Scott is an intelligent boy who, among other things, is fully au fait with basic theories of Physics. His younger brother Gavin, however is vain and idle, who whiles away his time practising his fake magical powers. Gavin also believes himself superior to Scott in intelligence.
Accordingly, Gavin challenges Scott to a little competition whereby, the first to get 5 ounces of water to freeze will be declared the most intelligent guy at their residence. They set up some rules as follows:
- Each of them can only use normal water that comes out of their stainless steel faucet.
- They both must use identical containers.
- They both must use the same freezer at the same time.
Now, it is a do-or-die situation for Scott as his prowess in Physics has been called into question.
How will Scott have the best chance of winning over Gavin?
While superficially it seems like the time to freeze is simply proportional to the mass of the water being frozen, there are heat transfer effects that cause it to also depend on the shape of the vessel. Specifically, the freezing time scales like the square of the thickness of the shape (and also is inversely proportional to the surface area where freezing occurs.) So water on, say, a cookie sheet, would freeze faster than the same mass of water as a single cube.
Now, Scott and Gavin are constrained to use the same containers, so Scott needs a slightly different approach -- layers.
The two must use the same freezer at the same time, but there's no constraint that they must put all their water into that freezer at once. So Scott can win by splitting his water into, say, N portions, and freezing them sequentially rather than all at once. Assuming a relatively ordinary container of constant cross-sectional area, each portion would freeze faster by a factor of N^2, but there are N of them now, so the overall increase should be on the order of a factor of N.
Two important caveats:
1) Of course opening and closing the freezer so much will make the freezing process take longer since the freezer itself will have to re-equilibrate each time, but since they both are using the same freezer at the same time, that applies equally to both competitors and so won't change the outcome.
2) The dependency of freezing time on thickness is not actually direct proportionality -- the dependency is on the sum of a linear and a quadratic term with coefficients that depend on the geometry of the container. As a result, it's only approximately true that 1/Nth the thickness freezes in 1/N^2 the time -- the linear term reduces the effect. BUT, it's still true that it freezes in *better* than 1/Nth the time and so freezing N layers of 1/N thickness sequentially is still faster than putting all the water in at once.
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Posted by Paul
on 2022-04-12 11:38:50 |
Another possibility?
