By Chris M. Hart
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14. Ex. 25. Find the velocity at which the relativistic momentum of a particle exceeds its Newtonian momentum by n fold. Sol. pr = pc 1 1− v2 c2 = n c n2 − 1 . ∴v = n 1 The Special Theory of Relativity 39 Ex. 26. An electron is accelerated through a potential difference of 1 million volts. What is the speed of electron? Sol. Kinetic energy of electron 1 2 − 1 T = m0 c 1− β 2 2 \ m0 c2 . 1− T + m c2 0 b = Putting m0 c2= 0. 51 MeV and T = 1 MeV, we get b = 0. 9988.
I) Suppose that the particle under study is at rest in frame S then u = 0 and u' = – v. Substituting these values in Eqn. 8) (ii) If the particle is at rest in frame S' then u' = 0 and u = v. Substituting these values in Eqn. 9) 12 Introduction to Modern Physics Fig. 2 (iii) Instead of mechanical particle, let the observers see photon or light wave front. , u = u' = c. Hence from Eqn. 10) Substituting the values of constants a12, a22 and a21 in Eqn. 11) (iv) According to the first postulate both the frames S and S' are equally suitable for the description of physical phenomena.
Experiment and discuss the significance of this result in the development of special theory of relativity. 5. What are Lorentz transformations? Show that two events, which are simultaneous in one frame of reference, are not simultaneous in other frame of reference in relative motion with the first. 6. Write down the Lorentz transformation equations. Explain the phenomenon of time dilation and length contraction. 7. State the postulates of special theory of relativity and show how Lorentz transformations have been obtained from them.