Two clocks are positioned at the ends of a train of length L (as measured in its own frame). They are synchronized in the train frame. The train travels past you at speed v. If you observe the clocks simultaneously in your frame what is the difference in reading of the two clocks?
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Answer:
In our frame, the light beam approaches the rear end of the train at speed c+v, because both are moving and in opposite directions. Note that this does not mean anything is traveling faster than light. Similarly, the light beam approaches the front of the train at speed c−v. We want to choose the position of emission so that they arrive at the same time. If df is the distance to the front of the train from the emission point in our frame and dr is the distance from the rear of the train to the emission point in our frame, we have
df+dr=Ldfc−v=drc+vdf+dfc+vc−v=L2cdf=L(c−v)df=L(c−v)2c
as required. The 2 comes from solving the equations. Note that if v=0 this simplifies to df=L/2 because the light source is at the center of the train. This is the same 2.