Professor Olsen,
I’ve been doing some thinking about the race to the Ford at the end of Book 1, and I’ve done some graphing and measuring to try and explain the Ringwraiths’ movements when trying to catch Frodo. I’ve tried to do this to scale, which has lead to some interesting/nonstandard units of measurement. I've attached a pdf with both a Map-view representing the movements of the Riders and Frodo as well as a Graph-view representing the Riders' distance from Frodo over time. As I wanted to focus on the movements of the non-witch-king groups, the Witch King starts off the map in the map-view.
First, start with the facts we know:
Second, allow me to make some assumptions; for the purpose of ‘easy’ math:
Third, I think we need to think about the Riders’ intentions:
I had always assumed that the Riders intended to grab Frodo and take him to Mordor and that they were trying to come into physical contact with Frodo; however, I no longer think this is true (as we've discussed). The effect of the Black Riders is proximity-based: which is to say that the closer a Rider is, the stronger its effect. Also, the effect of the Black Riders is cumulative: which is to say that the more Riders there are, the stronger their effect. And, the effect of the Black Riders takes time: which is to say that the longer you are exposed to a Rider, the more potent the effect (this is why they take their time invading Crickhollow, unaware that Fredegar Bolger has already fled).
They need to dominate Frodo’s will through fear. This does not require physical contact with Frodo but is more effective at shorter distances and over longer periods of time.
I’ve drawn two views of this: A map-view and a graph-view. Let’s examine the movement patterns:
The Witch King: The Witch King (and posse) is represented as a ‘w’ in the views. He starts 24gs (½ mile) from Frodo. He gallops in a straight line and is unable to get closer to Frodo. His line on the graph is a straight line, always 24gs (½ mile) from Frodo. The Witch King’s group ensures that Frodo rides straight for the Ford.
Toward Ford: The two Riders who ‘galloped madly towards the Ford to cut off his escape’ are represented as a ‘+’ in the views. This group starts about 20gs (0.42 miles) from Frodo. They gallop in a straight line and close the gap between themselves and Frodo at a constant rate, reaching the Ford at the same time as Frodo. Their line on the graph is also a straight line, descending diagonally from 20gs (0.42 miles) to 0gs over the course of the 24ut (45 seconds). The goal of this group is to maximize proximity. This group, by the end of the chase, is the closest to Frodo.
Toward Frodo: The two Riders who ‘rode towards Frodo’ are represented as an ‘x’ in the views. This group starts with the Toward Ford group, about 20gs (0.42 miles) from Frodo. They continuously readjust their course to gallop toward Frodo. They do not gallop in a straight line, rather they gallop in an arc. This arc does not put them in the best position to intercept Frodo: the ‘Toward Ford’ group is already taking that path. Instead, the path this group takes is the past optimized for getting as close to Frodo as possible as quickly as they can, even though they don’t have a chance of physically catching him. On the graph, their path becomes a parabolic arc that steadily approaches 6gs (660 ft) from Frodo. The goal of this group is to maximize time. For the first ¾ of the ‘race’ (approximately the first 30 seconds) they are consistently closer to Frodo than the ‘Toward Ford’ group which allows them to have more influence over him during that time. This group gets the closest to Frodo early in the race then begins to fall behind, which facilitates him being under more influence for longer.
It seems to me that this would be an effective tactic for the Ringwraiths to take when setting up an ambush like this. One group herds the target toward the ambush point, another group attempts to intercept the fleeing target (it is worth pausing to remember that if Asfaloth had not been faster than the black horses the Riders would have reached the Ford before Frodo. I think we can fairly safely assume that, with the exception of elf-horses like Asfaloth, Sauron wants his Riders to be riding the fastest horses around; so it isn’t unreasonable to think that the Riders assumed they would be successful in cutting off Frodo’s escape), and one group races toward the target to attempt to get it under their influence as quickly as possible.
Yours asynchronously,
John O’Connor
I’ve been doing some thinking about the race to the Ford at the end of Book 1, and I’ve done some graphing and measuring to try and explain the Ringwraiths’ movements when trying to catch Frodo. I’ve tried to do this to scale, which has lead to some interesting/nonstandard units of measurement. I've attached a pdf with both a Map-view representing the movements of the Riders and Frodo as well as a Graph-view representing the Riders' distance from Frodo over time. As I wanted to focus on the movements of the non-witch-king groups, the Witch King starts off the map in the map-view.
First, start with the facts we know:
- Frodo has a head start; he is about halfway between the ‘Gate in the Trees’ and the Ford.
- The distance from the Gate to the Ford is roughly 1 mile.
- Frodo starts about ½ mile from the Ford
- The Witch King & the four Riders with him start from the Gate.
- The Witch King starts 1 mile from the Ford
- Four more Riders start from an indeterminate point off to the side, somewhere between the Gate and the Ford.
- Frodo travels in a straight line to the Ford.
- The Witch King follows the same straight line toward the Ford.
- The Four additional Riders split up as follows:
- ‘Two rode towards Frodo…’
- ‘two galloped madly towards the Ford…’
- The Two galloping toward the Ford arrive at roughly the same time as Frodo.
