Essay: Scale Railroading (Part 2: Time)

Model railroaders frequently like to talk about scale speeds, and often quote the scale speeds at which their models run. However, it is very clear there are no physicists amongst them, because they way they scale speed is incorrect by most measures.

How It's Done

Let me back-up. I should first explain how modelers currently do it, and the reason why they do it the way they do.

To do this, I need to do a little math. Bare with me, because I don't need a lot of math, but for this article to be meaningful to anyone, a little bit of simple algebra needs to be invoked.

Speed is calculated by the equation
    v = d/t,
where d is the distance travelled and t is the time it took.

Additionally, we will introuce an equation to take a real distance d and change it into a scale distance d' using the scale factor m
    d' = m*d.

Now if I want to calculate the scale speed v'--according to the vast majority of modelers--I can do it by cleverly combining these too formulas. I must first observe that I can change out the scalable quantities in the velocity equation to produce
    v' = d'/t,
and then combine that with the scaled distance equation to produce the relationship
    v' = m*v.

For those of you that love examples, an HO scale railroad would give m a value of 1/87.1, or approximately 0.0115. Thus a track speed of 60 mph would be equivlent to very nearly 0.7 mph, which is 1 ft/sec.

What About Time?

Of course, the question we should ask ourselves, is why do we scale both the velocity and the distance in the above equation, but not time? Is there a fundamental reason why time should not be scaled?

To most, the reason is intuitive. It just seems absurd. And if you were to naively scale time in the same way we scale distance, you'd be right.

Going back to the ugly math, let us write the velocity equation with all the coordinates scaled, and then scale both distance and time by the factor m. Doing so produces the equation for scaled velocity as     v' = d'/t', and after relating it back to the original velocity through the scaling equations, d'=m*d and t'=m*t, we get
    v' = v,
which is, of course, an absurd result!

Imagine having to run your model trains at an actual 60mph around the track! Your very expensive train, if it could even produce that much speed, would go flying off the track and probably injure someone in the process!

But this is not the end of the story for time scaling; Indeed, this is just the beginning.

One Man's Scale Clock

One of the justifications one hears for measuring scaled time the way it does, is because our concept of time is so thoroughly rooted in our own wrist watches. Indeed, when timing out the distance a model train goes, we run off to our wrist watches, happily counting off non-scaled seconds, and never even give it a second thought.

Human cognitive function is so rooted in this absolute concept of time, that the theory or relativity (which dictates that time runs at seemingly contradictorily different rates depending on where you are) is still impossible for most scientists to intuitively work with.

That being said, let's devise an interesting thought experiment that messes with our concept of a uniform clock.

In the real world, I set up a simple pendulum clock next to the train tracks, and seeing how far the next locomotive to pas the clock gets in one cycle of the pendulum.

Now, if we made a scale model of the simple pendulum clock, and put it next to our model train tracks, how fast (in the real world) would we need to run that model train to cover the scaled distance the real train traveled in one cycle of the pendulum?

In other words, given the scaled distance and the scaled concept of time from the ticking pendulum, how does the speed scale?

To solve this problem, one must (once again) invoke some math. More specifically, the first step is to find the equation that relates the length of one tick of the pendulum, T, with the length of the pendulum L. A little looking around the web will show you that the period of a pendulum's swing is given by
    T = 2*pi*sqrt(L/g),
where g is the acceleration of gravity, which is a constant at any position on the Earth. Combining that with the speed equation, we find that the speed of a the train is related to the pendulum's length via
    v = (d/(2*pi)) * sqrt(g/L).

If one scales the distance L by the factor m, and uses the techniques presented above, it is very easy to show the surprising result that the scaled velocity v' is related to the real velocity v by the relationship:     v' = v * sqrt(m).

Surprise!

So what does that mean?

