№ 
Condition 
free/or 1.5$ 
341  In a triangle the proper length of each side equals a. Find the perimeter of this triangle in the reference frame moving relative to it with a constant velocity V along one of its (a) bisectors; (b) sides. Investigate the results obtained at V<<c and V>c, where c is the velocity of light. 

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°. 

343  A stationary upright cone has a taper angle T=45°, and the area of the lateral surface S0 = 4.0 m2. Find: (a) its taper angle; (b) its lateral surface area, in the reference frame moving with a velocity v = (4/5)c along the axis of the cone. 

344  With what velocity (relative to the reference frame K) did the clock move, if during the time interval t = 5.0 s, measured by the clock of the frame K, it became slow by dt = 0.10 s? 

345  A rod flies with constant velocity past a mark which is stationary in the reference frame K. In the frame K it takes At 20 ns for the rod to fly past the mark. In the reference frame fixed to the rod the mark moves past the rod for dt' = 25 ns. Find the proper length of the rod. 

346  The proper lifetime of an unstable particle is equal to Δt0 = 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. 

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". 

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. 

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 Δx1 = 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 Δx2 = 9.0 m. Find the proper length of the rod and its velocity relative to the ruler. 

350  Two rods of the same proper length L0 move toward each other parallel to a common horizontal axis. In the reference frame fixed to one of the rods the time interval between the moments, when the right and left ends of the rods coincide, is equal to At. What is the velocity of one rod relative to the other? 

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? 

352  A rod AB oriented along the x axis of the reference frame K moves in the positive direction of the x axis with a constant velocity v. The point A is the forward end of the rod, and the point B its rear end. Find: (a) the proper length of the rod, if at the moment to the coordinate of the point A is equal to xA, and at the moment tB the coordinate of the point B is equal to xB; (b) what time interval should separate the markings of coordinates of the rod's ends in the frame K for the difference of coordinates to become equal to the proper length of the rod. 

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 l0 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'. 

354  There are two groups of mutually synchronized clocks K and K' moving relative to each other with a velocity v as shown in Fig. 1.92. The moment when the clock A' gets opposite the clock A is taken for the beginning of the time count. Draw the approximate position of hands of all the clocks at this moment "in terms of the K clocks"; "in terms of the K' clocks". 

355  The reference frame K' moves in the positive direction of the x axis of the frame K with a relative velocity V. Suppose that at the moment when the origins of coordinates O and O' coincide, the clock readings at these points are equal to zero in both frames. Find the displacement velocity x of the point (in the frame K) at which the readings of the clocks of both reference frames will be permanently identical. Demonstrate that x < V. 

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 causeandeffect 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'. 

357  The spacetime 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. 

358  The velocity components of a particle moving in the xy plane of the reference frame K are equal to vx and vy. Find the velocity v* of this particle in the frame K' which moves with the velocity V relative to the frame K in the positive direction of its x axis. 

359  Two particles move toward each other with velocities v1 = 0.50c and v2 = 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. 

360  Two rods having the same proper length L0 move lengthwise toward each other parallel to a common axis with the same velocity v relative to the laboratory frame of reference. What is the length of each rod in the reference frame fixed to the other rod? 
