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 № Condition free/or 1.5\$ 361 Two relativistic particles move at right angles to each other in a laboratory frame of reference, one with the velocity v1 and the other with the velocity v2. Find their relative velocity. 362 An unstable particle moves in the reference frame K' along its y' axis with a velocity v*. In its turn, the frame K' moves relative to the frame K in the positive direction of its x axis with a velocity V. The x' and x axes of the two reference frames coincide, the y' and y axes are parallel. Find the distance which the particle traverses in the frame K, if its proper lifetime is equal to dt0 . 363 A particle moves in the frame K with a velocity v at an angle θ to the x axis. Find the corresponding angle in the frame K' moving with a velocity V relative to the frame K in the positive direction of its x axis, if the x and x' axes of the two frames coincide. 364 The rod AB oriented parallel to the x' axis of the reference frame K' moves in this frame with a velocity v* along its y' axis. In its turn, the frame K' moves with a velocity V relative to the frame K as shown in Fig. 1.94. Find the angle θ between the rod and the x axis in the frame K. 365 The frame K' moves with a constant velocity V relative to the frame K. Find the acceleration w' of a particle in the frame K', if in the frame K this particle moves with a velocity v and acceleration w along a straight line (a) in the direction of the vector V; (b) perpendicular to the vector V. 366 An imaginary space rocket launched from the Earth moves with an acceleration w' = 10g which is the same in every instanta-neous co-moving inertial reference frame. The boost stage lasted t=1.0 year of terrestrial time. Find how much (in per cent) does the rocket velocity differ from the velocity of light at the end of the boost stage. What distance does the rocket cover by that moment? 367 From the conditions of the foregoing problem determine the boost time to in the reference frame fixed to the rocket. Remember that this time is defined by the formula t0=int(sqrt(1-(v/c^2)))dt, where dt is the time in the geocentric reference frame. 368 How many times does the relativistic mass of a particle whose velocity differs from the velocity of light by 0.010% exceed its rest mass? 369 The density of a stationary body is equal to ρ0. Find the velocity (relative to the body) of the reference frame in which the density of the body is η = 25% greater than ρ0. 370 A proton moves with a momentum p = 10.0 GeV/c, where c is the velocity of light. How much (in per cent) does the proton velocity differ from the velocity of light? 371 Find the velocity at which the relativistic momentum of a particle exceeds its Newtonian momentum n = 2 times. 372 What work has to be performed in order to increase the velocity of a particle of rest mass mo from 0.60 c to 0.80 c? Compare the result obtained with the value calculated from the classical formula. 373 Find the velocity at which the kinetic energy of a particle equals its rest energy. 374 At what values of the ratio of the kinetic energy to rest energy can the velocity of a particle be calculated from the classical formula with the relative error less than e = 0.010? 375 Find how the momentum of a particle of rest mass m 0 depends on its kinetic energy. Calculate the momentum of a proton whose kinetic energy equals 500 MeV. 376 A beam of relativistic particles with kinetic energy T strikes against an absorbing target. The beam current equals I, the charge and rest mass of each particle are equal to e and m0 respectively. Find the pressure developed by the beam on the target surface, and the power liberated there. 377 A sphere moves with a relativistic velocity v through a gas whose unit volume contains n slowly moving particles, each of mass m. Find the pressure p exerted by the gas on a spherical surface element perpendicular to the velocity of the sphere, provided that the particles scatter elastically. Show that the pressure is the same both in the reference frame fixed to the sphere and in the reference frame fixed to the gas. 378 A particle of rest mass mo starts moving at a moment t = 0 due to a constant force F. Find the time dependence of the particle s velocity and of the distance covered. 379 A particle of rest mass m o moves along the x axis of the frame K in accordance with the law x = sqrt(a^2+c^2t^2) where a is a constant, c is the velocity of light, and t is time. Find the force acting on the particle in this reference frame. 380 Proceeding from the fundamental equation of relativistic dynamics, find: (a) under what circumstances the acceleration of a particle coincides in direction with the force F acting on it; (b) the proportionality factors relating the force F and the acceleration w in the cases when F_|_v and F||v, where v is the velocity of the particle.

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