2440.79 – There's a Train in that Tunnel

A freight train 1 mile long goes through a tunnel 2 miles long. The train is traveling 15 mph. How long does it take to pass entirely through the tunnel?

Generalize your answer, that is, find a formula that gives the time for a train nn miles long to pass through a tunnel mm miles long traveling at rr miles per hour.

Solution

The engine of the train travels the two miles of the tunnel in 215\dfrac{2}{15} of an hour. At the moment the engine begins to emerge, the end of the train is halfway through. It goes the extra mile in 15\dfrac{1}{5} of an hour. The whole train is out of the tunnel in 315\dfrac{3}{15} of an hour, or 1212 minutes.

The same reasoning works in general. For the engine to begin to emerge from the tunnel takes mr\dfrac{m}{r} hours: mm miles at rr mph. For the end of the train to emerge takes another nr\dfrac{n}{r} hours: nn miles at rr mph. The total time required: m+nr\dfrac{m + n}{r}.