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.


The engine of the train travels the two miles of the tunnel in 2/152/15 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 1/51/5 of an hour. The whole train is out of the tunnel in 3/153/15 of an hour, or 1212 minutes.

The same reasoning works in general. For the engine to begin to emerge from the tunnel takes m/rm/r hours: mm miles at rr mph. For the end of the train to emerge takes another n/rn/r hours: nn miles at rr mph. The total time required: (m+n)/r(m + n)/r.