Everything is complicated or not

I got this from the Physics Forum. It reads like a Zen story:

Two trains, each having a speed of 50 km/hr, are headed at each other on the same straight track. A bird that can fly 75 km/hr flies off the front of one train when they are 100 km apart and heads directly for the other train. On reaching the other train it flies directly back to the first train, and so forth. What is the total distance the bird travels?

As the speed of the bird is already known, the easy way to solve the problem of course is to figure out for how long the bird was flying (which is basically the time it takes for the trains to eventually come together): the product (speed x time) gives the total distance flown.

Of course one could try to do it the HARD way: calculate each leg back and forth and do it as an infinite series!

There is an old story, about a famous mathematician (John Von Neumann?), that a man presented him with a variant of this question (involving a fly and two bicycles), he thought for a few seconds and immediately gave the correct answer. The man chuckled and said "A lot of people try to do that by summing the infinite series." The mathematician looked puzzled and said "But I did sum an infinite series!"

For those of you into that sort of things, the math goes as follow:

Easy way:

The trains will meet in the middle since both trains are traveling at the same speed
 (i.e. total distance divided by two = 100/2 = 50 km.)

It follows that at a speed of 50 km/hr, it would have taken the trains one hour to travel that distance.
Mathematically, it goes something like this:

     The general formula is: speed x time = distance

       - d1 : distance traveled by train 1 =  50
          - d2 : distance traveled by train 2 = 50
          - D : total distance = 100
          - r : speed (km/hr) of either train = 50
          - T: time it takes the two trains to come together

           d1 + d2 = D
                 r T + r T = D
                T (r + r) = D
                T (2r) = D
                T = D/2r = 100/(2 x 50) = 100/100 = 1
                T = 1

One hour: meaning that the bird will be flying for 1 hour before the two trains meet. At a speed of 75 Km/hr, the bird will have then flown (flying back and forth for one hour) a total distance of 75 Km by the time the trains come together.

Hard Way:

         Summing the series: 

More here about that one.

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