SOLUTION: this is actually a common topic. 2 trains (A and B) apart of each other 90km moving towards each other on the same straight track. train A with speed 45km/h, and train B with

Question 173940: this is actually a common topic.
2 trains (A and B) apart of each other 90km moving towards each other on the same straight track.
train A with speed 45km/h, and train B with speed 45km/h
a bee is in between the 2 trains flying from train A to train B and back and forth continuosly with speed 60km/h
assume that we do not consider the bee's mass and length, assume it's just a dot
the usual questions will ask what is the total distance traveled by the bee, but i would like to know "how many times can the bee fly back and forth before the 2 trains crash??"
there is also a picture of the situation in the URL below,
http://polymathprogrammer.com/2007/08/13/two-trains-and-a-bumblebee-problem/
a friend once said the answer is "the bee can fly infinite times before the 2 trains crash"
but I doubt that, because the 2 trains will crash within a fix amount of time, and therefore there is only a fix amount of times the bee can fly back and forth
can someone clarify this? thank you.

Answer by vleith(2517) (Show Source):
You can put this solution on YOUR website!
If you calculate how long it takes the bee to get from the starting place to the other train, you solve
t = 0.857
Then turn around and solve it going the other way, with new values for the positions of he two trains

t = 0.1226
The ratio of these t values is 0.1429. this same ratio happens for each pass
That is, the next t is 0.1226*0.1429
And so on.
The 'fallacy' about taking an infitite amount of time is a variant of the 'if i am one step away from the goalline and take a step that is half of the distance", then I can take helf distance stesp forever and never get there.
In reality, the bee will fly for an hour. and then it becomes beejam between the two trains.
How many trips?
solve
...
By the fourth trip you are already at 0.9996, each trip after that adds in something at successivley less significant powers of 10. The engineer in me says, "the math tells you it is 'infinite', the reality says 5 trips and the bee is toast.