# Train on Bridge

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

It takes t_{1} seconds for a train to pass an observer and t_{2} seconds to clear a bridge of length L meters. Find the length and the speed of the train.

**Note**: t_{2} is the time between the train enters the bridge and the moment the last car gets off the bridge.

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

It takes t_{1} seconds for a train to pass an observer and t_{2} 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 _{1} = S/V,S/V,SV,V/S._{2} = (L + S)/V,L/V,L + V,(L + S)/V.

Subtracting the two gives L/V = t_{2} - t_{1}, or _{2} - t_{1})._{1}._{1} / (t_{2} - t_{1}).

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Copyright © 1996-2018 Alexander Bogomolny