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)`.

`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:

- 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.
- 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.

## 5 comments:

There are a ton of things that don't work in any scaled model. The only things we can truly count on is the scale size and nothing more at least not on this world (I will go on into details later). I have been asked many times why real trains derail when going to fast around a corner of 4-8 degrees curve in the real world when in models they can sail around a 20 - 40 degree or more curve traveling at 5 times the speed. The answer you'll find in all the problems in the scaled world is our planets gravity (g). For everything to work like it should you would have to have a scaled world to build it on. Also if someone could also figure out how to manipulate gravity in an efficient way that could help too. We can't use scaled time because we live on a planet that has too much mass. If you scale built a HO planet you would find it would rotate the same scaled distance in our time factor to equal the scaled (g) of our own planet. I believe time would be the same on both if everything was truly scaled down. The only time you would want to scale time is if gravity in our world were to take over, like when falling off the table. But still time shouldn't be messed with and at this point the only logical thing to do is scale (g). Astronauts do this all the time when doing their physicals before entering space. They do this by fling in a jet in a negative g maneuver. Following are some examples of problems and facts in an HO world.

About Weight and LengthIf you have a SD40-2 @ 68 ft 10 in (21.0 m) over the coupler pulling faces when you scale it by a factor of 1/87th you will get a scaled size of 68 ft 10 in (21.0 m) over the coupler pulling faces. We also know that a mile is 5280ft, in the real world we can fit 76.7 SD40-2's all coupled up in both the HO and the real worlds mile. Now that we know these things match our real world 1:1 measurements we can move on and (g)& (t) never said a word of influence. When dealing with scaled speeds (g) is not a factor since we're just looking at how many feet we have traveled in the given time frame. We scaled the model down but we still live in the same big world, so time is a constant for us and the model due to our world's (g).

With (g) this measurement is constant in our scale 1:1 and the model scale being 1/87 with g = 9.8 m/s^2 you will find there is a weight problem. The problem here is if we measure this engine in scale weight on our planets (g) it would crush the rail it would sit on due to the fact it would now weigh not 368,000 lb (167,000 kg) but 3,201,600 lb (1,452,900 kg) scaled weight on our planet. With a scaled 110 lb or 130 lb rail it would crush it like it was a pop can. Now take that same engine and scale the planet it sits on to the same scaled (g), you would find the weight would be the same.

We have taken a model and built it on our planets mass and not on the scaled planets mass. Therefore to make this work we would have to scale (g) down to the scale we are modeling. Good luck with that one!

About pendulum clocks and time (The Tompion clocks at Greenwich)Richard Towneley and with the help from John Flamsteed's, the pendulum clock was only to prove to their own satisfaction that the Earth rotated at a constant speed. They didn't create it for the time piece we use it for today. To scale a pendulum clock and make it work correctly to the specs of Towneley. It would only work if the (g) in the HO model world was 1/87th of our own planet's (g).

If you were to put that same clock 1:1 on a different planet you would get a very different time due to gravity and mass and the rotation of that planet telling us (g) drives this clock. You could however calibrate the clock by converting its pendulum length and mass to match that of the new planets (g) so it would track time the same as it did on earth. Scaling the pendulum clock here on earth on a layout doesn't work as the earth is way to big for it to function the same way, in fact it will run faster by a factor of 8.7 times or maybe more. You would have to put that clock on a planet that was 1/87th scaled in everyway to make it work and at that point time would equal our time.

So really (g) could be described as the motor for the clock. We made the object it is designed to propel a big and heavy one so that the motor runs it at the speed we need it to. We didn't change the power of the motor to make it work, we changed what it is driving. Now when we scaled the clock down but forgot to scale the motor that powers it down you will find this will makes the clock run way too fast. To scale one thing down you must do it for all the components. We don't leave the 3200HP prime mover EMD 16-645-E3 motor in the scaled model train, doing so would cause us some big problems.

Falling off the tracks and why (g) is so importantTake a jet, put it in a negative 8.7 of gravity spin on it. The model now would fall off the table at the same rate scaled speed feet per second as the prototype would if it were to fall off a cliff. You also would find it derails a lot easier like the prototype when going around a curve.

