Montgomery’s Gedanken Experiment – at the molecular level – illustrating that a smaller body elastically rebounding between two much larger bodies can actually impart large (but opposite) momenta to them while still conserving the energy and momentum of the three-body system.
Gedanken Experiment: A thought experiment that considers some hypothesis, theory, or principle for the purpose of thinking through its consequences. Given the structure of the experiment, it may or may not be possible to actually perform it, and if it can be performed, there need be no intention of any kind to actually perform the experiment in question. (Wikipedia)
Gedanken Experiment: A thought experiment that considers some hypothesis, theory, or principle for the purpose of thinking through its consequences. Given the structure of the experiment, it may or may not be possible to actually perform it, and if it can be performed, there need be no intention of any kind to actually perform the experiment in question. (Wikipedia)
John Montgomery, “The Aeroplane: A Scientific Study” (1905):
"My course of study and experiment has forced me to conclude that flight is based upon unrecognized mechanical principles, arising directly from the impact of elastic bodies; and in its essence is the exchange of momenta between two bodies by the action of an elastic medium. …
"If two large elastic bodies (A and B) without motion are suspended in space (thus being free to follow any impulse) and a third, but smaller body, between them is given an impulse towards one of the larger (A), it will, at the instant of impact, transmit all its motion to A, and rebounding from it, exert a reaction due to its recoil. And traveling back towards the other large body, B, it will repeat the same process. And thus, the continued rebounding of the smaller body between the two larger ones will gradually give them movements in opposite directions. In other words they will exchange their momenta. An interesting element in this consideration is that either of the larger bodies may, after a time, have a greater momentum than was given originally to the smaller body. Yet if we take the algebraic sum of the momenta of the three bodies, we find it is always equal to the original impulse."
"My course of study and experiment has forced me to conclude that flight is based upon unrecognized mechanical principles, arising directly from the impact of elastic bodies; and in its essence is the exchange of momenta between two bodies by the action of an elastic medium. …
"If two large elastic bodies (A and B) without motion are suspended in space (thus being free to follow any impulse) and a third, but smaller body, between them is given an impulse towards one of the larger (A), it will, at the instant of impact, transmit all its motion to A, and rebounding from it, exert a reaction due to its recoil. And traveling back towards the other large body, B, it will repeat the same process. And thus, the continued rebounding of the smaller body between the two larger ones will gradually give them movements in opposite directions. In other words they will exchange their momenta. An interesting element in this consideration is that either of the larger bodies may, after a time, have a greater momentum than was given originally to the smaller body. Yet if we take the algebraic sum of the momenta of the three bodies, we find it is always equal to the original impulse."
We developed a simulation of a small body (representing an air molecule) elastically rebounding back and forth between two much larger bodies (representing a bird and the surrounding air treated as a single object). Our analysis reveals that the small body can indeed impart large momenta to the two much larger bodies while conserving the energy and momentum of the three-body system.
We also discovered that, as the masses of the larger bodies are increased, the momentum transferred to them by the small body is also increased. That is quite a revelation! No wonder this was difficult for some people to accept or comprehend; it certainly seems counter-intuitive.
The following animations will make this clear.
We also discovered that, as the masses of the larger bodies are increased, the momentum transferred to them by the small body is also increased. That is quite a revelation! No wonder this was difficult for some people to accept or comprehend; it certainly seems counter-intuitive.
The following animations will make this clear.
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View Animations:
small medium large Masses = 1 10,000 10,000 Masses = 1 1,000 10,000 Masses = 1 100 10,000 Masses = 1 10 10,000 Masses = 1 1 10,000 Animations created by Charles P. Plum & Bernard J. Burdick |
View MATLAB® Simulation
Simulation performed by William H. Schoendorf
Another example of momentum transfer:
Stacked Ball Drop
Simulation performed by William H. Schoendorf
Another example of momentum transfer:
Stacked Ball Drop
Analysis
The average momentum transferred to the large bodies increases with the size of the large bodies and the number of collisions between the small body and the large bodies (see figures below). One might expect (incorrectly) that, as the large bodies become more massive, the small body (being less massive) should have less influence on them. Quite the contrary: as the large bodies become more massive, they are not as easily moved away from the small body, giving it more time to rebound back and forth between them, influencing them more.
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John J. Montgomery, "The Aeroplane, A Scientific Study." Redwood Magazine, Volume IV, issues of April and May 1905:
"An interesting element in this consideration is that either of the larger bodies may, after a time, have a greater momentum than was given originally to the smaller body. Yet if we take the algebraic sum of the momenta of the three bodies, we find it is always equal to the original impulse.
"The three bodies constitute a system and any motion imparted to one is given to the system; and while this may result in opposite movements of the parts, the total movement in the whole system cannot exceed that originally imparted to it. These statements are made because the soaring of a bird has remained a mystery, so profound, that one of the foremost in scientific investigation has said “it seems to be in defiance of all known laws of nature," and this is so, because of a too restricted comprehension of the phenomenon presented. Most, if not all, have considered a bird's wings as being surfaces suitably formed and adjusted to receive a pressure from a stream of moving air. While this is true in a certain limited sense, it is very far from being a proper statement of the case.
"The phenomenon, in its essence, is only a development of the problems just stated, viz., the exchange of momenta between two bodies by the action of an elastic medium between them; the wings being of such formation and adjustment as to induce a series of impacts of the adjacent air, between their surface and the surrounding air; in consequence of which the bird and the air receive equal and opposite momenta. And hence we must conceive that the bird is not in itself a complete working device, but only one element of a system: the other elements of the system being the great mass of the surrounding air and the moving particles of that adjacent."