Think about it for a moment. Let's say, in the real world, we have a clock with a pendulum that swings a complete cycle once every second. This allows a train passing at 60mph to travel 88ft during that one second. Now if we made an HO scale model of this clock, and placed it next to our model train, we'd have to run the train at about 9.43 ft/sec in order to cover a scaled 88 ft distance (12 1/8 inches) in one tick of the pendulum! That is nearly 10 times faster than the speed of 1 ft/sec we calculated using the classic relationship.

Of course, the reason for this is that the clock itself runs faster. Indeed, a pendulum that is shorter by a factor of 100 will run at a rate that is 10 times faster than that of the longer pendulum.

Thus, in this example at least, it is clear that we have scaled not just distance, but also time.

Superelevation

Of course, I hear you say, this is just one example. One example that could have been carefully constructed to give the desired result. And I'm happy to hear that skepticism, because that is what real science is all about.

So let's try another situation entirely. Train curves are superelevated, just like roads, in order to make corners smoother. Ideally, this is done by finding the elevation angle such that the rails tilt the coach in a way that exactly balances the forces so that they pull straight down through the bottom of the coach, causing you to feel no lateral forces at all (and keeping the tea in your cup from spilling).

So the question is, if we built a scale model of a real world curve that is ideally superelevated, and we perfectly model that curve in our model train set, what speed would we have to run the model train at so that the forces are balanced for the model passengers with their model tea cups in the model coach?

Of course, to answer that, we need to know that the equation for relating the superelevation angle theta of the coach (and hence the rails it is on), to the curve's radius R and the design speed of the train around the curve v.

Fortunately, this is a problem that is solved in the first few months of class by every student of physics in the world, and the correct answer is always
   v^2 = g*R*tan(theta).

Applying the same techniques to this equations as the examples before it, we find something surprising: yet again, the relationship between the scale velocity v' and the real world velocity v, is given by
    v' = v * sqrt(m).

In other words, if a real world curve has its superelevation perfectly designed to a 60mph train, then an HO scale model train would have go around that curve at 9.43 ft/sec to keep the tea in the model cup from spilling while going around the corner. (This, of course, being a good deal faster than the 1 ft/sec scale speed that modelers would calculate!)

Welcome to the Movies

It's hard to argue with physics, and so far I've given two entirely unrelated physical situations in which I have no choice but to produce the same, very surprising result.

But, being the stubborn kind of people we are, it is hard to accept radically different ideas (indeed cognitive dissonance is a very fascinating field of psychology), and so I fear I must produce one more example before you actually believe what it is I have to say.

Imagine you are a special effects artist working for Robert Zemeckis and Bob Gale, and are in charge of the crash sequence at the end of Back to the Future III, where the beautiful steam engine drives off the cliff at 88mph, and plummets into the Earth below.

Obviously, Sierra Railway will not let you drive their prized engine number 3 off a cliff, so you must resort to models to do it!

Being the sort of person you are, you want the shot to look realistic. This requires you to satisfy two criteria:

1. You must find the correct speed to drive your model train off the cliff so that it follows the exact same parabolic path through the air as the real thing would, except scaled down according to the scaling equations we now know and love.
2. You must run the film through the camera at a faster rate so that, when slowed back down to the standard 24 fps of a theater projector, it looks like it took the same amount of time to hit the ground as the real thing, even though it actually hit much sooner due to the much shorter distance.

Now the math for this one is much too ugly to show here, but rest assured that over the last year I have calculated it several different times, several different ways, and always come to the same conclusions:

• Velociy scales by the same sqrt(m) factor as we found in all of the previous examples.
• Time scales by the sqrt(m) as well!

Time Does Scale

So we've determined that velocity scaled according to the square root of the scale factor, and that time also scales by the square root of the scale factor. These are not independent results! Indeed, following the lead of theories such as special relativity, the fundamental equations are best taken as atransformation of the basic coordinates; that is to say, the basic equations are:
    d' = m*d
    t' = t*sqrt(m)

One can then easily derive the velocity transormation equations by applying these transforms to the scaled velocity equations v'=d'/t' to get the velocity transform equation v'=v*sqrt(m).

But Why?

But why does time scale in this seemingly absurd way? It is very counter intuitive, even to me, that this would be the case.

