toothpaste asymptote

I've noticed that automobile gas tanks and batteries of all shapes and sizes follow the same general principle: they seem to last forever until they reach roughly the halfway point, after which they deplete at a logarithmic rate. It's a long time from "full" to "half" in my car, but no time at all from "half" to "fuel light."

Toothpaste tubes, thankfully, seem to follow the opposite rule: they deplete rapidly until you reach that final glob residing right next to the opening, then suddenly act as if they contain an infinite supply of toothpaste. A mere hint of a squeeze is enough to send out enough dentifrice to cover the bristles of one's toothbrush. It's not much, but it's always enough, and this state of affairs can last for weeks—even months. Do toothpaste tubes contain little wormholes into a universe filled with the stuff? The mind boggles at the implications.

Or perhaps a single universe is sufficient to explain the problem. Squeezing your toothpaste down to zero is a lot like accelerating an object to the speed of light: it takes an infinite amount of force to reach the speed of light, and the graph of the force necessary to push an object to that goal veers upward to avoid the asymptote set in place by Mother Nature. The same may apply to toothpaste in a tube: it may never be possible to empty any given tube of its paste; the force required would approach infinity, and this is why we always end up throwing away so much toothpaste.

It could be that the interior of a toothpaste tube begins to behave strangely as the toothpaste approaches zero. I'm sure some sort of experiment, performed at CERN, for example, could show us what we're up against.


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