1.340. A rod moves lengthwise with a constant velocity v relative to the inertial reference frame K. At what value of v will the length of the rod in this frame be η = 0.5% less than its proper length?

1.342. Find the proper length of a rod if in the laboratory frame of reference its velocity is v = c/2, the length l = 1.00 m, and the angle between the rod and its direction of motion is θ = 45°.

1.346. The proper lifetime of an unstable particle is equal to Δt_{0} = 10 ns. Find the distance this particle will traverse till its decay in the laboratory frame of reference, where its lifetime is equal to Δt = 20 ns.

1.347. In the reference frame K a muou moving with a velocity v = 0.990c travelled a distance l = 3.0 km from its birthplace to the point where it decayed. Find:
(a) the proper lifetime of this muon;
(b) the distance travelled by the muon in the frame K "from the muon's standpoint".

1.348. Two particles moving in a laboratory frame of reference along the same straight line with the same velocity v = (3/4)c strike against a stationary target with the time interval Δt = 50 ns. Find the proper distance between the particles prior to their hitting the target.

1.349. A rod moves along a ruler with a constant velocity. When the positions of both ends of the rod are marked simultaneously in the reference frame fixed to the ruler, the difference of readings on the ruler is equal to Δx_{1} = 4.0 m. But when the positions of the rod's ends are marked simultaneously in the reference frame fixed to the rod, the difference of readings on the same ruler is equal to Δx_{2} = 9.0 m. Find the proper length of the rod and its velocity relative to the ruler.

1.351. Two unstable particles move in the reference frame K along a straight line in the same direction with a velocity v = 0.990c. The distance between them in this reference frame is equal to l = 120 m. At a certain moment both particles decay simultaneously in the reference frame fixed to them. What time interval between the moments of decay of the two particles will be observed in the frame K? Which particle decays later in the frame K?

1.353. The rod A'B' moves with a constant velocity v relative to the rod AB (Fig. 1.91). Both rods have the same proper length l_{0} and at the ends of each of them clocks are mounted, which are synchronized pairwise: A with B and A' with B'. Suppose the moment when the clock B' gets opposite the clock A is taken for the beginning of the time count in the reference frames fixed to each of the rods. Determine:
(a) the readings of the clocks B and B' at the moment when they are opposite each other;
(b) the same for the clocks A and A'.

1.356. At two points of the reference frame K two events occurred separated by a time interval Δt. Demonstrate that if these events obey the cause-and-effect relationship in the frame K (e.g. a shot fired and a bullet hitting a target), they obey that relationship in any other inertial reference frame K'.

1.357. The space-time diagram of Fig. 1.93 shows three events A, B, and C which occurred on the x axis of some inertial reference frame. Find:
(a) the time interval between the events A and B in the reference frame where the two events occurred at the same point;
(b) the distance between the points at which the events A and C occurred in the reference frame where these two events are simultaneous.

1.359. Two particles move toward each other with velocities v_{1} = 0.50c and v_{2} = 0.75c relative to a laboratory frame of reference. Find:
(a) the approach velocity of the particles in the laboratory frame of reference;
(b) their relative velocity.

1.361. Two relativistic particles move at right angles to each other in a laboratory frame of reference, one with the velocity v_{1} and the other with the velocity v_{2}. Find their relative velocity.

1.363. 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.

1.364. 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.

1.365. 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.

1.369. The density of a stationary body is equal to ρ_{0}. Find the velocity (relative to the body) of the reference frame in which the density of the body is η = 25% greater than ρ_{0}.

1.370. 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?

1.376. 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 m_{0} respectively. Find the pressure developed by the beam on the target surface, and the power liberated there.

1.385. A particle of rest mass m_{0} with kinetic energy T strikes a stationary particle of the same rest mass. Find the rest mass and the velocity of the compound particle formed as a result of the collision.

1.386. How high must be the kinetic energy of a proton striking another, stationary, proton for their combined kinetic energy in the frame of the centre of inertia to be equal to the total kinetic energy of two protons moving toward each other with individual kinetic energies T = 25.0 GeV?