Built on Facts

Bouncing Birds

Challenge question! It’s either very easy or somewhat difficult depending on how clever you are at approaching it. No fair answering if you’ve already seen the problem before, though if no one’s managed it in a few hours I’d say it’s fair game to post the solution if you already know it.

Two trains are 100 miles apart on the same track, headed on a collision course towards each other. Both are traveling 50 miles per hour. A very speedy bird takes off from the first train and flies at 75 miles per hour toward the second train. The bird then immediately turns around and flies back to the first train. Then he flies back to the second train, and repeats the process over and over as the distance between the trains diminishes. How far will he have flown before the trains collide?

Famously John von Neumann was given this problem by a colleague, and he solved it instantly. His colleague remarked on how clever John had been to see the trick and not have to work out the sum of the infinite series. “There’s a trick?”, John asked. Von Neumann, however, was one of those higher-order geniuses – brilliant even in comparison to many of the other 20th century mathematical geniuses. Don’t feel so bad if the problem is more difficult for you.

For the record, I was posed this question on my first or second day of class as a physics undergrad. I didn’t see the shortcut.


  1. #1 Brandon Reese
    December 5, 2008

    My guess: 25 Miles

  2. #2 Beddrumiu
    December 5, 2008

    75 miles

  3. #3 Lucas
    December 5, 2008

    I find this problem interesting because the first time I heard it, I started working out the infinite series (which, let’s be honest, isn’t really that hard). The person who told it me said that there’s a better way. Just that knowledge allowed me to almost instantly see how to do it. For this reason, when something isn’t working I now say to myself “there is a better way, what is it?” rather than “is there a better way, and, if so, what might it be?” This seems to be a much more effective strategy in math.

  4. #4 Matthew
    December 5, 2008

    Well, assuming the bird has infinite acceleration abilities and can change direction instantaneously, it’s just travelling at 75mph for 1 hour (the time it takes for the trains to meet), so 75 miles. It took a second to notice the trick, but knowing that a trick exists helps.

  5. #5 Blaise Pascal
    December 5, 2008

    Is this one of these “show your work” type things? So far I’ve seen two answers thrown out with no explanation.

    (For the record, I know the trick (it’s easy… look at the bird’s pedometer after you find its body in the train wreckage), so I’m not going to answer.)

  6. #6 foole
    December 5, 2008

    The bird flies 75 miles (as the crow flies? 😉 ).

  7. #7 foole
    December 5, 2008

    Oh, and as for the show your work: the trains will collide in one hour (both travel 50 mph and start out 100 miles apart). Since the bird travels at 75 mph, he’ll have traveled 75 miles in that hour (and I gather that we assume that the bird’s speed doesn’t change).

  8. #8 Karl Withakay
    December 5, 2008

    66.666666666…. Miles ?

    Or Zeno’s paradox, the trains will never collide since the bird is always traveling faster than the trains.

  9. #9 peter
    December 5, 2008

    for those of us who are good at trick questions, (and not so hot on infinite series,) could someone elaborate on the ‘real’ method of getting the answer?

  10. #10 ossicle
    December 5, 2008

    Wow, that’s interesting. Seconding Lucas (though not tarring him with the brush I’m using on myself), I have a history of being awful at problems and puzzles. I am NOT good at that stuff. However, simply the fact that Matt said there was a trick made me think about the matter for a moment, and I was quickly able to get the correct answer.

    Psychology am interesting!


  11. #11 Uncle Al
    December 5, 2008

    Trains accumulate time interval to impact. Bird travels at constant speed given stated approximation. (time)(speed)=distance

    For the anally rententive, make it two mirrors and a lucky photon (no absorption, scattering, or bad angle reflection). Extra credit: If the photon begins with wavelength lambda, what is it Doppler shifted wavelength at collision?

  12. #12 Eric
    December 5, 2008

    I read this, and was sitting there going “Crap… I forget the trick.” Then I remembered it an hour later and was really excited.

