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Flying to the Moon

If you follows the suggestion of Francis Godwin in 1638 and harnesses some swans to a chair and directs them toward the Moon, you might or might not reach it. If the swans always travel toward the current position of the Moon, at constant speed, then the swans will fail to reach it unless their speed is at least as great as that of the Moon. In this Demonstration, the basic unit is 1 lunar distance and the swans start at the surface of the Earth, whose radius is 0.0167. The Moon is assumed to travel in a circle, with an orbital period (with respect to the fixed stars, not the Sun) of 27.3 days. In the event of failure, the swans continue to orbit around the dashed red path.

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DETAILS

The differential equation for the motion of the swans is easily set up in vector form as , where is the speed of the swans and is the position of the Moon. This equation states that the swans' velocity vector has constant length and is always directed toward the Moon. Note that the gravitational attraction of the Moon and Earth are ignored. If the swans' speed is not greater than that of the Moon, then the swans approach a circular path, shown as a dashed red curve (see second and third snapshots). This example is discussed in more detail in [1].
Reference
[1] A. Simoson, "Pursuit Curves for the Man in the Moone," College Mathematics Journal, 38(5), 2007 pp. 330–338.
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