As it turns out, the common thread is gravity. The pendulum uses gravity to drive it. Superelevation is determined as balancing gravity with centrifugal aceleration. And, of course, a projectile motion is the quintessential problem of gravity.

More specifically, gravity is scale invariant. No matter how you scale your model railraod, the gravitational acceleration roots you down to a very specific scale of motion. In other words, you have no choice but to accelerate a real 9.8 meters per second per second, no matter how you scale!

This scale invariance, then, gives a way for both the brain, and for measuring devices, to determine the scale factors involved, unless you change the rate of time!

So why does the rate of time change the way it does? The easiest way is to use a bit of differential calculus. But that is beyond the scope of this article. Instead, I will note that the constraint on how to scale time is imposed by the quadratic relationship between distance and time under the effects of gravity. More technically, since d is proportional to t^2, and we know that d scales according to d'=m*d, it necessarily constrains t to scale by t'=t*sqrt(m).

So Now Then

While modelers as a whole have determined what feels to a reasonable scaling methodology (space scales, but time does not). But to anyone serious about modeling the physics of trains correctly, it is important to realize that it isn't a matter of what feels right, but instead is determined by what is right.

And as much as everyone feels that time does not scale, the physics does not agree. Indeed, time must scale in order to replicate the balance of the physical forces involved.

Essay: Scale Railroading (Part 1: Space)

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.

Article: Egos Stop Innovation; How to Have a Discussion

Any academic professional, from a software architect to a physicists, is at their peak innovative performance when they can effectively communicate, discuss, and refine their ideas with others.

Unfortunately, it seems that a large number of people are more concerned with their own egos than with innovation, as evidenced by their inability to communicate with others. It seem that these people are always irrationally attacking ideas that are not their own while taking an emotional bias towards ideas that are their own.

This is a natural thing for people to do. It is in our blood. We evolved from the genes of the top-dog alpha-males and their mating successes.

But today should be different. The human race is now capable of attaining much greater heights if we work with others instead of against them.

Take Quantum Mechanics, for example. Quantum Mechanics was not the invention of a single mind quietly working away. No. It is the hard won innovation that resulted from many great minds working together to solve a common goal.

So how can we keep from being the jerk down the hall that no one wants to work with, and help to further the innovations of yourself and your company, making your managers happy and helping you to attain popularity, love, and wealth?

I Like Friends

Let's try to learn by example.

I had two friends that, through countless discussions and debates, showed me most of what I know about a successful exchange of ideas.

One of these friends was infuriating clam and methodical in his approach, but his goal was always to lead to a common understanding of the truths behind the material we discussed.

The other of these friends was irrational and stubborn, always hanging onto his idea no matter how well it could be proved false, and then would stomp off in a hissy fit whenever he was defeated.

What I learned from all this is contained in the ten rules below; but before I go there I wanted to follow my own rules and define two terms. These definitions aren't the dictionary definitions of these terms, but as long as you can understand my definition then you can follow what it is I'm trying to say.

Defining Talking

The way I see it, there are essentially two ways to exchange and refine ideas with others: discussion and debate.

I define "discussion" to be the friendly and logical open exchange of ideas, where the goal of everyone involved is to reach a new, common understanding of the material, knowing that this will most likely be different than any of the ideas brought to the table by anyone there.

On the otherhand, I define "debate" to be what happens when a discussion breaks down into egos and arguing, caused by even just one person to not want to budge from their flawed arguments, leading to an overall breakdown of the process of innovation.

The Ten Commandments

These rules take practice and hard work to follow, but following them is important not just to others, but to yourself as well.

One last thing before I start, though: I should note that rules 1-3 are mostly concerned with how to hold yourself, rules 4-7 are about communication, rules 8 and 9 are about arriving at a conclusion, and rule 10 is stating an obvious fact that people seem to forget about in the heat of a debate.

So without further ado, on with the show!

1.Be civil; always treat other people with respect and dignity.

