Speed adjustment considerations

[RulerofRails]

So, I thought I would just do something simple. Test the time to reach top speed on an empty train on flat track in slow time hours (half- normal days).

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Hrs:	Level:
4	Instant
15	Virtually Instant
25	Ultra Fast
36	Very Fast
47	Fast
58	Above Average
69	Average
80	Below Average
90	Poor
101	Very Poor
112	Extremely Poor

The settings in major use in the middle of the scale got the most testing. The extreme ones I only tested once. This is pretty bad really in terms of progression. Once a consist is added the maximum speed for that train will drop depending on weight (and, in real-life, grades). This should be differentiated from the top speed that I tested above. I have the flat track so have removed the effect of grades, but last time I tried I didn't see conclusive evidence that acceleration is a constant to reach maximum speed on each level. Will persevere.

[Gumboots]

It's not a bad progression at all. It's perfectly linear if you graph it.

[RulerofRails]

True, it makes a nice straight line. Depends how you look at it. Fuel cost settings also look this way, BTW. Here there is a difference of almost exactly 11 between each step, but because we have modifiers acting on each value (top speed and distance), IMO the realtionship between the value of each individual step matters more.

Here's the change in time to accelerate if jumping up one level:

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9.8%		Extremely Poor to Very Poor
10.9%	Very Poor to Poor
12.2%	Poor to Below Average
13.8%	Below Average to Average
15.9%	Average to Above Average
19.0%	Above Average to Fast
23.4%	Fast to Very Fast
30.6%	Very Fast to Ultra Fast
44.0%	Ultra Fast to Virtually Instant
73.3%	Virtually Instant to Instant

It's not too bad below Fast, but much beyond that it's broken. Generally, the faster locos also have higher acceleration settings. In fact as you found with the Schools class many locos are quite unlikely to reach their maximum speed below a certain acceleration level. We could basically discount a fair bit of some locos maximum speed on this account. IMO, this effect is covering up the erroneous weights quoted in the game's loco stats display. So that with the lower acceleration settings commonly used before 1900, it's hard to notice. I have, however slowly realized this with some locos like the E18 and Zephyr which look like rubbish in their stats after 1950, but still can do useful service in-game.

A pretty good job has been done of adjusting the stats so that the trains do useful service. It's something to keep in mind when re-balancing locos.

[Gumboots]

TBH I view the top few settings as whiz-bang stuff for 12 year olds, that isn't really relevant to quasi-realistic railways. No train would have instant acceleration anyway. It'd kill the passengers.

[RulerofRails]

I agree that anything above Ultra Fast is too much even to represent the fastest electric locos such as the Eurostar.

I did a few quickish tests for passengers rot factor starting 6 or so new games in the years before and after 1970. I didn't find a big jump. Sure looks like the increase is applied gradually.

A few basics made around the $40 price point:
1990: 6.2%
1975: 5.6%
1965: 5.5%
1935: 4.8%

[Gumboots]

Ok. Methinks that's indicative of another simple formula. I have a suspicion it'll just be linear, from something like 2% in 1830 up to 6% in 1980 or whatever.

[RulerofRails]

I made a chart that attempts to predict the minimum percentage that a steam train will be in the state of acceleration when it travels the standard distance before running out of water. Values over 100% mean that maximum speed is never reached. This is assuming linear acceleration from the start without interruption. So, in the reality of normal service values will always be greater, maybe even markedly so.

What I did to make it:
I ran a train at a constant speed for one month. I then calculated the distance it travelled in track miles. I came up with a value of 2.35 track miles covered per mph per month. I only did one test to get it, but I can easily substitute a refined value later. Then I worked with the calculation that a train will cover 225* track miles before running out of water. In reality most journeys are shorter. Shorter distances increase the percentage of the trip that a train will spend accelerating. A new distance can be added at will.

(EDITED:) I assumed linear (uniform, where speed is described by a right triangle) acceleration so that where x is the time spent accelerating, the time the train will spend at top speed will be x/2 less than if it were running at full speed the entire distance. I calculated total trip time by adding time at full speed with x. Then I calculated what percentage of the total trip time was spent accerating. Any values over 100% mean that the train doesn't get a chance to reach top speed.

