Two bombs lie on a train platform, a distance L apart. As a train passes by at s

Question

Two bombs lie on a train platform, a distance L apart. As a train passes by at speed v, the bombs explode simultaneously (in the platform frame) and leave marks on the train. Due to the length contraction of the train, we know that the marks on the train will be a distance gamma*L apart when viewed in the train's frame (since this distance is what is length-contracted down to the given distance L in the platform frame).

How would someone on the train quantitatively explain to you why the marks are gamma*L apart, considering the bombs are only a distance L/gamma apart in the train frame?

Explanation / Answer

   Although two explosions are simultaneous in platform frame, there exist a time difference in train frame.    Time of explosion for first bomb   t1   =   {t0 - (v/c2) * x1} / v(1 - v2 / c2)    Time of explosion for second bomb   t2   =   {t0 - (v/c2) * x2} / v(1 - v2 / c2)    where x1 and x2 are positions of bombs at the platform such that   x2   -   x1   =   L    Hence time difference in train frame   ?t   =   t2   -   t1    ?t   =   {t0 - (v/c2) * x2} / v(1 - v2 / c2)   -   {t0 - (v/c2) * x1} / v(1 - v2 / c2)          =   { t0 - (v/c2) * x2   -   t0 + (v/c2) * x1} / v(1 - v2 / c2)          =   (v/c2) * (x2   -   x1) / v(1 - v2 / c2)          =   (v/c2) * L / v(1 - v2 / c2)    In this time the train would have moved through ditance   ?L   =   v * ?t    ?L   =   (v2/c2) * L / v(1 - v2 / c2)    Hence the marks on the train would be closer    L'   =   L   -   ?L    L'   =   L   -    (v2/c2) * L / v(1 - v2 / c2)     L'     =   L * v(1 - v2 / c2)    it explains the difference in distance of marks on the train.    it explains the difference in distance of marks on the train.

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