322 6 Special Theory of Relativity
6.29 A spaceship is moving away from earth with speedν= 0. 6 c. When the ship
is at a distanced= 5 × 108 km from earth. A radio signal is sent to the ship
by the observers on earth.
How long does the signal take to reach the ship as measured by the scientist
on earth?
6.30 In Problem 6.29, how long does the signal take to reach the ship as measured
by the crew of the spaceship?
6.31 If the mean track length of 100 MeVπmesons is 4.88 m up to the point of
decay, calculate their mean lifetime.
[University of Durham 1962]
6.32 A beam ofπ+mesons of energy 1 GeV has an intensity of 10^6 particles per sec
at the beginning of a 10 m flight path. Calculate the intensity of the neutrino
flux at the end of the flight path (mass ofπmeson=139 MeV/c^2 , lifetime=
2. 56 × 10 −^8 s)
[University of Durham 1961]
6.33 Aπ+meson at rest decays into aμ+meson and a neutrino in 2. 5 × 10 −^8 s.
Assuming that theπ+meson has kinetic energy equal to its rest energy. What
distance would the meson travel before decaying as seen by an observer at rest?
6.34 Beams of high-energy muon neutrinos can be obtained by generating intense
beams ofπ+mesons and allowing them to decay while in flight. What frac-
tion of theπ+mesons in a beam of momentum 200 GeV/c will decay while
travelling a distance of 300 m?
At the end of the decay path (an evacuated tunnel) the beam is a mixture of
π+-mesons, muons and neutrinos. What distinguishes these particles in their
interactions with matter, and how is a neutrino beam free of contamination by
π-mesons and muons obtained? [π+-meson mean-life: 2. 6 × 10 −^8 s]
[Osmania University]
6.35 A beam of 140 MeV kinetic energyπ+mesons through three counters A, B,
C spaced 10 m apart. If 1,000 pions pass through counter A and 470 in B. (a)
how many pions are expected to be recorded in C? (b) Find the mean life time
of pions (Take mass of pion as 140 MeV/c^2 )
6.36 A moving object heads toward a stationary one with a velocityαc.Atwhat
velocityβcwould an observer have to move so that in his frame of reference
the objects would have equal and opposite velocities?
6.37 A beam of identical unstable particles flying at a speedβcis sent through
two counters separated by a distanceL. It is observed thatN 1 particles are
recorded at the first counter andN 2 at the second counter, the reduction being
solely due to the decay of the particles in flight.
(i) Show that the lifetime of the particles at rest is given by
τ=
L
ln(
N 1
N 2)
(
√
γ^2 − 1 c