Mighty Mosquito and the Runaway Train

(Picture from the 1985 movie, “Runaway Train”)

An out-of-control freight train with no brakes is hurtling towards an infinitely massive dead end. A lone Mighty Mosquito flies out to save the day, and put itself between the train and the dead end, and waits motionless. The freight train is \(1,000,000,000,000\) times more massive than Mighty Mosquito. What happens next is a sight to behold!

Mighty Mosquito is infinitely elastic, acting like a perfect tiny hard ball, so that when the train collides with the insect, it slows down a bit, but sends the insect hurtling towards the infinitely massive dead end, which is also perfectly elastic. The insect bounces back, and hurtles towards the train, only to collide with it again, which slows it down further. This repeats many, many times, the insect bouncing back and forth between the train and the dead end, until finally, not only it actually reverses the motion of the train, but sends it hurtling towards the other direction!

In doing this mighty feat, Mighty Mosquito endured \(X\) number of collisions, all the times he hit the train and hit the dead end, until finally the train is travelling faster than Mighty Mosquito, both going in the same direction, so that there are no further collisions.

Find the 7 digit number X

Note It's assumed that the runaway train is moving freely and under no power, and we ignore friction, air resistance, etc., and that the train itself is also perfectly elastic. Also, disregard relativistic effects.


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