NYJC_EJC 2026 Special Relativity Extra Practice

1 Additional Problems for Special Relativity 1 (a) With reference to the rectangular coordinates system ( x,y,z,t), write down the Galilean Transformation equations. (b) Show that the Galilean Transformation equations are not consistent with the second postulate of the Special Theory of Relativity. (c) Consider two objects that interact through a force that is dependent on their separation and not on their velocities and accelerations (e.g., the gravitational force). Show that under Galilean transformation, the force law (e.g. 2F GMm r= ) remains invariant. 2 (a) Two events occur at the same place in a certain inertial frame and are separated by a time interval of 4 seconds. What is the spatial separation between these two events in an inertial frame in which the events are separated by a time interval of 6 seconds? (b) Two events occur at the same time in a certain inertial frame S and are separated by a distance of 1 km along the x axis. What is the time difference between these two events as measured in a frame S’ moving with constant velocity along x and in which their separation is measured as 2 km? Ans: 1.34×109 m, 5.77×10−6 s 3 (a) An archer releases an arrow that travels at a speed of 50 m s −1 towards a target 25 m away. At what velocity must an observer move so as to see the releasing of the arrow and the arrow hitting the target happening at the same position? (b) Two stars that are 3 × 1010 m apart are observed to undergo nova explosions at times t1 and t2, with t2 = t1 + 1 sec. With what velocity must a space shuttle be moving so that the astronaut on board will observe the two nova explosions happening simultaneously? Ans: 4 A physics professor on Earth gives an examination to students who are on a rocket ship traveling at speed v relative to Earth. The moment the ship passes the professor, he signals the start of the exam. If he wishes his students to have time T o (rocket time) to complete the exam, How long does he have to wait before sending a light signal telling them to stop. Ans: 1 1 o vcTT vc −= + 5 A rocketship of proper length lo travels at constant velocity v relative to frame S (see diagram below). The nose of the ship ( A’) passes the point A in S at t = t’ = 0, and at this instant a light signal is sent out from A’ to B’.

2 (a) When, by rocketship time (t’), does the signal reach the tail B’ of the ship? (b) At what time t1, as measured in S, does the signal reach the tail of the ship? (c) At what time t2, as measured in S, does the tail of the ship pass the point A? Ans: t’ = lo/c, 1 1 1 ol vct c vc −= + , t2 = lo/γv 6 At noon a rocketship passes the Earth with a velocity 0.8 c. Observers on the ship and on Earth agree that it is noon. (a) At 12.30 pm as read by a rocketship clock, the ship passes an interplanetary navigational station that is fixed relative to the Earth and whose clock read Earth time. What time is