Thursday, September 25, 2014

As the wolf runs or the crow flies: Gandalf estimates distances to Moria

In the Lord of the Rings we read the following lines:
"There was a door south-west of Caradhras, some fifteen miles as the crow flies, and maybe twenty as the wolf runs," answered Gandalf grimly.  (Lord of the Rings, p. 298)
Distances measured by the flight of crows is the shortest distance between two points.  If the two points are in a line of sight, then the shortest distance is a straight line.  If the two points are in the ocean and not in the line of sight, then the shortest distance is a geodesic arc of the great circle of the Earth, assuming, of course, that the Earth is spherical.  Distances measured by the flight of crows assume that the air is not turbulent but calm--an assumption that may not be valid because of the winds, but suffices to make the point.

Distances measured as the wolf runs is still a geodesic, but the "geo" here refers now to the land--sand, grass, woods, hills, and mountains.  The wolf will try to find the path that makes its travel time from one point to another the least.  In this sense, the wolf will try to apply Fermat's Principle of Least Time.

As a rule of thumb, distances measured as the wolf runs is longer than those measured as the crow flies.

Now, we ask: who arrives first at the Mines of Moria? Is it the crow that flies15 miles or the wolf that runs 20 miles?  The flight speed of crows is about 30 to 60 miles per hour, so the crow will reach Moria in 15 to 30 minutes.  On the other hand, since the running speed of wolves is about 8 to 35 miles, the wolf will reach its Moria away in 34 minutes to 2.5 hours.  So it will be the crow that flies and not the wolf that runs that will reach Moriah first.

If Gandalf and company travel the 20 miles like a wolf and the average walking speed of humans is about 3 miles per hour (assuming that halflings are nimble enough to keep pace), then they will reach the mines of Moriah after 6.3 hours.  The wolf's howl can be heard as far as 6 miles.  So if the company has a 6 mile advantage over  wolves, then the wolves need to run 26 miles or  44 min to 3.25 hours.  But hearing a sound is equivalent to measuring distances as the crow flies. If we use Gandalf's 15:20 or 3:4 rule for crow to wolf distances, then  the 6 miles advantage of Gandalf and company may translate to 8 miles, so that the wolves have to travel 28 miles.  This means that the wolves will reach the Moriah in 48 minutes to 3.5 hours, which is shorter than the 6.3 hours that the company needs to take.  Well, Gandalf and company were really overtaken by wolves at night:
The night was old, and westward the waning moon was setting, gleaming fitfully through the breaking clouds.  Suddenly Frodo started from sleep.  Without warning a storm of howls broke out fierce and wild all abou the camp.  A great host of Wargs had gathered silently and was now attacking them from every side at once. (p. 299)
Roommates RMK2160GM The Hobbit Gandalf Giant Peel and Stick Wall Decals
Roommates RMK2160GM The Hobbit Gandalf Giant Peel and Stick Wall Decals
YOU SHALL BE THE FELLOWSHIP OF THE RING... LOTR Master Works Lithograph Art Print
YOU SHALL BE THE FELLOWSHIP OF THE RING... LOTR Master Works Lithograph Art Print
Rubie's Costume Men's Lord Of The Rings Deluxe Adult Uruk-Hai Mask, Multicolor, One Size
Rubie's Costume Men's Lord Of The Rings Deluxe Adult Uruk-Hai Mask, Multicolor, One Size