitic energy can be expressed by

$$E\approx m{c}^2+\frac{p^2}{2m},$$

(10.21)

with an error of one procent.
b) find the minimum value of $v$ and $\gamma$ so that the relativisitic energy can be expressed by

$$E\approx pc,$$

(10.22)

again with an error of one percent.