Two Trains and a Fly:
We are given the following:
Two trains are traveling towards each other on the same track.
One train is traveling at 30 miles per hour, the other at 20 mph.
When the trains are 100 miles apart, a fly leaves one and flies
towards the other. When it reaches the other it immediately travels
back to the first. Upon returning, it immediately starts back to
the second, and so on, until the trains crash.
Question: When the trains crash, how far has the fly travelled,
if it flies continuously at 50 mph?
Note that when we say 'immediately' above, we mean that there was
no time whatever lost in turning around -- that there was an
instantaneous change of motion.
[I don't remember when I first encountered this problem]
SOLUTION:
Let's forget about the fly for a moment, and just consider the
trains. The trains start out 100 miles apart at time t0. At
time t, they crash. t0 and t are the same for both trains.
Let t0=0.
d = distance = rate * time = r * t
Between them the two trains cover a distance of 100 miles, and
t = t = t so we have
1 2
d = d + d = r t + r t = (r + r ) t
1 2 1 1 2 2 1 2
100 mi = (30 mi/hr + 20 mi/hr) t
100 mi 100 mi
t = --------------------- = -------- = 2 hr
(30 mi/hr + 20 mi/hr) 50 mi/hr
So the trains crash at t = 2 hours (when the first train has
travelled 60 miles and the second 40 miles).
The fly has travelled 50 mi/hr * 2 hr = 100 miles in that same time.
-------
Once again, I don't remember when I first came across this problem.
Actually, I just made up the numbers, since I don't remember the original
ones. My recollection is that the original 'seemed' more difficult.
In any case, the basics are there. There's a story about this one.
Apparently, a fellow was walking with John von Neumann (or maybe it was
Norbert Wiener) and posed the problem to JVN. JVN responded very quickly
with the correct answer and the fellow said to him something along the
lines of "You know most people try to solve it with the infinite sequence."
To which our hero responded (something like), "You mean there is another
way?"
Hehehe. Who knows what really happened. But it IS a cute story.
(
geocities.com/athens)