330 6 Special Theory of Relativity
6.108 A linear accelerator produces a beam of excited carbon atoms of kinetic
energy 120 MeV. Light emitted on de-excitation is viewed at right angles
to the beam and has a wavelengthλ′.Ifλis the wavelength emitted by a
stationary atom, what is the value of (λ′−λ)/λ? (Take the rest energies of
both protons and neutrons to be 10^9 eV)
[University of Manchester 1970]
6.109 A certain spectral line of a star has natural frequency of 5× 1017 c/s. If the
star is approaching the earth at 300 km/s, what would be the fractional change
of frequency?
6.110 A neutral pion (mass 135 MeV/c^2 ) travelling with speedβ=v/c= 0. 8
decays into two photons at right angle to the line of flight. Find the angle
between the two photons as observed in the lab system.
6.111 An observer O sights light coming to him by an objectXat 45◦to its path as
in the diagram. If the corresponding angle of emission of light in the frame
of reference of the object is 60◦, calculate the velocity of the object.
Fig. 6.3Aberration of light
6.112 In an inertial frameSa rod of proper lengthLis at rest and at an angleθ
with respect to thex-axis with relativistic speed relative toS
(a) Show that the product tanθ.Lxis independent of which frame it is evalu-
ated in:
tanθ.Lx=tanθ′.Lx′
whereLxandLx′are the projections of the rod length onto thex-axis in
framesSandS′, respectively.
(b) In frameSthe rod is at an angleθ= 30 ◦knowing that an observer in
frameS′measures the rod to be at an angleθ′= 45 ◦with respect to the
x-axis, determine the speed at which the rod is moving with respect to the
observer.
[adapted from University of London 2004 Royal Holloway]
6.2.6 Threshold of Particle Production .....................
6.113 Show that the threshold energy for the production of a proton–antiproton pair
in the collision of a proton with hydrogen target (P+P→P+P+P+P−)
is 6 Mc^2 , whereMis the mass of a proton or antiproton.