– Total energy and momentum of a relativistie particle:
where $T$ is the kinetic energy of the partiele.
– When eremisine collinions of partietes it pays to us the isvariant:
whers $E$ asd $p$ are the total energy asd the lotal momentum of the system prior to collision,, is the rest masergy the tormed particle.
* Threshold (minisaly kinetic energy of : partiele $m$ atriking a statiosary particle $M$ and activating the endoergic Faction $n+M^{\prime} \rightarrow m_{4}+m_{4}+\ldots$,
\[
r_{\mathrm{m}}=\frac{\left(m_{1}+m_{1}+\ldots\right)^{n}-\left(n+m^{2}\right.}{2 M} e^{n} \text {, }
\]
where $m_{3}, M_{1}, m_{1}, m_{1}, \ldots$ ary the reat mases of the reapective particles.
$\dot{O}$, Quantuin numbers classifying elementary particles:
Q. electrie charg.
B. baryo chars.
$r$. istopic spin, $r_{z}$, its projection,
s. strangenesis, $s^{2}=2.0,-B$.
$Y$, hypertharge, $Y=B+s$.
– Relation betwees quantan numbers of strongly interseting partieles:
\[
Q=T_{3}+\frac{r}{2}=T_{3}+\frac{B+s}{2} .
\]
– Intersetions of particles abey the laws of censervation of the $Q, L$ and $B$ charges. Is strong interactions the laws of conservation of $S$ (or $Y$ ), $T$, and its propection $r_{\text {, }}$ aris also valid.
6.291. Calculate the kinetic energies of protons whose momenta are $0.10,1.0$, and $10 \mathrm{GeV} / c$, where $c$ is the velocity of light.
6.292. Find the mean path travelled by pions whose kinetic energy exceeds their rest energy $\eta=1.2$ times. The mean lifetime of very slow pions is $\tau_{4}=25.5$ ns.
6.293. Negative pions with kinetic energy $T=100 \mathrm{MeV}$ travel an average distance $t=11 \mathrm{~m}$ Irom their origin to decay. Find the proper lifetime of these pions.
6.294. There is a narrow beam of negative pions with kinetic energy $T$ equal to the rest energy of these particles. Find the ratio of fuxes at the sections of the beam separated by a distance $t$ $-20 \mathrm{~m}$. The proper mean lifetime of these pions is $\tau_{4}=25.5 \mathrm{~ns}$.
6.295. A stationary positive pion disintegrated into a muon and a neutrino. Find the kinetic energy of the muon and the energy of the neutrino.
6.296. Find the kinetic energy of a neutron emerging as a result of the decay of a stationary $\mathbf{Z}^{-}$hyperon $\left(\Sigma^{-} \rightarrow n+\pi^{-}\right)$.
6.297. A stationary positive muon disintegrated into a positron and two neutrinos. Find the greatest possible kinetic energy of the positron.
6.298. A stationary neutral particle disintegrated inte a proton with kinetic energy $T=5.3 \mathrm{MeV}$ and a negative pion. Find the mass of that particle. What is its name?
6.299. A negative pion with kinetic energy $T=50 \mathrm{MeV}$ disintegrated during its flight inte a muon and a neutrino. Find the energy of the neutrino outgoing at right angles to the pion’s motion direction.
6.300. A $\Sigma$ ‘ hyperon with kinetic energy $T_{z}=320 \mathrm{MeV}$ disintegrated during its flight into a neutral particle and a positive pion outgoing with kinetie energy $T_{\mathrm{n}}=42 \mathrm{MeV}$ at right angles to the hyperon’s motion direction. Find the rest mass of the neutral particle (in MeV units).
6.301. A neutral pion disintegrated during its alight into two gamms quants with equal energies. The sngle of divergence of gamma quanta is $\theta=60$. Find the kinetic energy of the pion and of each gamma quantum.
6.302. A relativistic particle with rest mass $m$ collides with a stationary particle of mass $M$ and activates a reaction leading to formation of new particles: $m+M \rightarrow m_{1}+m_{2}+\ldots$, where the rest masses of newly formed partieles are written on the right-hand side. Making use of the invariance of the quantity $E^{x}-p^{2} c^{2}$, demonstrate that the threshold kinetic energy of the particle $m$ required for this reaction is defined by Eq. $(6.7 \mathrm{c})$.
6.303. A positron with kinetic energy $T=750 \mathrm{keV}$ strikes a stationary free electron. As a result of annihilation, two gamma quanta with equal energies appear. Find the angle of divergence between them.
6.304. Find the threshold energy of gamma quantum required to form
(a) an electron-positron pair in the field of a stationary electron;
(b) a pair of pions of opposite signs in the field of a stationary proton.
6.305. Protons with kinetic energy $T$ strike a stationary hydrogen target. Find the threshold values of $T$ for the following reactions:
(a) $p+p \rightarrow p+p+p+\tilde{p}$; (b) $p+p \rightarrow p+p+\pi^{0}$.
6.306. A hydrogen target is bombarded by pions. Calculate the threshold values of kinetic energies of these pions making possible the following reactions:
(a) $\pi^{-}+p \rightarrow K^{+}+\Sigma^{-}$; (b) $\pi^{0}+p \rightarrow K^{+}+\Lambda^{0}$.
6.307. Find the strangeness $S$ and the hypercharge $Y$ of a neutral elementary particle whose isotopic spin projection is $T_{z}=+1 / 2$ and baryon charge $B=+1$. What particle is this?
6.308. Which of the following processes are forbidden by the law of conservation of lepton charge:
(1) $n \rightarrow p+e^{-}+v$;
(4) $p+e^{-} \rightarrow n+v$;
(2) $\pi^{+} \rightarrow \mu^{+}+e^{-}+e^{+}$;
(5) $\mu^{+} \rightarrow e^{+}+v+\stackrel{\tilde{v}}{v}$;
(3) $\pi^{-} \rightarrow \mu^{-}+v$;
(6) $K^{-} \rightarrow \mu^{-}+\tilde{v}$ ?
6.309. Which of the following processes are forbidden by the law of conservation of strangeness:
(1) $\pi^{-}+p \rightarrow \Sigma^{-}+K^{+}$;
(2) $\pi^{-}+p \rightarrow \Sigma^{+}+K^{-}$;
(4) $n+p \rightarrow \Lambda^{0}+\Sigma^{+}$;
(5) $\pi^{-}+n \rightarrow \Xi^{-}+K^{+}+K^{-}$;
(3) $\pi^{-}+p \rightarrow K^{+}+K^{-}+n$;
(6) $K^{-}+p \rightarrow \Omega^{-}+K^{+}+K^{0}$ ?
6.310. Indicate the reasons why the following processes are forbidden:
(1) $\Sigma^{-} \rightarrow \Lambda^{0}+\pi^{-}$;
(2) $\pi^{-}+p \rightarrow K^{+}+K^{-}$;
(3) $K^{-}+n \rightarrow \Omega^{-}+K^{+}+K^{0}$;
(4) $n+p \rightarrow \Sigma^{+}+\Lambda^{0}$;
(5) $\pi^{-} \rightarrow \mu^{-}+e^{+}+e^{-}$;
(6) $\mu^{-} \rightarrow e^{-}+v_{e}+\tilde{v}_{\mu}$.