I am not sure the best way to use this data. Currently, making allowances that this set shows a best-case scenario, I think that a level of maybe 50% or greater is where a train will in practice be so unlikely to reach maximum speed that engine adjustment allowances could be made that assume that only a percentage of maximum speed is usable. So much for data, the trick is in the interpretation. A start on comparing the acceleration levels anyway.

ETA: I corrected the graph where I had mistakenly used 2x instead of 1.5x to get the top speed. The other error was in the explanation. At least, I hope it was.

Edit: *This is my mistake. The correct figure is double that. See OP. I decided here to leave the 225 graph to represent those journeys that are out-and-back or one service stop per return trip. I have added a second one to correctly represent a single run until water goes dry.

[Gumboots]

I assumed linear acceleration so that where x is the time spent accelerating, the time the train will spend at top speed will be x/2 less than if it were running at full speed the entire distance. I calculated total trip time by adding time at full speed with 1.5x. Then I divided total trip time by 1.5x to give a percentage of the trip under acceleration.

You've lost me here, but I do remember that time and distance calculations involving some time spent accelerating, and some time spent at a constant speed, was the sort of example problem they used in school to teach us kids why calculus was invented. IOW, I haven't been through the logic but I have a feeling your arithmetic isn't right.

[RulerofRails]

You're right. I made some mistakes. Thanks for picking that up. I corrected the sheet. It's saner now.

My trains usually spend only about half of their time actually carrying goods. The rest is (un)loading and maintenance. The actual average speed of my trains is somewhere on the order of 10mph, going a bit faster on some part of the track does little to raise that figure.

This statement in the thread "Why people start with industries" confirms my testing that I would expect an average steamer to be stopped for 5 months of the year. This is somewhat dependent on speed, so will affect slower trains less.


I am failing at working out how the pulling power formula works. I had a thought, hey, some people have examined speed and pulling power a fair bit in the past. So I went looking at some of the old spreadsheets that people have made. Of interest is Low_grade's one from the tips/tutorials download section. He used a simple scale of 1 to 10 to relate most of the levels i.e. Good to Very Good for example. He made pretty complicated formulas, but I wonder if we could update the scales to nearer their true relationships. What do you think?

[Gumboots]

Yeah we could have a go at it. My guess is it will be linear. Something like (pulling power x speed on flat)/(grade percentage x weight) = speed up grade. More or less. If we stick to round numbers for weight and pulling power and free weight, it probably won't be too hard to nail it down. I'm not sure how useful it'd be though.

And one catch with slower trains is that they'll usually be the ones you set to lower priority, which means they'll probably spend more time stopped than you think. Sometimes I think it's a wonder they move at all.

[Gumboots]

Crap. We've been misled for years.

This whole thing about free weight affecting speed on the flat and pulling power affecting speed up grades is dead wrong.

I just did a simple and obvious test: see what happens if you set free weight and pulling power to zero. Result?

For any consist at all, from 1 car to 8 cars, the speed is only 1 mph either on the flat or up any grade.

Obvious conclusion: despite what people have been saying for years, pulling power is critically important for speed on dead flat track.

So there ya go.

Next test: leave free weight at 0 and set pulling power to 1. Result?

For a mixed consist (34 tons cars) in the C era, a simple inverse power of 2 progression. Speeds go like this:

0 cars: 95 mph
1 car: 48 mph (1/2 top speed)
2 cars: 24 mph (1/4 top speed)
3 cars: 12 mph (1/8 top speed)
4 cars: 6 mph (1/16 top speed)
5 cars: 3 mph (1/32 top speed)
6 cars: 2 mph (1/64 top speed)
7 cars: 1 mph (1/128 top speed)
8 cars: 1 mph (1/256 top speed)

In other words, speed with a mixed consist is (Nominal top speed)/(2^n) where n is the number of cars.

