An express train roars through the train station, taking 5 seconds to pass a waiting passenger, and 19 seconds to clear the whole station. The platform is 280 meters long. How long is the train, and at what speed is it moving?

The train takes 5 seconds to pass any given point — the passenger P, the starting point S of the station platform, or its ending point E:

It took 19 seconds between the time the front of the train passed S and the rear of the train passed E. We know that it took 5 seconds for the whole train to clear point S, at which point the back of the train was at S. So it took 19 – 5 = 14 more seconds for the back of the train to travel the 280m of the station. Well, 280 meters in 14 seconds is 20 meters per second.

So the speed of the train is 20 meters per second.

Now, if it takes 5 seconds for the whole train to pass a given point at 20 m/sec, then the train is 100 meters long.

Let’s see what this means in the English system: 20 meters/sec = 72 km/hr, or about 45 miles/hr. Not terribly fast.

The classy French TGV trains travel up to 200 mph in commerical use. They really howl along. (TGV: tres a’grande vitesse: “high speed train,” see Wikipedia).