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        riddles >> easy >> Shadow of a 747
        
(Message started by: Speaker on Nov 29^th, 2002, 1:17am)

Title: Shadow of a 747
Post by Speaker on Nov 29^th, 2002, 1:17am

A Boeing 747-300 measures 59.6 m from wingtip to wingtip, and 70.6 m from stem to stern. The aircraft stands 19.3 meters tall when on the ground.

How large is the shadow cast by the plane at high noon on the fourth of July while it is at rest on the runway?

Is the shadow cast on to a level plain by this plane while it is flying at its normal cruising speed of Mach 0.85 at an altitude of 35,000 feet larger or smaller than when it is at rest on the runway? (For calculation purposes, assume its a plane:)

What is the degree of change in the size of the shadow?

Title: Re: Shadow of a 747
Post by Kozo Morimoto on Nov 29^th, 2002, 5:34am

I suppose 4July question depends on where on Earth the runway is. If it is at the south pole, it would be winter, and 4July is pretty close to the winter solstice, so the Sun would be pretty close to the horizon at midday creating a long stretched out shadow. But if you are around the Tropic of Cancer, the Sun would be pretty much directly overhead, which would mean the shadow would be pretty much the same size as the plane.

I'm not sure on the flying plane shadow... I presume that since the moon casts a shadow that is smaller in size than itself on the Earth's surface during a Solar Eclipse and that the Earth casts a shadow smaller than itself during a Lunar Eclipse, that the plane would cast a shadow smaller than itself while above the surface of Earth.

Title: Re: Shadow of a 747
Post by wowbagger on Nov 29^th, 2002, 8:15am

on 11/29/02 at 05:34:37, Kozo Morimoto wrote:

I doubt that you could see any shadow (cast by the sun) near the south pole on the fourth of July. Why should the sun be above the horizon in the midst of polar winter?

Title: Re: Shadow of a 747
Post by Icarus on Nov 29^th, 2002, 11:20am

This question also depends on how you define the shadow. Is a particular point in shadow if the sun is partially occluded, or do you demand it to be fully occluded? In the former case, the closer to the sun you are, the larger the shadow is. In the latter case, for objects so much smaller than the sun, the closer they get, the smaller the shadow is. At 35000 ft, the plane does not cast a fully occluded shadow at all.

Title: Re: Shadow of a 747
Post by Chronos on Dec 1^st, 2002, 10:35pm

If we assume that the Sun is a point (so the umbra and penumbra are identical, a lousy assumption in the real world), and we're only interested in the area of the projection of the shadow, not the volume, and we're neglecting the curvature of the Earth over the horizontal distance from the plane to its shadow on the ground, and we're considering cruising altitude to be a negligible proportion of the distance to the Sun, then they will be the exact same size and shape. Relax the first assumption, and the penumbra gets huge and the umbra disappears. Relax the second assumption, and the volume is much larger for the flying plane. Relax the third assumption while keeping the other three, and the answer will depend on the exact shape of the plane, not just its extreme dimensions. Relax the fourth assumption while keeping the other three, and the shadow will be larger when it's closer to the Sun.