For freight (40 ton cars) it's:

0 cars: 95 mph
1 car: 42 mph (1/2 top speed x 7/8)
2 cars: 19 mph (1/4 top speed x 7/8 x 7/8)
3 cars: 8 mph (1/8 top speed x 7/8 x 7/8 x 7/8)
4 cars: 4 mph (1/16 top speed x 7/8 x 7/8 x 7/8)
5 cars: 2 mph
6 cars: 1 mph
7 cars: 1 mph
8 cars: 1 mph

In other words, speed with freight is (Speed with mixed)(7/8)^n where n is the number of cars.

For express it's:

0 cars: 95 mph
1 car: 55 mph (1/2 top speed x 8/7)
2 cars: 32 mph (1/4 top speed x 8/7 x 8/7)
3 cars: 18 mph (1/8 top speed x blah blah bah)
4 cars: 11 mph
5 cars: 6 mph
6 cars: 4 mph
7 cars: 2 mph
8 cars: 1 mph

In other words, speed with express is (Speed with mixed)(8/7)^n where n is the number of cars.

This is what is shown in the in-game popup anyway. Haven't yet tested for how it relates to actual speeds when playing.

[Gumboots]

Next test: leave free weight at 0 and set pulling power to 2. Result?

For a mixed consist (34 tons cars) in the C era:

0 cars: 95 mph
1 car: 67 mph (1/2 top speed)^0.5
2 cars: 48 mph (1/4 top speed)^0.5
3 cars: 34 mph (1/8 top speed)^0.5
4 cars: 24 mph (1/16 top speed)^0.5
5 cars: 17 mph (1/32 top speed)^0.5
6 cars: 12 mph (1/64 top speed)^0.5
7 cars: 9 mph (1/128 top speed)^0.5
8 cars: 6 mph (1/256 top speed)^0.5

So, if pulling power = 2 you take the square root of the fraction of top speed you had when pulling power was equal to 1.
In other words, speed with a mixed consist and pulling power of 1 was (Nominal top speed)/(2^n) where n is the number of cars.
Speed with pulling power of 2 is [(Nominal top speed)/(2^n)]^(1/2).
So the generic formula for all cases becomes (Speed with n cars) = [(Nominal top speed)/(2^n)]^(1/P) where P is the pulling power.

I haven't yet tested for higher values of pulling power, but this formula should stand up. The mathematical basis is clear and simple.

For freight (40 ton cars) it's:

0 cars: 95 mph
1 car: 63 mph (1/2 top speed x 7/8)^0.5
2 cars: 42 mph (1/4 top speed x 7/8 x 7/8)^0.5
3 cars: 28 mph (1/8 top speed x 7/8 x 7/8 x 7/8)^0.5
4 cars: 19 mph (1/16 top speed x 7/8 x 7/8 x 7/8)^0.5
5 cars: 13 mph
6 cars: 8 mph
7 cars: 6 mph
8 cars: 4 mph

In other words, speed with freight is [(Speed with mixed)(7/8)^n]^0.5 where n is the number of cars.

For express it's:

0 cars: 95 mph
1 car: 55 mph (1/2 top speed x 8/7)^0.5
2 cars: 32 mph (1/4 top speed x 8/7 x 8/7)^0.5
3 cars: 18 mph (1/8 top speed x blah blah bah)^0.5
4 cars: 11 mph
5 cars: 6 mph
6 cars: 4 mph
7 cars: 2 mph
8 cars: 1 mph

In other words, speed with express is [(Speed with mixed)(8/7)^n]^0.5 where n is the number of cars.

What this makes clear is that the inverse of pulling power sets an exponent that is applied to the fraction of (speed with x cars/top speed with no cars). You then multiply the result of that to get your actual speed on flat terrain with that many cars.

[Gumboots]

Ok, next bit: performance up grades. Still with the free weight set to 0.

For pulling power of 1, and a mixed consist, speeds with 1 car are:

0% grade: 48 mph
2% grade: 6 mph
4% grade: 1 mph
6% grade: 1 mph

For pulling power of 2, speeds are:

0% grade: 67 mph
2% grade: 24 mph
4% grade: 6 mph
6% grade: 1 mph

Will do more testing later.

