## Posts Tagged ‘drinking problem’

March 7, 2009

The original problem asked: “How high above a lake can Superman drink with a straw?”

Several readers gave the solution and pointed out some interesting subtleties. (Thanks, all!) However, no one pointed out the particular trick I had in mind when posing the question. The answer is: twice as high as you thought!

First, one might wonder why there should be a maximum height at all. However high he is, couldn’t Superman drink from even higher by sucking a little harder? The answer is no, because Superman is not pulling the water up from above so much as letting the atmosphere push it up from below, and he can’t make the atmosphere push any harder.

Air isn’t very heavy, and cotton candy isn’t very filling. But a lot of air (a hundred miles’ worth) or a lot of cotton candy (not necessarily a hundred miles of it) can be oppressive. The weight of the atmosphere pushes down on the lake with lots of pressure – 15 pounds per square inch. You may not notice this seemingly-high pressure on an everyday basis because you’re well-adapted to it. However, when the pressure changes by a few percent, as when you drive over a mountain pass or dive down in a pool, you’ll feel the difference.

When Superman sticks his straw in the lake and sucks, what he functionally does is thin out the atmosphere right in that little area above the lake where the end of the straw is. With all the pressure taken off it, water shoots up into the air and towards Superman’s mouth.

However, there’s a limit to this process. Suppose the water has risen a little way up the straw. Now imagine you are the thin layer of water at the bottom of the straw, and further suppose the straw is one square inch in area (not that this last bit matters, but we already know the pressure in pounds per square inch, so we’ll keep things in those units). If you are that little film down there, you don’t care whether the pressure on you comes from air or water. You just care how big the pressure is. Superman can make the pressure from air drop to zero in the straw by sucking everything out, but he can’t get rid of the pressure from the water. If the pressure from the water that’s already climbed up the straw is equal to the normal air pressure, the bottom of the straw will no longer know that someone up top is thirsty, and Superman won’t be able to suck water up any higher.

A square inch of air weighs about 15 pounds. How tall a column of water one inch square do we need to equal that same 15 pounds? Turns out to be about 34 feet. (Honestly I’m not sure what the point of these barbarian units is, but Superman was American.)

So is the highest height superman can get a drink 34 feet? No – that’s the highest height he can sustain a column of water above the surface. He can actually get a tiny little drink from just below double that – 67 feet, say.

The way he does this is as follows: he sucks really hard, creating a vacuum all at once. Then the water will accelerate all the way up to 34 feet. The net force on the column of water will be zero at that point, but it has upward velocity. It keeps rushing upward, but with negative acceleration, until the water reaches the apex of its trajectory at (ideally) 68 feet. If Superman wants more than just a dribble, he’ll need to drop down a little below that. He’ll also need a straw with extremely low friction, but in imaginationland, zero-friction is EVERYWHERE!

Of course, I’ve made ridiculous simplifications and assumptions. Kenneth pointed out some interesting thermodynamics considerations. Nikita mentioned capillary action, the understanding up which requires going into surface tension and intermolecular forces. Ken mentioned several important fluid dynamics concepts, none of which I feel bold enough to attack quantitatively just now (but Ken, if you wanted to come back and give it a shot of your own, I’d love to hear from you.)

There’s really no sharp line in things like this between when you’re honestly working on a problem and when you’re just being silly. For example, I could say that superman could blast little puffs of air down the tube with just the right timing to excite resonances in the lake so that after some minutes (or hours) he builds up a hugely constructive pressure wave at location of the straw’s mouth, and water comes shooting out like oil from a dinosaur’s graveyard. Or I could say he uses his heat vision to superheat the water right below the straw and send its pressure way up like a geyser or something (however those things work.) But that would be unrealistic, and Superman can’t stand for unrealistic scenarios.