Anne and Mike have an argument about the hands of the clock:

Anne: "The minute hand takes one hour to make a complete circle. Therefore, it will pass the hour hand once every hour!"

Mike: "But by the time the minute hand has made a full circle, the hour hand will have moved 1/12th of the circle ahead. And by the time the minute hand gets to that spot, the hour hand will have moved forward yet again. This will continue indefinitely, so it is obvious that contrary to what it may seem like, the minute hand will never pass the hour hand."

In reality, how often does the minute hand pass the hour hand?

Ahhh, yes, the minute hand will pass the hour hand every one hour and 5 minutes, BUT, to get more specific, the hour hand is still moving even as the minute hand approaches, so it has moved another 5 minutes' worth while the minute hand tries to reach it. 5 minutes = 1/12 hour, which translates to 1/12 of 5 minutes' worth of distance for the minute hand--or 25 seconds. And of course, in THOSE 25 seconds, the hour hand has further retreated (ahhhh, what a futile attempt at escape) another 25 seconds' worth. That translates to 1/144 hr, which is equivalent to 1/144 of the 5 minute period that the hour hand is moving in, which equals about 2 more seconds. So the minute hand crosses the hour hand once every 1 hr, 5 min, 27 sec. We could take this infinitely further (which is exactly what confused poor Mikey Boy), but any further iterations will result in fractions of a second, and this is an analog clock, so lets be sane! =0)

Posted by Jim C on 2003-06-05 09:33:32