[RulerofRails]

Good to see you found a relationship. The values displayed correspond to the .car file values rock-solid (sometimes when rounding occurs trains will never reach the final mph of maximum speed). I tested this over and over on the acceleration tests. Which means the game has been misleading us all this time. Out of interest here are the real values for the Mallard as PopTop released it (this is before 1950):

After 1950 the stats that we normally see in the C-era are in use. So, it doesn't look that bad, except to consider that even for flat ground locos, an average of the 0 and 2% grade values is probably a better indicator of speed on plains than top speed. After all plains regularly have 1% grades and in normal games it's difficult to make bridge ramps under 3%.

You are dead right about pulling power. I noticed that too. There is no way to make a loco better on grades. All we can do is:
Increase the free weight to make it better on the flat ground with a tiny boost to graded performance.
Increase the engine or tender weight which will make the loco worse on grades with no effect on flat-ground performance.
Increase pulling power which increases all values with flat-ground performance obviously benefiting more from higher values.

---

My time is going to be limited for the next little bit. I didn't work out anything exact for speed. With 80 ton cars a "1" Pulling Power with 0 engine or free weight = one car maximum of 20% of top speed. "2" gives 45%. "4" gives 67%. "8" gives 83%, and "16": 90%. That progression left me clueless. I'm sure you'll have it any minute now. I even tried using the square root of inverse of pulling power but couldn't work out how to fit it perfectly.

I didn't even start trying to figure out how to work out graded performance. Working out the formulas would make adjustments easiest. If you can do that, I'm glad. If this fails, here's a work-around method that is slightly tedious but works.

I decided to look into the effects of changing the values. After testing for verification, I worked out that we can create a loco that performs according to any given "visible" specs, as long as we take the ratio of "displayed" car weight to the desired custom car weight and apply it to combined engine and tender weight, free weight, and pulling power.

We have 12 different weight "display" settings available: 7, 9, 10, 13, 20, 27, 34, 40, 53, 67, and 80 tons. Picking one that is a simple multiple: double, triple, half, etc. of desired custom weight should make this process pretty easy.

For example to get 5 tons, adjust specs to what looks nice at 10 tons and then halve everything when you are happy and ready to test. This means 1/2 pulling power, 1/2 free weight, 1/2 of the combined (engine + tender) weight.

The other development from this is the possibility to just use blanket modifiers to keep current loco specs in-tact and apply them to the new custom weights. That would be the quick method. Taking more time for finer adjustments into the new classes would yield better results especially seeing that the displayed "stats" have been misleading us.

---

Few thoughts: changing the modifiers to numbers sounds good to me. This way freights can have higher (one level or at least equal) acceleration to mixed without things looking too weird (simply because freights stop more and suffer from slower acceleration in using less of their maximum speeds between stops).

It's my opinion that a mixed class should have higher fuel costs per mile than heavy freight for two reasons. First, the dedicated freighter will carry higher tonnages and be a heavier loco giving it higher fuel costs. Secondly, the mixed loco will be faster giving it an advantage in less rot factor and ROI when considered as an individual investment (such as when you start a company and buy one or two locos). Passenger appeal settings alone may be enough to keep passengers useful, but we need to have a reason to have dedicated freight as well. We need to answer the question: Why should I buy a slow, dedicated freight hauler over a faster mixed consist loco?

I believe relying on a better graded performance isn't going to be enough as in most games less than 10% of my locos travel over sustained grades.

[Gumboots]

You are dead right about pulling power. I noticed that too. There is no way to make a loco better on grades. All we can do is:
Increase the free weight to make it better on the flat ground with a tiny boost to graded performance.
Increase the engine or tender weight which will make the loco worse on grades with no effect on flat-ground performance.
Increase pulling power which increases all values with flat-ground performance obviously benefiting more from higher values.

Beats me why nobody noticed the obvious before. I had assumed that out of everyone pontificating about pulling power being only related to grade performance, at least one person would have actually checked it. Anyway at least we know now.

About this though: "Increase the engine or tender weight which will make the loco worse on grades with no effect on flat-ground performance."

