Train on Bridge

The following problem was offered at the very first Moscow Mathematical Olympiad in 1935


It takes t1 seconds for a train to pass an observer and t2 seconds to clear a bridge of length L meters. Find the length and the speed of the train.


Note: t2 is the time between the train enters the bridge and the moment the last car gets off the bridge.

Solution

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Copyright © 1996-2017 Alexander Bogomolny

Solution

It takes t1 seconds for a train to pass an observer and t2 seconds to clear a bridge of length L meters. Find the length and the speed of the train.


Let V be the train's speed and S its length.

The first conditon of the problem tells us that t1 = S/V,S/V,SV,V/S. The second is t2 = (L + S)/V,L/V,L + V,(L + S)/V.

Subtracting the two gives L/V = t2 - t1, or V = L / (t2 - t1). The first condition is equivaelent to S = Vt1. In other words, S = Lt1 / (t2 - t1).

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Copyright © 1996-2017 Alexander Bogomolny

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