new fly and train math problem

Discussion in 'Physics & Math' started by Jeff 152, Aug 23, 2007.

  1. Jeff 152 Registered Senior Member

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    364
    that other thread about the fly and the train made me think of a math problem my multivariable calculus teacher asked my high school class one day when we were working on infinite series and zeno's paradox and such. I remember I solved it but many people in the class could not figure it out. Im curious as to what the brains on sciforums can do with this problem. and if you figure it out try not to ruin it for those that havent.

    Ok so the problem:

    2 trains are 100 miles apart traveling towards each other on the same track. Each train tavels at 10 miles per hour. A fly leaves the first train heading towards the second train the instant they are 100 miles apart. The fly travels at 30 miles per hour (relative to the ground not relative to the train he left). When the fly reaches the second train, it turns and heads back to the first train at 30 miles per hour (assume that the change in direction takes zero time). when the fly reaches the first train, he turns again. this process continues with the fly zipping back and forth between the trains as they come ever closer, until the two trains colide.

    The question: How far does the fly travel until he is crushed? (total distance traveled not displacement from original position)
     
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  3. Pete It's not rocket surgery Registered Senior Member

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    I've always wondered how to do this problem the hard way. What does the infinite series look like, and how is it summed?
     
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  5. camilus the villain with x-ray glasses Registered Senior Member

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    sounds hard but i think i can do it. give me a minute, ima try it right now
     
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  7. camilus the villain with x-ray glasses Registered Senior Member

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    I got the answer, but believe it or not I got it a geometric way, and I even found a geometric explanation for Zeno's Paradoxes. Thanks man!
     
  8. D H Some other guy Valued Senior Member

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    The problem as phrased by the OP is simple to do the hard way. Suppose the trains are separated by a distance d at the instant that the fly leaves one of the trains. Since the fly moves three times faster than the trains, the fly will meet the other train after traveling 3/4 of the initial separation. Both trains have traveled 1/4 of the initial separation: halving the distance between the trains. This leads to a very familiar sum. If the initial separation between the trains is d[sub]0[/sub], the total distance traveled by the fly is
    \(d_f = \frac 3 4 \, d_0\, \sum_{r=0}^{\infty}2^{-r} = \frac 3 2\, d_0\)
    Since the trains were initially separated by 100 miles, the fly travels 150 miles.

    The problem is a bit harder if the fly travels at some velocity other than three times the train velocity. Denote the fly's speed as \(v_f\) and a train's speed as\(v_t\). Now when the trains are separated by a distance d at the instant that the fly leaves one of the trains, the fly travels a fraction \(\frac{v_f}{v_f+v_t}\) of the separation distance before meeting the other train. At this time, the trains will have reduced the separation distance by \(2\,\frac{v_t}{v_f+v_t}\). The series becomes
    \(d_f = \frac {v_f}{v_f+v_t} \, d_0\, \sum_{r=0}^{\infty}\left(1-2\frac {v_t}{v_f+v_t}\right)^r\)
    Using the expansion of \(\frac{1}{1-x}\), the above becomes
    \(d_f = \frac 1 2\, \frac {v_f}{v_t}\, d_0\)
     
  9. Pete It's not rocket surgery Registered Senior Member

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    Wonderful!
     
  10. camilus the villain with x-ray glasses Registered Senior Member

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    895
    hmm.. I got something else besides 150 miles, what answer do you have Jeff?
     
  11. 2inquisitive The Devil is in the details Registered Senior Member

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    Why would anyone want to do the problem the hard way? Is it just a pretense to learn more math? I mean, the problem can be solved entirely in your head in just a few seconds, without having to write anything down. 150 miles.
     
  12. Atom Registered Senior Member

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    928
    << zeno's paradox >>


    You gottit!

    Please Register or Log in to view the hidden image!

     
  13. 2inquisitive The Devil is in the details Registered Senior Member

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    No, the OP is not the same thing as zeno's paradox. Infinity does not come into play with this gedanken.
     