When the train is on the track it is as close to 1/87th (scale) as we can get it here on earth. No one can afford to build a layout in a manipulated-gravity room to make it truly correct.

In the real world when it leaves the track and it falls off the table you then transfer that 1/87th scale model while on the rails to our world and at a rate of g = 9.8 m/s^2 = 32.2 ft/s^2. This is not of the scaled (g) for the model, its our worlds (g) and speed. If someone had the time they could do the math and figure what (g) would be in a 1/87th scaled world. I'm going to guess it would be in the neighborhood of g = 1.25 m/s^2 but remember this is a guess and not fact.

The strength of the gravitational field is numerically equal to the acceleration of objects under its influence, and its value at the objects surface. With 9.8 m/s (32.2ft/s or 22 mph) for each second of its descent we cannot use this equation for our 1/87th world cause (g) is not scaled down.

If you decrease (g) to the scale you will find that all the math will then work out. With the correct scaled down (g) model trains derail just like the real ones do when they're going too fast around a curve. They fall at the same scaled speed even in our time. And last they weight the same without crushing the rail at where they sit. By converting gravity you will find it fixes every problem in a scaled world. With the scaling of (t) time you can compensate (g) and the rate of the train falling like in Back to the Future III. It will even prove that the time speed increase of the same will cause the train to derail going around a 4-8 degree curve if speeding. It does not and will not make a change in the weight of the locomotive and with that your pendulum clock is still running to fast. (t) doesn't solve the problem but (g) does.

They slowed the film back down to our time because it didn't look realistic and if I remember correctly they filmed the layout section on a angle so the train wouldn't travel a ridicules distance once it fell off the end of the tracks. By doing this the added (g) due to the fact they added (t). Once it was slowed down it looked great. Just like a bullet you have speed over time equals fall rate or Spd/t = fr. When your object is moving faster it tends to fly a bit longer making its falling out of the sky rate too slow for the distants it just traveled. This proves once again we see gravity took place and if they had filmed it in a jet with the right -g they wouldn't have had to speed up the train to make it look right.

I so need to get to bed, great topic though!

Ross: That's a wonderfully written assessment that comes to (mostly) the exact same conclusions I did, except written in a more intuitive,and less mathematical way. (Something tells me I should re-write my blog article to follow some of your lead!) :)

Indeed, this morning I was thinking about the conclusion that should really be drawn from all this, now that I've had time to see other people's thoughts on the matter, and I think I came to two things:

1. Modelers should understand that there is

notone way to scale speed and/or time, but instead many different ways to do it based on the requirements at hand.2. A modeler should understand

whythey use the model that they do, andwhatthe problems are.I plan on supplementing and/or re-writing my blog entry to reflect these conclusions.

It also dawned on me why,

exactly, the standard transformation is used. I had the right idea before with it "feeling" right, but I think I understand more scientifically now.It is rooted in an interesting asymmetry in the way our brain works. We are able to perceive symbols. For example, a map. It is a symbolic miniature representation of a real place. Our brain has no problem with that. But with time, we are not. Conceiving time running at a different rate is something that our brain is not wired to do.

So when we are in our basements with our model trains, we have no trouble perceiving that the distance is, in an abstract sense, a symbolic representation of the real, bigger space. However our perception of time (and hence speed) is not so flexible, and results in us needing to

notscale time in any way in order to make thingsfeelcorrect to your brain.Jeff,

We're not trying to be physicists (or spellers), just model railroaders. We are simple minded enough to go with RATE x TIME = DISTANCE all over 87.1.

Thanks for chuckle, Steve (wesolint)

I can't comment on your post, only because I was lost after the second sentence! I wanted to thank you for the comment on my blog and for the suggestion to add the link to the white pigment.

Fascinating post, Jeff. I especially like your thoughts about how we are happy with scaled lengths but not scaled durations.

I guess it makes sense from an evolutionary perspective. We're used to scaled lengths just because we often see the same thing from different distances. But we didn't evolve in an environment where traveling at relativistic speeds was routine, so we're not used to seeing the same events take different subjective durations.

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