If you spend much time around railroad modelers, they tend to talk a lot about scale, and particularly in reference to speed.
Being a physicist, I've put some thought into scaling things, and don't agree with the reasoning for how to derive scale speed. But that is for Part 2: Time.
In this part, I want to cover some basics about pure spacial scaling, and how standard layouts and the real world already agree very poorly.
Length is very important. But how big is a model layout in the realworld? How long would a real North American train be in a scaled layout?
Prototypicality
Train Length
Let's answer the latter question first. In North America, trains are often 100+ cars long, resulting in lengths of around a mile long. Given the most favored modeling scale, HO, which has a 1:87.1 scale factor, 1 mile is a hair under 61 feet long!
This means that a scale model railroad of North American trains needs 60 feet just for a single train!
To put this in perspective, an 8'x4' layout, which is fairly common to find in spare bedrooms, only gives about 20' on an oval track around the edge, which is only about 1/3rd of a mile!
So how does this scale relate to the real world?
The Shopping Mall
Doing even a small real life layout would take a space about the size of a shopping mall. Indeed, let's look at a mall of about average mid-sized town length. Spokane Valley, WA, and Santa Rosa, CA, each serve communities of around 1/4 million, and each have a mall that supports three department stores, and are of a length-wise construction. In each case, if you ran an HO scale layout from the far end of one department store, through the main corridor of shops, and to the far end of the opposite department store, you'd have about 1400ft in which to model. Translate this to miles, and you get very nearly only 24 scale miles of layout! This is smaller than many steam excursion branch railway lines!
Trees Are How Tall?
While this puts a lot into perspective, I do find the most interesting case--and the one that will actually surprise a lot of modellers--to not be a geographic distance, but instead the scale modeling of an evergreen tree.
The Northern California coast is filled with dense redwood forests which were, and still are, logged of their trees. These trees, the tallest in the world, can grow over 300 feet tall and become over 26 feet in diameter, and in the time in which it was logged, it was common to find trees 200 feet tall and 18 feet in diameter.
Thus, if you were modeling a logging railroad, you'd need redwood trees that are nearly 3.5 feet tall whose trunks are made from dowels about 2.5" in diameter! I dare you to build a layout with trees that large and not have all your friends call it absurd.
So how big are those 6" tall model evergreens you probably have on your layout? A measly 44 feet, which is about 1/5 the maximum height of a Pacific Coast Ponderosa Pine tree!
Curves
Lengths are all fine and dandy, but something that modelers tend to pay less attention to is how tight those curves really are. This is compounded by the fact that track manufacturers tend to sell pre-molded curve pieces that are sizes like 15", 18" and 24".
But Just How tight are those curves? To answer that, we must first learn how curves are measured.
Surveying Curves
Railroads survey their curves by using what is called the "degree" of the curve. According to an excellent article by Robers S. McGonigal in TRAINS Magazine, this is done by measuring the change in heading of a piece of curved track that is 100ft in length. Expressed mathematically, one can find that the radius of a curve in feet can be found by dividing 5729 by the degree of a curve.
Furthermore, the article goes on to explain that for a typical mainline, curves are limited to only 1-2 degrees, but that mountainous territory dictates curves of around 5-10 degrees. Furthermore, the limit for a four-axle diesel with rolling stock is about 20 degrees and the locomotive by itself can handle curves up to about 40 degrees.
Scaling Curves Down
So how do these real world curves relate to an HO scale model?
As modelers tend to like the beauty of mountainous scenery, let's start with those 5-10 degree curves. In the real world, these correspond to a curve radius of about 570-1150 feet. Scaled down, these correspond to a relatively giant 79"-158" curve radii!
But if those curves are so tight, how tight are those model curves that the track manufacturers make?
As it turns out, the standard 18" curve is already nearly a 44 degree, 130 scale foot curve, which is already in excess of the maximum for a four-axle engine with out any rolling stock!
So what are the limits? For a 20 degree curve (287ft radius), you do not want to have smaller than a 39.5" curve radius, and for a 40 degree curve (144ft radius), you do not want to have smaller than a 20" curve radius
Indeed, as a rule of thumb, an HO scale modeler can divide 800" by the degree of curve you wish to obtain, and you will obtain a pretty good estimate of the necessary curve radius you need for your model. Equivalently, you can divide 800" by the radius of a curve on your layout and get a fairly good estimate of the curve degree.
Of course, in real life, if you plan on having speed, you'll want to model those curve radiuses of only 1-2 degrees, which correspond to around 1/2-1 mile. But most people don't have the room to construct a prototpyical curve in their layout, as a 30'-60' curve radius is bigger than pretty much any layout.
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