I don't think that's the case. From the relationship I found, loco and tender weight isn't counted in grade performance at all. All that counts is the number of cars, and how heavy each car is (in its .car file) relative to a baseline weight of 17 tons.

My time is going to be limited for the next little bit. I didn't work out anything exact for speed. With 80 ton cars a "1" Pulling Power with 0 engine or free weight = one car maximum of 20% of top speed. "2" gives 45%. "4" gives 67%. "8" gives 83%, and "16": 90%. That progression left me clueless. I'm sure you'll have it any minute now. I even tried using the square root of inverse of pulling power but couldn't work out how to fit it perfectly.

Ok, first point: is this taking your "80 ton cars" from the in-game stats pop-up? Because if it is then I'd totally ignore it. If you mean "actual 80 ton .car file values" then yes, that fits the formula I posted before. AFAICT it works like this:

Speed on flat terrain = (Nominal top speed)[(17n/2(total consist weight))^n/(pulling power)] where n is the number of cars. IOW, if you have 80 ton .car values then for one car n =1, and if pulling power is also 1 then formula reduces to Speed on flat terrain = (Nominal top speed)(17/80) = 21.25% (in theory).

For 1 car and a pulling power of 2: Speed on flat terrain = (Nominal top speed)[(17/80)^0.5] = 46.1% (in theory).

For 1 car and a pulling power of 4: Speed on flat terrain = (Nominal top speed)[(17/80)^0.25] = 67.9% (in theory).

Etc, etc.

I decided to look into the effects of changing the values. After testing for verification, I worked out that we can create a loco that performs according to any given "visible" specs, as long as we take the ratio of "displayed" car weight to the desired custom car weight and apply it to combined engine and tender weight, free weight, and pulling power.

Forget the visible specs compared to custom car weights. Too much drama, and far too limiting. Even if you try fudging the figures as per your idea, it will only work for the four default eras and it will only work if you double car weights at every era change. If we're going to do custom weights and era dates we should just bite the bullet and say the stats pop-up will not be anywhere near accurate if you are running non-default cars. It isn't anyway, so who cares? We can just give out accurate stats in a supplement instead, and people will quickly figure out which locos are good for which conditions.

It's my opinion that a mixed class should have higher fuel costs per mile than heavy freight for two reasons.

I thought we had already agreed on that and settled it.

[Gumboots]

Re the game's stats pop-up, we could just edit the language file.

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	;@gumbootz: Hard-coded weights are used here!
	711 "Cars @ %1 tons each"

Change to:

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	;@gumbootz: Hard-coded weights are used here!
	711 "Do not trust these stats any more than you would trust a used car salesman with a serious cocaine habit"

Oh and I edited my previous post after re-reading it. Should make more sense now.

[RulerofRails]

When free weight is 0, try increasing engine or tender weight and see what happens. There is always the chance that I was mistaken, but I have observed this while testing multiple times.

Yes, I was using "80 ton cars" from the in-game status. From my acceleration testing I am sure that the in-game stats using "80 ton cars" are true for real 80 ton (in the hex) .car file car values on flat ground (I didn't do much testing on hills). This is why I posted the shot of the Mallard. Basically the stats for the B-era (shown for 20 tons) apply to the C-era, and those of the C-era apply to the D-era. I am fairly sure about this, but test it for yourself. It's always possible that I messed up.

Because the game has 7 and 29 ton ratings in the in-game stats, I also succesfully tested that dividing all the figures (at 29 tons) by 3 would give identical stats for 7 ton cars. I assume that this principle applies to any other relationship like a quarter, a sixth, etc..

I agree that making a supplement is a better idea. This is only a workaround method in case graded percentages become too hard to find a formula for. I'm quite sure it's easier than testing 24 grades per loco even if acceleration was set to instant for that test.

Sorry to go over old stuff. I saw the U1 (is it mixed class?) and it's proposed fuel costs seemed really cheap. I failed to notice the mixed class being more expensive than the others, but I only had a quick glance at the new fuel sheet. I am away for a couple days now, but will look at it more closely later. Good luck on the testing in the meantime.

