Train Speed

Copyright 1997 by Paul Niquette.  All rights reserved.

Whatever the total distance to be traveled, the train in this puzzle will have used up all of the time allowed for completing the trip at 60 miles per hour during the first half at 30 miles per hour. So the solution is -- well, there is no solution.
  • Most people guess 90 mph, inasmuch as the average of 30 mph and 90 mph is given by (30 + 90) / 2 = 60. 
A train that does 30 mph for the first half of a trip and 90 mph for the second half will average only 45 mph for the whole trip.
  • Here are other popular guesses: 120 (48 mph), 180 (51 mph), 240 (53 mph).
Try modifying the problem so that the train is required to average only 59 miles per hour for the whole trip after traveling at 30 mph during the first half.
Surprise: The train would have to go 1,770 mph!
Supersonic trains are still the stuff of science fiction.

Time in the Numerator

The sophisticated person puts time in the numerator (see Bucketing Bandwidth).  Examples abound.  Here is a message received in 2003 from Ketan Bhaidasna that makes the point...

Ben and Carl were in a 100-meter race.  When Ben crossed the finish line, Carl was only at the 90-meter mark. 

Ben suggested they run another race. This time Ben would start ten meters behind the starting line.  Will Carl win or lose -- or will the second race end in a tie? 

Ketan's solution...
Carl will lose again.  In the second race, Ben started ten meters back.  By the time Carl reaches the 90-meter mark, Ben will have caught up to him.  Therefore, the final ten meters will belong to the faster of the two.  Since Ben is faster than Carl, he will win the final 10 meters and the race.
The analytic approach...

Let the respective velocities be VB and V
        with the first race ending at time T.
We have that T = 100 / VB = 90 / VC.
Thus, VB = 100/T and VC = 90/T.
Ben finishes the second race at 110 / VB = 110/100 / T = 1.1T
Carl finishes the second race at 100 / VC = 100/90/T = 1.111T, 
       slower by about 1%.

So then, how much of a head-start should Ben give Carl?
This question will be pertinent in the World's Fastest Dragster.


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