  13. #13 Eric Lund
    December 5, 2008

    @Peter: The “series” method involves calculating where the westbound bird meets the eastbound train, and vice versa. You get a geometric series out of it, and since the ratio of successive terms has an absolute value less than 1, you can use a standard technique to sum the series, namely that the sum from n = 1 to infinity of x^n is 1 / (1 – x). Of course, it takes at least as long to calculate the first term of the series (let alone enough additional terms to notice that it is in fact a geometric series) as it does to arrive at the correct answer via the trick (if you spot the trick right away).

  14. #14 Abby Normal
    December 5, 2008

    Even with instatnt acceleration and such, this is unsolvable unless you know the shape of the track. For example, if the track is curved the trains may be 100 miles apart but have to travel 200 miles worth of track before they collide. 😉

  15. #15 Matt Springer
    December 5, 2008

    In lieu of typesetting the series myself, here’s a good explanation I located online for those interested in the longer method.

  16. #16 meichenl
    December 5, 2008

    The distance the bird flies must be proportional to distance between the trains, because if we doubled that distance, the bird would make the same infinite series of trips, but each leg would be twice as long. The answer has the form

    x = c*d,

    where “x” is the distance the bird travels, “c” is some constant, and “d” is the initial separation.

    Examine the first flight:
    The bird and train have relative velocity 125mph, so they meet in 100/125 = 4/5 hours. In 4/5 hours, the trains, traveling at relative velocity 100mph, go 80 miles, and are now 20 miles apart. Also, in 4/5 hours, the bird flies 75*4/5 = 60 miles.

    So now you have the exact same problem, except the 100 has been changed to 20, and the distance the bird will fly has been reduced by 60 miles. Using that earlier relation, we have two equations for the total distance the bird flies.

    x = c*100
    (x-60) = c*20

    5x – 300 = 100c = x
    4x = 300

  17. #17 meichenl
    December 5, 2008

    for Uncle Al’s problem:

    first find the doppler shift in a given reflection. Imagine two photons chasing each other, separated by a distance dx. They will hit a stationary mirror separated by a time dx/c, but a moving mirror will rush towards the second photon after hitting the first, and reduce that time to dx/(c+v), with “v” the speed of the mirror. So the frequency must go up by a factor (c+v)/c with every reflection.

    Every time the photon hits a mirror, it pushes back on the mirror a little. So the photon energy keeps increasing while the mirror kinetic energy decreases. The mirrors can’t collide, because to do so the photon would have to bounce off the mirrors an infinite number of times, and each time its energy would be multiplied by some constant greater than (c+v_f)/c, where “v_f” is the final speed of the mirrors as they collide.

    Instead, the photon’s energy increases until it has all its original energy, plus the original kinetic energy of the mirrors. Then the photon starts pushing the mirrors apart. Eventually, the mirrors are shooting away from each other faster than they were originally approaching, until the energy of the photon, now being depleted with every reflection, ultimately goes asymptotically towards being transferred completely to the kinetic energy of the mirrors.

  18. #18 Roger Sweeny
    December 5, 2008

    Lewis Carroll Epstein has this problem (with different movers and numbers) in his wonderful Thinking Physics.”

  19. #19 Matthias
    December 5, 2008

    @meichenl this would make for great bumpers you should get a patent.

  20. #20 Anonymous
    December 5, 2008

    This is the first physics problem that I was assigned in the very first physics class that I ever took. Except it was a bee and two bicycles, not a bird and a train. I am happy to say that I DID see the shortcut.

  21. #21 Anonymous
    December 5, 2008

    The Correct Answer is A))”Infinity”. The bird was Polish A)
    and got lost and just kept flying around
    or B) the bird was a blond and died in the collision.

  22. #22 IBY
    December 5, 2008

    I didn’t get it right away, but still, too easy: 75 miles.

  23. #23 nanoAl
    December 6, 2008

    I definitely switched it to the reference frame of the first train. “the other train is really going 100mph and is 100-“*Facepalm* 75!
    technically, the bird only has a displacement of 50 miles.

  24. #24 scotth
    December 6, 2008

    Pretty easy.

    50 + 50 mph gives closing distance of 100 mph.

    100 miles of track / 100 mph = 1 hour until collision.

    Bird travels 75mph for 1 hour = 75 miles travel.

  25. #25 dWj
    December 6, 2008

    Eugene Wigner is reported to have said, “There are two kinds of people in the world: Johnny von Neumann and everybody else.”

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