Egg Sandwiches

Dropping a ball bearing off a 747 and hitting an egg sandwich is not a common occourance. Physics has not devoted a lot of effort to measuring endevours of this sort, but we can make some educated guesses. We know how big a sandwich is.. say 15 cm x 15 cm, but how many of them are there to hit?

Well, let's say that 1 billion people on earth routinely eat egg sandwiches, which counts a good part of the western world. Let's further guess that each of these people eats an egg sandwich once every sixty days. (We may be wrong by a factor of ten either way here. That is, perhaps only 100 million people eat them, or maybe they eat them every 6 days, or whatever. Still, we have a ballpark number to work fro.) The number of sandwiches made every day, on average is:

$$1\, 000\, 000\, 000\, people\times \frac{1\, sandwich}{60\, days}=1.67\times 10^{7}sandwiches/day$$

Now, these sandwiches are not all outside. Let's say that for 3 months of the year is good picnic weather, and that everyone eats outside one day every three weeks in good weather.

$$1.67\times 10^{7}sandwiches/day\times \frac{3\, months}{12\, ... ...times \frac{1\, day}{21\, days}=1.94\times 10^{5}sandwiches/day$$

This new number is the number of sandwiches that are brought outside every day. They don't last forever, though; let's say we eat our sandwich in half an hour, on average. So, the average number of sandwiches that are outside at any given moment, on average, is:

$$1.94\times 10^{5}sandwiches/day\times 0.5hours\times \frac{1\, day}{24\, hours}=4.13\times 10^{3}sandwiches$$

So, there are about four thousand sandwiches exposed to the sky, on average. (This number is probably a bit high; people don't picnic that much, and egg sandwiches aren't that popular.)

Now, each of these four thousand sandwiches measures so the total sandwich-area-coverage is .

Finally, the possible area in which we could drop the ball bearing is the size of the surface of the earth, or

$$\frac{4}{3}\pi r^{2}=\frac{4}{3}\times 3.14\times \left( 6.3\times 10^{6}m\right) ^{2}=1.66\times 10^{14}m^{2}$$

and the net probablility of the ball bearing hitting the egg sandwich is thus

$$\frac{Sandwich\, Area}{Area\, of\, Earth}=\frac{93\, m^{2}}{1.66\times 10^{14}\, m^{2}}=5.6\times 10^{-13}$$

This calculation implicitly assumes that the 747 in question could be flying anywhere on earth. This is not strictly a good assumption, since we know that 747 flights tend to be flying between cities mainly in North America and Europe. Coincidentally, these same areas also tend, for cultural reasons, to be the largest consumers of egg sandwiches. This correlation may actually increase the eg-bb (egg sandwich - ball bearing) interaction rate. This effect, however can be ignored to first order, and may help mitigate the optimism earlier in choosing egg consumption rates.