A passing stick
(a) The stick has length L/γ in your frame, and it moves with speed v. Therefore, the time taken in your frame to cover the distance L/γ is L/γv.
(b) The stick sees you fly by at speed v. The stick has length L in its own frame, so the time elapsed in the stick frame is L/v. During this time, the stick sees the watch on your wrist run slow by a factor γ. Therefore, a time of L/γv elapses on your watch, in agreement with part (a).
Logically, the two solutions in (a) and (b) differ in that one uses length contraction and the other uses time dilation. Mathematically, they differ simply in the order in which the divisions by γ and v occur.
(c) You see the rear clock on the stick showing a time of Lv/c² more than the front clock. In addition to this head start, more time will of course elapse on the rear clock by the time it reaches you. The time in your frame is L/γv (because the stick has length L/γ in your frame). But the stick's clocks run slow, so a time of only L/γ²v will elapse on the rear clock by the time it reaches you. The total additional time (compared with the front clock's reading when it passed you) that the rear clock shows is therefore
c2Lv+γ2vL=vL(c2v2+γ21)=vL(c2v2+1−c2v2)=vL, (11.79)
in agreement with the quick calculation that follows in part (d).
(d) The stick sees you fly by at speed v. The stick has length L in its own frame, so the time elapsed in the stick frame is L/v.
⁴¹ Since we'll be working in the train frame here, we'll use the words "rear" and "front" in the way that someone on the train uses them as she watches the platform rush by. That is, if the train is heading east with respect to the platform, then from the point of view of the train, the platform is heading west, so the eastern bomb on the platform is the rear one, and the western bomb is the front one. So they have the opposite orientation compared with the way that someone on the platform labels the rear and front of the train. Using the same orientation would entail writing the phrase "front clock ahead" below, which would make me cringe.
11.13 Solutions 567