[Gumboots]

Yes, I was using "80 ton cars" from the in-game status. From my acceleration testing I am sure that the in-game stats using "80 ton cars" are true for real 80 ton (in the hex) .car file car values on flat ground (I didn't do much testing on hills). This is why I posted the shot of the Mallard. Basically the stats for the B-era (shown for 20 tons) apply to the C-era, and those of the C-era apply to the D-era.

Oh crap. I hope this isn't true as it would be really stupid.

It's probably true.

Anyway I'm sure we can find a formula for performance on grades. All the maths we've found in the game so far is really simple, so I can't see this aspect being any more complicated.

Sorry to go over old stuff. I saw the U1 (is it mixed class?) and it's proposed fuel costs seemed really cheap. I failed to notice the mixed class being more expensive than the others, but I only had a quick glance at the new fuel sheet. I am away for a couple days now, but will look at it more closely later. Good luck on the testing in the meantime.

The mixed class is meant to be more expensive, but for mixed or express the U1 should have its fuel rating changed. Its default fuel rating is more suitable for heavy freight. I was just comparing with default fuel ratings to get a better idea of the exact effect of the proposed weight multiplication scale itself, before fuel rating changes were thrown into the mix.

[Gumboots]

When free weight is 0, try increasing engine or tender weight and see what happens. There is always the chance that I was mistaken, but I have observed this while testing multiple times.

Yup, you're right. I need to get more data points to fully sort this. Setting the loco weight to 0 makes it go a lot faster up hills, and doubling its weight makes it go a lot slower up hills.

For loco weight of 0 and pulling power of 2, speeds with 1 car are:

0% grade: 67 mph
2% grade: 53 mph
4% grade: 38 mph
6% grade: 26 mph

For loco weight of 50 and pulling power of 2, speeds with 1 car are:

0% grade: 67 mph
2% grade: 37 mph
4% grade: 17 mph
6% grade: 7 mph

For loco weight of 100 and pulling power of 2, speeds with 1 car are:

0% grade: 67 mph
2% grade: 25 mph
4% grade: 7 mph
6% grade: 2 mph

For loco weight of 200 and pulling power of 2, speeds with 1 car are:

0% grade: 67 mph
2% grade: 12 mph
4% grade: 1 mph
6% grade: 1 mph


At least we know loco weight doesn't affect performance on flat terrain (apart from fuel cost).

I'll play around with it some more. I think testing with loco weight as multiples of car weight is going to be the easiest way to figure this out. Using freight-only consists would be the way to go, because they're all round numbers (10, 20, 40) once you get past the A era. I'm confident we can still nail the formula for grades.

[Gumboots]

Ok, I think I'm onto something. Mind you, I've thought that before and been wrong, but let's not worry about that.

So I was trying to fit various basic functions to what is available in the way of data points, which aint much. We only have data points for 0%, 2%, 4% and 6% grades, but nothing for 1%, 3%, 5%, 7%, or anything higher than that. This means any attempt to extract a mathematical basis for performance on grades is going to have to be a bit of a guesstimate, but I've found something that fits nicely and is very simple. This means it's likely to be how they actually did it. It should be close enough that even if the actual algorithm is a bit different, it won't matter for the grades any sane person will be dealing with.

So keeping things simple while I nail down the basics, this is still for pulling power of 1, free weight of 0, locomotive weight of 0, and one cargo car of 34 tons. The speeds are:

0% grade: 48 mph
2% grade: 29 mph
4% grade: 16 mph
6% grade: 7 mph

If we forget about fitting a continuous function between those four points and adopt a different approach, it starts to make sense. We know from real life rail that there's a sudden jump in required power for a given speed when you go from 0% to 1% grade. The required power to maintain the same speed goes from 100% to 350% (see lotsa numbers here). So it makes sense that the RT3 dudes would have had a sudden (discontinuous) drop in speed from 0% to 1% grade. If they were being totally realistic they would have had the speed on 1% grade drop to only 28% of the speed on the flat, but they didn't. That was probably far too severe for a playable and enjoyable game.

