The question of the shadow of the airplane (or the airship), raised last week by a book by Yakov Perelman, has sparked a long and intense debate (see comments in the previous installment). Let’s start at the end:
The “meta-problem” of the airplane-airship duality has to do with the fact that the prologue corresponds to a much later edition than the first, from the 1920s, when it was still common to see airships in the sky. It is assumed that Perelman later raised the issue with an airplane, in the prologue of an edition from the late 1930s, to update the book a bit. In any case, the airship is more suitable to raise the issue, because it flies (it flew: they still exist, but in a vestigial form) lower than an airplane and always moves horizontally, and it is also much larger, so it casts a considerable shadow. Unlike an airplane, which in many cases does not cast any shadow, and if it does cast one, it is much smaller than the aircraft (unless it flies very low). Let us see why:
It is true that the Sun is so far away (about 150 million kilometres) that we can consider its rays to be parallel, so we must rule out the “divergence effect” that enlarges shadows when illuminating something with a point light, such as a flashlight; but the Sun is very large (approximately 1,400,000 km in diameter): its distance from the Earth is equivalent to only about 100 solar diameters, so an observer on Earth forms with a solar diameter an isosceles triangle similar to one with a base of 1 cm and a height of 1 metre; a very narrow and elongated triangle, but recognisable to the naked eye and not at all irrelevant. Objects receive light from the Sun from their entire surface, and so form a “shadow cone”. That is why the Moon, during a solar eclipse, projects on the Earth a shadow of a few hundred kilometres wide at most, when its diameter is about 3,500 km. And that’s why the shadow of the airship will be smaller than that of the aircraft, and the shadow of the plane will probably not even form.
To verify this phenomenon on a small scale, as Ramón Jaraba points out, all you have to do is throw a tennis ball upwards on a sunny day: its shadow decreases in size, until it disappears completely, as the ball rises.
Famous problems
One of Yakov Perelman’s many contributions to recreational mathematics was to popularize some classical problems and/or problems linked to great figures of thought, such as Newton, Tolstoy or Einstein. In his book Recreational AlgebraPerelman mentions, among other illustrious problems, one that was apparently Tolstoy’s favorite, in which the author of War and Peace’s fondness for mathematics is combined with his interest in agricultural planning:
An artel (voluntary association of workers) of reapers has to mow two fields, one of which is twice as large as the other. For half a day all the reapers work on the large field, and after lunch half of them remain on the large field and the other half work on the small one. In the afternoon they finish mowing both fields almost completely, except for a small section of the small field, the mowing of which occupies only one reaper for the whole of the following day. How many reapers were there in the artel?
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#Tolstoy #diligent #reapers