  14. Pete It's not rocket surgery Registered Senior Member

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    How many times does the fly turn around?

    Why would anyone want to do the problem at all, if not to be entertained by the trick? Well, I've been entertained by the trick of doing it the easy way plenty of times before... so this time I sought a different sort of entertainment

    Please Register or Log in to view the hidden image!

     
  15. temur man of no words Registered Senior Member

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    Once von Neumann was given this problem and he answered in a few seconds. The challenger said "I knew you would find the easy way", and then von Neumann answered "What easy method?". Apparently, von Neumann added the infinite series in his head.
     
  16. 2inquisitive The Devil is in the details Registered Senior Member

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    Pete,
    The fly does not turn around an infinite number of times, so your question is completely irrelevant.
    Why do you refer to an efficient method as a 'trick'? I have never seen the gedanken before, let alone an 'easy' method of solving it. I thought the point of solving any problem is to determine most accurate and efficient method. I am not certain if I am using the same method you have seen written in a book, my method was just a matter of logic for me. Perhaps you should attempt using a little logic sometimes yourself, Pete.
     
  17. D H Some other guy Valued Senior Member

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    2,257
    The trick in calculating the infinite series quickly is recognizing that the series is a constant times 1/2+1/4+1/8+..., which is of course one.

    The trick in doing solving it the "fast way" is to focus on the trains. The fly's journey ends when the trains meet head-on. Simply multiply the time it takes for this to happen by the fly's speed to get the distance flown by the fly.
     
  18. D H Some other guy Valued Senior Member

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    2,257
    Yes, it does. It flies 75 miles, turns around and flies another 37.5, turns around and flies another 18.75 miles, turns around and flies another 9.375 miles, and so on. Each flight is half as long as the previous flight.
     
  19. 2inquisitive The Devil is in the details Registered Senior Member

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    No, it does not turn around an infinite number of times, which was my statement. Each flight is half as long as the previous flight only until the trains meet after five hours. The fly is crushed between the two trains at that point stopping all turn arounds. In a zeno paradox, there is no destructive event that ends the cycles.
     
  20. Pete It's not rocket surgery Registered Senior Member

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    In the mathematical idealisation (which is what this puzzle is about), the fly turns around an infinite number of times.
    In Zeno's paradox, there is certainly an event that ends the cycles. Achilles does indeed pass the tortoise. The arrow does in fact reach the target.

    My apologies. It's a well-known puzzle sometimes called the mathematician's trap.

    Either way... the puzzle has no real point except education and entertainment. As such, the efficiency or otherwise of a method used is secondary to its educational and entertainment value.
     
  21. 2inquisitive The Devil is in the details Registered Senior Member

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    Well, I did not arrive at a solution by the method described in your link. I used simple logic to arrive at the second method proposed by DH. The trains will collide after five hours, stopping all motion by the fly. If the fly is travelling at 30 mph, it will cover 150 miles in the five hours. All speeds can change and this method will work for all speeds. Example, if the fly is travelling at 25 mph, it will fly 125 miles before the collision. Change the speed of the trains and it is still very easy to solve, no calculus or analysis needed.

    Also, your Zeno's paradox is different than the one I am know about, the one where a person is crossing a room, each new step half as long as the previous step. He can never reach the other wall, only get closer as the series stretches out to infinity. There is no 'event' that stops the exercise as he never reaches the wall.
     
  22. Pete It's not rocket surgery Registered Senior Member

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    You're correct - the person will never reach the wall. But that's not a paradox.

    Zeno's paradoxes
     
  23. 2inquisitive The Devil is in the details Registered Senior Member

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    Well, that's debatable. If we add a time element to it, the person can cross the room. Assume the person walks at the same speed throughout the exercise. Suppose it takes the person ten seconds to reach the halfway point. It would take 5 seconds to cross half the remaining distance, then two and a half seconds for the next half, etc. Add all the times together and it will take the person a total of twenty seconds to cross the room, not an infinite amount of time.
     

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