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.

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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