Q.291. Calculate the kinetic energies of protons whose momenta are 0.10, 1.0, and 10 GeVic, where c is the velocity of light.
Ans. The formula is
Thus T = 5.3 MeV for
T = 0.433 GeV for
T = 9.106 GeV for
Here we have used m_{o} c^{2} = 0.938 GeV
Q.292. Find the mean path travelled by pions whose kinetic energy exceeds their rest energy η = 1.2 times. The mean lifetime of very slow pions is ζ_{o} = 25.5 ns.
Ans. Energy of pions is (1 + η) m_{o} c^{2} so
Hence or
Here of pion. Hence time dilation factor is 1 + η and the distance traversed by the pion in its lifetime will be
15.0 metres
on substituting the values of various quantities. (Note. The factor can be looked at as a time dilation effect in the laboratory frame or as length contraction factor brought to the other side in the proper frame of the pion).
Q.293. Negative pions with kinetic energy T = 100 MeV travel an average distance l = 11 m from their origin to decay. Find the proper lifetime of these pions.
Ans. From the previous problem
where is the rest mass of pions.
substitution gives
where we have used
Q.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 fluxes at the sections of the beam separated by a distance l = = 20 m. The proper mean lifetime of these pions is ζ_{o}= 25.5 ns.
Ans. here so the life time of the pion in the laboratory frame is
The law of radioactive decay implies that the flux decrease by the factor.
Q.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.
Ans. Energymomentum conservation implies
But
Hence
So
Substituting
Also
Q.296. Find the kinetic energy of a neutron emerging as a result of the decay of a stationary ∑  hyperon (∑  →n + π^{}).
Ans. We have
(1)
or
or
because (1) implies
Hence
and
Substitution gives T_{n }= 19.55 MeV
Q.297. A stationary positive muon disintegrated into a positron and two neutrinos. Find the greatest possible kinetic energy of the positron.
Ans. The reaction is
The neutrinoes are massless. The positron will carry largest momentum if both neutriones move in the same direction in the rest frame of the nuon. Then the final product is effectively a two body system and we get from problem (295)
Substitution gives
Q.298. A stationary neutral particle disintegrated into a proton with kinetic energy T = 5.3 MeV and a negative pion. Find the mass of that particle. What is its name?
Ans. By conservation of energymomentum
Then
This is a quadratic equation in M
or using and solving
Hence,
taking the positive sign. Thus
Substitution gives
From the table of masses we identify the particle as a ∧ particle
Q.299. A negative pion with kinetic energy T = 50 MeV disintegrated during its flight into a muon and a neutrino. Find the energy of the neutrino outgoing at right angles to the pion's motion direction.
Ans. See the diagram. By conservation of eneigy
or
or
Hence the eneigy of the neutrino is
on writing
Substitution gives E_{v} = 21.93 MeV
Q.300. A ∑^{+} hyperon with kinetic energy T_{∑ }= 320 MeV disintegrated during its flight into a neutral particle and a positive pion outgoing with kinetic energy T_{π} = 42 MeV at right angles to the hyperon's motion direction. Find the rest mass of the neutral particle (in MeV units).
Ans. By eneigy conservation
or
or
or using the K.E. of ∑ & π
and
Q.301. A neutral pion disintegrated during its flight into two gamma quanta with equal energies. The angle of divergence of gamma quanta is Find the kinetic energy of the pion and of each gamma quantum.
Ans. Here by conservation of momentum
Thus
or
and
substitution gives
Also
= m_{x} c^{2} in this case (θ = 60°)
Q.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 →m_{1 +}m_{2} . ..,where the rest masses of newly formed particles are written on the righthand side. Making use of the invariance of the quantity E^{2} — p^{2}c^{2} , demonstrate that the threshold kinetic energy of the particle m required for this reaction is defined by Eq. (6.7c).
Ans. With particle masses standing for the names of the particles, the reaction is
On R.H.S. let the energy momenta beetc. On the left the energy momentum o f the particle m is and that of the other particle is where, of course, the usual relations
hold. From the conservation of energy momentum we see that
Left hand side is
We evaluate the R.H.S. in the frame where (CM frame of the decay product).
Then
because all energies are +ve. Therefore we have the result
or since E = mc^{2} + T, we see that T ≤ T_{th} where
Q.303. A positron with kinetic energy T = 750 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.
Ans. By momentum conservation
or
Substitution gives
θ = 98.8°
Q.304. Find the threshold energy of gamma quantum required to form (a) an electronpositron pair in the field of a stationary electron; (b) a pair of pions of opposite signs in the field of a stationary proton.
Ans. The formula of problem 3.02 gives
when the projectile is a photon
(a) For
(b) For
Q.305. Protons with kinetic energy T strike a stationary hydrogen target. Find the threshold values of T for the following reactions: (a) (b)
Ans. (a) For
(b) For
Q.306. A hydrogen target is bombarded by pions. Calculate the threshold values of kinetic energies of these pions making possible the following reactions: (a) (b)
Ans. (a) Here
Substitution gives T_{th =} 0.904 GeV
Substitution gives T_{th =} 0.77 GeV.
Q.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?
Ans. From the GellMann Nishijima formula
we get
Also . Thus the particle is =° 0.
Q.308. Which of the following processes are forbidden by the law of conservation of lepton charge:
Ans. (1) The process cannot occur as there are 2 more leptons on the right comopared to zero on the left
(2 ) The process is forbidden because this corresponds to a change of lepton number by, (0 on the left  1 on the right)
(3) The process is forbidden because being both leptons ΔL = 2hre.
(4) , (5), (6) are allowed (except that one must distinguish between muon neutrinoes and electron neutrinoes). The correct names would be
Q.309. Which of the following processes are forbidden by the law of conservation of strangeness:
Ans. (1)
SO
(2)
SO
(3)
SO
(4)
SO
(5)
SO
(6)
SO
Q.310. Indicate the reasons why the following processes are forbidden:
Ans. (1)
is forbidden by energy conservation. The mass difference
(The process 1 → 2 + 3 will be allowed only if m_{1} > m_{2 }+ m_{3}.)
(2)
is disallowed by conservation of baryon number.
(3)
is forbidden by conservation of charge
(4)
is forbidden by strangeness conservation.
(5)
is forbidden by conservation of muon number (or lepton number).
(6)
is forbidden by the separate conservation of muon number as well as lepton number.
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