People will only take you seriously if you treat them like an intelligent human being. If you let your frustration take over, you run the risk of insulting another person, causing them to close themselves from your point of view, destroying the whole process.

2. Place your ego aside; readily admit when you are wrong.

I doubt the knowledge you bring to a discussion is without flaws, inaccuracies, and other mistakes. Therefore you need to know and admit the limits of your knowledge. Admitting when you are wrong is probably the biggest and hardest step for people, but being ready to admit when your idea just isn't right is an important part of innovation. Put another way: don't let yourself look like an idiot by defending a lame-duck idea to the bitter end. People just won't ask for your input anymore because no one likes a self-centered, stubborn donkey!

3. Be open to new and different ideas; put yourself in the shoes of others.

Great thinkers are able to view things from many points of view other than their own. You do want to be like a great thinker, right? It is important, then, that you put yourself into the shoes of people presenting alternative (and usually contradictory) ideas and try hard to understand why they support that idea. This can help you either rebut their idea, accept their idea, or realize that there is no way to agree.

4. Make sure everyone agrees as to what the question really is.

As stupid as it seems, I have seen (and been in) many discussions or debates where each person thought a different question was trying to be answered! This, of course, causes much frustration. If it seems like the other person isn't understanding, try rephrasing what it is you are trying to find out, and see if they agree that that is the question at hand.

5. Define terms; be vigilant of disagreements caused by different definitions.

One of the funny things that often happens is that communication break downs can be the cause of many long discussions where everyone actually agreed the whole time. For example, I was in a debate with someone once where, after three hours, we found out that we were using slightly different definitions of the word "money". Once we hammered out a common definition, we suddenly found that we never disagreed on the real question at hand! This happens more often than one would think! So be vigilant of disagreements stemming from different definitions of terms and try to nip them in the bud.

6. Listen patiently and carefully to what others say.

This is really a two fold problem. One is that people get in such a hurry to say what is in their mind that they stop listening to what everyone else is saying and just want to blurt out their own thoughts. But listening turns out to be one of the most important skills in innovation. So don't be a jerk, listen up! The other part of this is that people naturally interpret, filter, and infer the words of others. It is important to pay attention to detail and make sure you understand what they mean and to ask questions when you don't understand.

7. Say what you mean.

There seems to be some sort of mangle-o-matic filter between the brain and the mouth. Be careful to say what you mean, try to make statements that don't leave anything to inference, and be willing to re-explain yourself in different terms if someone is confused as to what you meant. (Seems simple? It is harder than you'd think!)

8. Strive to reach the crux of any disagreement.

In order to reach resolution on a disagreement, it is important to find the crux of what it is, exactly, that you disagree on. It is no fun spending three hours hammering over a topic just to find that the crux of the disagreement lay in a misunderstanding of a word definition. Pealing away the layers to reveal the point of disagreement quickly will save everyone a lot of time, energy, frustration, and headache.

9. Discussions hinging on personal values are doomed to become debates.

Some discussions have no agreeable resolution. This is especially true of many socio-political discussions. When the crux of a disagreement hinges on a personal value or opinion, there is no way to agree. Whether it be a disagreement over something as stupid as the best flavor of ice-cream or the best band ever, or it be over deep issues such as abortion, gay marriage, and the validity of your own religion, there just isn't an answer that everyone can agree on. This doesn't mean that you can't understand and respect what the other person believes, but it means that you'll probably never agree, and so should agree to disagree.

10. Use logic, facts, and reasoning.

This should go without saying, but it doesn't seem to be the case. People cloud their reasoning with emotion. This is, once again, part of being human. But if you want to convince someone of the validity of a viewpoint, you must always support that with facts and logical reasoning, while being careful to avoid such traps as logical fallacy, inaccurate facts, and mis-representation of your knowledge limits. (But if you follow the other 9 rules, none of this should happen to you, right?)

Every Article Needs A Conclusion

So there they are, in all their glory. Some simple rules that take a lot of hard work to follow; but will quickly make you that innovative, team-playing, cool-guy, that everyone wants to have on their team and at their parties.