6.2 Problems 321
6.17 If a rod is to appear shrunk by half along its direction of motion, at what speed
should it travel?
6.18 Assuming that the rest radius of earth is 6,400 km and its orbital speed about
the sun is 30 km−^1 , how much does earth’s diameter appear to be shortened to
an observer on the sun, due to earth’s orbital motion?
6.19 The mean life-time of muons at rest is found to be about 2. 2 × 10 −^6 s, while
the mean life time in a burst of cosmic rays is found to be 1. 5 × 10 −^5 s. What
is the speed of these cosmic ray muons?
6.20 A beam of muons travels with a speed ofv = 0. 6 c. Their mean life-time
as observed in the laboratory is found to be 2. 9 × 10 −^6 s. What is the mean
life-time of muons when they decay at rest?
6.21 (a) If the mean proper life-time of muons is 2. 2 × 10 −^6 s, what average dis-
tance would they travel in vacuum before decaying in the reference frame
in which its velocity is measured as 0. 6 c?
(b) Compare this distance with the distance the muon sees while travelling.
6.22 With what constant velocity must a person travel from the centre to the edge
of our galaxy so that the trip may last 40 years (proper time)? Assume that the
radius of the galaxy is 3× 104 light years?
6.23 A pion is produced in a high energy collision of a primary cosmic ray particle
in the earth’s atmosphere 1 km above the sea level. It proceeds vertically down
at a speed of 0. 99 cand decays in its rest frame 2. 5 × 10 −^8 s after its production.
At what altitude above the sea level is it observed from the ground to decay?
6.24 One cosmic particle approaches the earth along its axis with a velocity of
0. 9 ctoward the North Pole and another one with a velocity of 0. 5 ctoward
the South Pole. Find the relative speed of approach of one particle with respect
to another.
6.25 A 100 MeV electron moves along the axis of an evacuated tube of length 4 m
fixed to the laboratory frame. What length of the tube would be measured by
the observer moving with the electron?
6.26 A man has a mass of 100 kg on the earth. When he is in the space-craft, an
observer from the earth registers his mass as 101 kg. Determine the speed of
the space-craft.
6.27 At the time a space ship moving with speedν= 0. 5 cpasses a space station
located near Mars, a radio signal is sent from the station to earth. This signal
is received on earth 1,125 s later.
How long does the spaceship take to reach the earth according to the observers
on earth?
6.28 In Problem 6.27, what is the duration according to the crew of the spaceship?