Math Chat

Old Milk Bottles and Hotel Expenses

Falling elevators

Suppose cartoon character Wile E. Coyote finds himself in a broken, brakeless elevator plummeting downward. Could he save himself by jumping up just before the elevator hit bottom? It might be hard for him to time his jump perfectly, but Lynne Lawson suggests that if he just keeps jumping up and down he'll probably be in the air when the elevator hits.

In a recent "Ask Marilyn" column (Parade Magazine, Sunday, Jan. 18), Marilyn Vos Savant says it is impossible to jump in an elevator in free fall. Actually, it is quite easy, since Wile E. would feel weightless, just as in a freely coasting orbiting spaceship. (Slight frictional breakage of the elevator would give him some small weight.) Pushing up onto his tip toes, he would find himself gently floating up toward the ceiling of the elevator.

Pushing off the ceiling, he would float gently back to the floor. From the start it would be hard to jump forcefully, since if he bent his knees, his feet would lift off the floor! Unfortunately, jumping would do little good. He would still be plummeting downward, just at a slightly slower speed than the elevator. (Perhaps this is what Marilyn meant.) His landing would be just a fraction of a second later, and almost as hard.

Old challenge (Michael Marcotty)

Take an old-fashioned bottle of nonhomogenized milk, with the cream risen to occupy the narrow neck at the top, shake it to mix in the cream, and put it back down on the table. Is the pressure of the milk on the bottom of the bottle the same as before mixing, greater, or less?

Answer

Greg Chapman and Robert Rein rightly answer that the pressure on the bottom will increase. The pressure at the bottom depends on the density of the fluid at all levels. Since a given amount of fluid in the narrow neck covers more levels, it influences the pressure more. When the cream, which is lighter, occupied the neck, the pressure was less.

Hotel $30 paradox

Three guys go into a hotel, each with $10 in his pocket. They book one room at $30 a night. A short while later a fax from headquarters directs the hotel to charge $25 a night. So the receptionist gives the bellhop $5 to take to the three guys sharing the room. Since the bellhop never got a tip from them and because he can't split $5 three ways, he decides to pocket $2 and give them each $1 back. So each of the three guys has now spent $9 and the bellhop has $2, for a total of $29. Where's the extra dollar? (Submitted by Thomas Linton, who is scripting such stories for a cartoon book, and also by Joe Herman.)

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