Friday, 25 December 2009

Solution to The Two Trains Puzzle

As promised I am going to post a solution to the two trains problem.
I restate the problem below:

Two trains are speeding towards each other on the same track both at a speed of 50 km/hr. At the instant they are 100 km apart Bumble Bee starts flying from the windscreen of the first train to the second train at a speed of 75 km/hr. When it reaches the windscreen of the second train it bounces back at the same speed to the other train. It continues repeating this cycle indefinitely. Determine the total. distance traveled by Bumble Bee until the instant it experiences an untimely death due to the catastrophic head on collision between these trains.

And now for the solution:

Here is a visualization of the problem:





And here is the solution:

First we take the instant the two trains are 100 km apart.
Bumble Bee will travel towards the second train and the second train will travel towards Bumble Bee for a time 't' until they meet. The distance travelled during this time must be 100 km.

75t +  50t = 100 
125t = 100
t = 4/5

Now Bumble Bee travels towards the other train but this time the trains are only 20 km apart. So:

75t + 50t = 20
125t = 20
t = 4/25

Now Bumble Bee again travels towards the other train. But this time the trains are only 4 km apart. So the time until they meet will be 4/125.

What I am trying to do is to find the total time Bumble Bee spends bouncing between the trains. If I have that figure I can multiply it by the speed of Bumble Bee and get the distance. If you glance at the time you can immediately comprehend that the time keeps reducing but does not become zero unless I repeat the process of finding the time an infinite number of times. Serendipitously there exists a method of finding the sum of all those times that is better than spending an infinite amount of time adding an infinite amount of numbers.

We can express the total time T as an Infinite Geometric Series.

T = 4/5 + 4/25 + 4/125 + .................... ad infinitum
or 

T =


It takes only a moron to realize that this Infinite Geometric Series converges to 1. So the total amount of time Bumble Bee spends bouncing between the trains is 1 hour.

The total distance traveled is therefore 1 x 75 = 75 km      
                                                                       ______
                                                                       --------
And that is a truly beautiful result.
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