Close Shave with a Train

Alan is traversing a railway track that spans over a treacherous ravine. As he reaches $\frac{3}{8}$ of the way across the track, he hears the whistle of a train approaching from behind him. He can run in either direction and just barely make it off the bridge without getting hit by the train. If his running speed is 45 km/h in either direction, how fast is the train travelling? Assume that the train’s speed is constant.

Solution

180 km/h.

If Alan runs back, both him and the train will reach the start of the bridge in the time it takes for Alan to run $\frac{3}{8}$ of the way across the bridge.

That means if Alan runs forward, he will likewise be able to run $\frac{3}{8}$ of the way across the bridge before the train reaches the start of the bridge. They will both make it to the end of the bridge at the same time, with Alan travelling the remaining $\frac{5}{8}$ of the bridge and the train travelling the full span of the bridge.

Since the train covers 4 times the distance of Alan in the same amount of time, it is 4 times faster than Alan. Therefore its speed is $4 \times 45 km/h = 180 km/h$ .

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