What they appear to have done is drop the speed to 71.43%, which may seem a daft number to pick but is simple when you look at it another way (it's 1/1.4). Going from that point, if you then use a divisor of 1.2 for the step up to 2% grade you'll get a speed of 48/(1.4 x 1.2) = 28.6 mph. If you take 1.3 for the next divisor, and 1.4 after that, and 1.5 after that, etc, it all fits.

1% grade: 48/1.4 = 34.3 mph
2% grade: 48/(1.4 x 1.2) = 28.6 mph, which rounds off to 29
3% grade: 48/(1.4 x 1.2 x 1.3) = 22.0 mph
4% grade: 48/(1.4 x 1.2 x 1.3 x 1.4) = 15.7 mph, which rounds off to 16
5% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5) = 10.5 mph
6% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5 x 1.6) = 6.5 mph, which rounds off to 7

So presumably for higher grades you get:

7% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5 x 1.6 x 1.7) = 3.8 mph
8% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5 x 1.6 x 1.7 x 1.8) = 2.1 mph
9% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5 x 1.6 x 1.7 x 1.8 x 1.9) = 1.1 mph
10% grade: 48/(1.4 x 1.2 x 1.3 x 1.4 x 1.5 x 1.6 x 1.7 x 1.8 x 1.9 x 2.0) = 0.6 mph

To bung it into a formula: Starting point on 1% grade is (flat terrain speed/1.4). For each additional percent of grade from there you simply divide your previous grade's speed by (1+(grade%/10)).

Ok, so that appears to be how it goes for one car and zero loco weight. Next thing is to start adding loco weight and see what happens. To keep things simple I added loco weight in multiples of car weight, so went 0, 34, 68, 102. That means total train weight goes 100%, 200%, 300%, 400%.

So total train weight of 100% gives:
0% grade: 48 mph
2% grade: 29 mph
4% grade: 16 mph
6% grade: 7 mph

Total train weight of 200% gives:
0% grade: 48 mph
2% grade: 18 mph
4% grade: 5 mph
6% grade: 1 mph

Total train weight of 300% gives:
0% grade: 48 mph
2% grade: 11 mph
4% grade: 2 mph
6% grade: 1 mph

Total train weight of 400% gives:
0% grade: 48 mph
2% grade: 7 mph
4% grade: 1 mph
6% grade: 1 mph

For the 2% grade the proportional speeds as weight increases are a consistent 5/8 for each 100% increase in train weight, while for the 4% grade they're a consistent 5/16, and for the 6% grade they're 5/32. There are no data points for 1% grade so an assumption has to be made there. If you assume that the divisor with increasing grade fits a basic power curve y = 1.4 + k(x-1)^2, and put the origin at x=1, y = 1.4 and set k to 1/5, then all of a sudden the weird fractions fit.

At 1% grade the speed change would be 1/1.4 for every 100% increase in total train weight. This just happens to fit with the 1/1.4 fraction they appear to have used for speed on 1% grade compared to speed on flat terrain, so it looks like this was one of their favourite fractions.

At 2% grade the speed change would be 1/(1.4 +0.2(2-1)^2) for every 100% increase in total train weight. This shortens down to 1/1.6 for a 2% grade, which fits the available data nicely (same as the 5/8 found earlier).

At 3% grade the speed change would be 1/(1.4 +0.2(3-1)^2) for every 100% increase in total train weight. This shortens down to 1/2.4 for a 3% grade.

At 4% grade the speed change would be 1/(1.4 +0.2(4-1)^2) for every 100% increase in total train weight. This shortens down to 1/3.2 for a 4% grade, which fits the available data nicely (same as the 5/16 found earlier).

At 5% grade the speed change would be 1/(1.4 +0.2(5-1)^2) for every 100% increase in total train weight. This shortens down to 1/4.6 for a 5% grade.

At 6% grade the speed change would be 1/(1.4 +0.2(6-1)^2) for every 100% increase in total train weight. This shortens down to 1/6.4 for a 6% grade, which fits the available data nicely (same as the 5/32 found earlier).

This is all looking close enough for me.