6.2 Problems 319
6.2 Problems..................................................
6.2.1 LorentzTransformations............................
6.1 In the inertial systemS, an event is observed to take place at pointAon the
x-axis and 10−^6 S later another event takes place at pointB,900 m further
down. Find the magnitude and direction of the velocity ofS′with respect toS
in which these two events appear simultaneous.
6.2 Show that the Lorentz-transformations connecting theS′andSsystems may
be expressed as
x 1 ′=x 1 coshα−ctsinhα
x 2 ′=x 2
x 3 ′=x 3
t′=tcoshα−(x 1 tsinhα)/c
where tanhα=ν/c. Also show that the Lorentz transformations correspond
to a rotation through an angleiαin four-dimensional space.
6.3 A pion moving alongx-axis withβ= 0 .8 in the lab system decays by emitting
a muon withβ′ = 0 .268 along the incident direction (x′-axis) in the rest
system of pion. Find the velocity of the muon (magnitude and direction) in the
lab system.
6.4 In Problem 6.3, the muon is emitted along they′-axis. Find the velocity of
muon in the lab frame
6.5 In Problem 6.3, the muon is emitted along the positivey-axis (i.e. perpendic-
ular to the incidental direction of pion in the lab frame). Find the speed of
muon in the lab frame and the direction of emission in the rest frame of pion.
Assumeβc= 0. 2
6.6 Show that Maxwell’s equations for the propagation of electromagnetic waves
are Lorentz invariant.
6.7 A neutralKmeson decays in flight viaK^0 →π+π−. If the negative pion is
produced at rest, calculate the kinetic energy of the positive pion.
[Mass ofK^0 is 498 MeV/c^2 ; that ofπ±is 140 MeV/c^2 ]
6.8 A pion travelling with speedν=|ν|in the laboratory decays viaπ→μ+ν.
If the neutrino emerges at right angles toν, find an expression for the angleθ
at which the muon emerges.
6.9 Determine the speed of the Lorentz transformation in thex-direction for which
the velocity in the frameSof a particle isu=(c/√
2 ,c/√
2) and the velocity
in frameS′is seen as
u′=(−c/√
2 ,c/√
2).
6.10 A particle decays into two particles of massm 1 andm 2 with a release of energy
Q. Calculate relativistically the energy carried by the decay products in the
rest frame of the decaying particle.