Q.5 S and S are in standard configuration with v= ac, (0 < a < 1). If a rod at rest in S makes an angle of 45 with Ox in S and 30 with O'x' in S', then find a. Q.6 The proper length of a rod is 10m. Calculate the length of the rod when it is moving with a velocity of 3000 km/s. What will be the change in length of the rod when it is moving with the velocity of (a) 3km/hr., (b) 0.99c. Q.7 Muon-particles are created at an altitude of more than 6000 m by cosmic rays coming from the outer space. These particles have an average life of 2 x 10-6 s in their rest frame and travel towards the earth with a typical speed of 2.994 x 108 m/s. Evaluate the lifetime of muons as measured by an observer on earth. Also find the distance travelled by the muon as measured by the same observer. Solve the problem of muon lifetime from its rest frame of reference. Q.8 Two electrons are moving in oppsite directions each with uniform speed of 0.7 c relative to the laboratory frame. What is the relative speed of the electrons as measured in the rest frame of one of the electrons? Q.9 A-hyperon has an average life-time of about 2.5 10-10 s in its rest frame. If it travels a distance of 0.05 m in a bubble chamber before decaying, calculate its velocity. Q.10 A particle has velocity u = (u, U2, U3) in S and u = (u, 2, 3) in S'. Prove from the velocity transformation foumulae that c u c (c-u) (c-v) (c + uv) Deduce that, if the speed of a particle is less than c in any one inertial frame, then it is less than c in every inertial frame.