Hopefully you've been following our recent rush of brain bogglers (see also Boggler #1 and Boggler #2). If so, let's think of them as mental "warming-up" exercises for the little beauty described below.
This is a puzzle of my own devising that's been lurking in the back of my mind for years, but I never seem to get the time to sit down and hammer it out, so I'm no closer to a solution than a was 20 or 30 years ago when I first started pondering this little rascal. Just to set the scene, imagine a guy (or a girl) standing with his legs apart holding a tennis ball as shown to the left of the following figure:
If this guy were to release the ball, it would fall to the floor and land halfway between his legs as illustrated to the right of the figure (we will, of course, assume "standard test conditions" with zero wind and so forth – I have no tricks up my sleeves).
Now, let's assume that the same person is transported into a cylindrical space station orbiting the earth and that we're in a "zero-gravity" condition. The cylinder has some radius r and it's rotating around its central axis at a rate of a radians-per-second as illustrated in the sketch below, thereby "holding" the guy to the floor by means of centrifugal or centripetal force (I always get those two mixed up).
The point to remember here is that the ball is actually rotating around the central axis along with the guy. For the purpose of these discussions, we'll assume that our hero's feet are glued to the floor (you'll see why in a moment).
So, let's consider what would happen if the guy were to release the ball at the exact same second that the cylinder "magically" stops revolving. In this case, the ball would continue moving in its current direction as shown in the figure below (the person would remain in place because – as you doubtless recall – we recently glued his feet to the floor).
OK, are we all in sync? If so, let's return to the real problem. Let's suppose that the cylinder is rotating again and that the guy is once again holding his ball. At some stage the guy releases the ball. Ignoring things like air resistance and so forth, where will the ball land in relation to his feet? Will it, for example, end up halfway between his legs as illustrated in the following figure?
Alternatively, will it land closer to one foot or the other, or to the left or the right of the person (let's say "ahead" or "behind" to avoid any confusion). Does the landing position depend on the radius of the cylinder and/or the release height of the ball and/or the rotational speed of the cylinder? Or will the ball always end up in the same relative position irrespective of these values?
Last but not least, if the position of the ball does depend on the values of the various variables, what angular rotation would be required to ensure the ball landed directly between the guy's feet if we assume that the radius of the cylinder is 10 meters and the guy starts by holding the ball 1 meter off the ground?
I await the results of your ponderings in dread anticipation...
Questions? Comments? Feel free to email me – Clive "Max" Maxfield – at firstname.lastname@example.org). And, of course, if you haven't already done so, don't forget to Sign Up for our weekly Programmable Logic DesignLine Newsletter.