Second, allow me to make some assumptions; for the purpose of ‘easy’ math:
- All of the horses travel at the same speed.
- This is demonstrably false as we are told Asfaloth is much faster than the black horses. However, the principals I am attempting to demonstrate will hold true no matter how fast the horses can move, but the math gets harder if the horses move at different speeds.
- Also, the Ringwraiths probably do not already know Asfaloth is faster than them. The Ringrwraiths, I believe, would probably assume their steeds are either the same speed as (or faster than) any opponent horses.
- On graph paper, a horse can traverse one grid-square in one unit-of-time.
- I’m using the scale 1 foot = 1 mile
- My graph paper has ¼-inch x ¼-inch squares
- Technically, this means 1 grid-square = .021 miles or 110 feet
- For ease of language, I'll be discussing in grid-squares (gs), and translating as best I can to miles or feet.
- IE: Frodo starts 24gs (½ mile) from the Ford.
- A horse can run a mile in 90 seconds (citing Wikipedia as a source here…)
- I’ve marked off 24 units-of-time on the graph.
- Technically, one unit of time is therefore equal to approximately 1.875 seconds
- For ease of language, I'll be discussing in units-of-time (ut), and translating as best I can to more standard units like seconds.
- IE: Frodo covers 1gs (110 feet) in 1ut (1.875 sec), therefore travelling at 1 gs/ut (about 40 mph).
- Remember, these are estimated speeds…
- IE: Frodo covers 1gs (110 feet) in 1ut (1.875 sec), therefore travelling at 1 gs/ut (about 40 mph).
- (This complicated math/unit-of-measurement is because I drew the picture first, then figured out the proportions later.)
- With these assumptions, the two Riders galloping toward the Ford also start about ½ mile from the Ford so that they will reach the Ford at approximately the same time as Frodo.
Third, I think we need to think about the Riders’ intentions:
I had always assumed that the Riders intended to grab Frodo and take him to Mordor and that they were trying to come into physical contact with Frodo; however, I no longer think this is true (as we've discussed). The effect of the Black Riders is proximity-based: which is to say that the closer a Rider is, the stronger its effect. Also, the effect of the Black Riders is cumulative: which is to say that the more Riders there are, the stronger their effect. And, the effect of the Black Riders takes time: which is to say that the longer you are exposed to a Rider, the more potent the effect (this is why they take their time invading Crickhollow, unaware that Fredegar Bolger has already fled).
They need to dominate Frodo’s will through fear. This does not require physical contact with Frodo but is more effective at shorter distances and over longer periods of time.
I’ve drawn two views of this: A map-view and a graph-view. Let’s examine the movement patterns:
The Witch King: The Witch King (and posse) is represented as a ‘w’ in the views. He starts 24gs (½ mile) from Frodo. He gallops in a straight line and is unable to get closer to Frodo. His line on the graph is a straight line, always 24gs (½ mile) from Frodo. The Witch King’s group ensures that Frodo rides straight for the Ford.
Toward Ford: The two Riders who ‘galloped madly towards the Ford to cut off his escape’ are represented as a ‘+’ in the views. This group starts about 20gs (0.42 miles) from Frodo. They gallop in a straight line and close the gap between themselves and Frodo at a constant rate, reaching the Ford at the same time as Frodo. Their line on the graph is also a straight line, descending diagonally from 20gs (0.42 miles) to 0gs over the course of the 24ut (45 seconds). The goal of this group is to maximize proximity. This group, by the end of the chase, is the closest to Frodo.
Toward Frodo: The two Riders who ‘rode towards Frodo’ are represented as an ‘x’ in the views. This group starts with the Toward Ford group, about 20gs (0.42 miles) from Frodo. They continuously readjust their course to gallop toward Frodo. They do not gallop in a straight line, rather they gallop in an arc. This arc does not put them in the best position to intercept Frodo: the ‘Toward Ford’ group is already taking that path. Instead, the path this group takes is the past optimized for getting as close to Frodo as possible as quickly as they can, even though they don’t have a chance of physically catching him. On the graph, their path becomes a parabolic arc that steadily approaches 6gs (660 ft) from Frodo. The goal of this group is to maximize time. For the first ¾ of the ‘race’ (approximately the first 30 seconds) they are consistently closer to Frodo than the ‘Toward Ford’ group which allows them to have more influence over him during that time. This group gets the closest to Frodo early in the race then begins to fall behind, which facilitates him being under more influence for longer.
It seems to me that this would be an effective tactic for the Ringwraiths to take when setting up an ambush like this. One group herds the target toward the ambush point, another group attempts to intercept the fleeing target (it is worth pausing to remember that if Asfaloth had not been faster than the black horses the Riders would have reached the Ford before Frodo. I think we can fairly safely assume that, with the exception of elf-horses like Asfaloth, Sauron wants his Riders to be riding the fastest horses around; so it isn’t unreasonable to think that the Riders assumed they would be successful in cutting off Frodo’s escape), and one group races toward the target to attempt to get it under their influence as quickly as possible.
Yours asynchronously,
John O’Connor