How Strong is Twilight? · 5:58pm Mar 7th, 2013
Hello once again to my faithful readers! Recently, there was a post on Equestria Daily regarding the Equestrian currency, specifically detailing how one might go about converting from Bits to USD. That got me thinking; in what other ways can I apply far too much thought to the show? The answer, naturally, is to attempt to find a quantitative measure of Twilight Sparkle's strength. Now, because I don't actually know the mechanics of unicorn magic, I've brought in the closest thing I could find to an expert on the subject—
The Great and Powerful Trixie! [fanfare]
Yes, thank you, Trixie. Ideally, of course, I would have brought Twilight herself in, but she's understandably busy at the moment, what with having recently become a princess and all.
What, so Trixie is just "the best you could get"?
I told you as much when I asked you to help. Anyway, the basic plan here is that I will attempt to use my own knowledge of Physics and Maths to gauge Twilight's power level. I'll be doing the calculations, with Trixie here to—
Trixie will call out Gizogin's mistakes and make him feel bad.
Basically, yeah. The primary example I'll focus on is Twilight beating the Ursa Minor in S1E06 "Boast Busters".
For the record, Trixie easily could have done that as well. She just felt it was more fair to give somepony else a share of her glory.
Truly, you are a paragon of humility. There are a few things to note before we get started: first, due to the paucity of numerical data, I'm going to be making a few broad assumptions. Second, for similar reasons, I'll be focusing on Twilight before her coronation. Thirdly, I'm a student of engineering, not a doctor of physics. Don't take this too seriously.
First assumption, and it's a biggie: Equestrian physics are generally equivalent to Earth physics.
This is the primary source I will be using, as it gives a few objects I can use as a baseline. It also shows Twilight at the limit of her power, at least at the time of this episode, so it's handy in that sense as well. Trixie, this is where I'm going to need your help. Is an Ursa Minor similar in density to the common bear?
No, actually. From what Trixie gathers, it's a bit lighter than it looks.
That's what I thought. By something known as the Square-Cube Law, strength goes up much more slowly than weight as something increases in size. For example, if you were to double my height, while keeping my proportions the same, I'd be four times as strong, but eight times as heavy. Upscale something too much and it'll collapse under its own weight. In order for something the size of an Ursa Minor to exist, it would either have to be radically less dense, or significantly squatter. In appearance, it looks broadly similar to a normal bear, so it must weigh less for its size.
This is my second assumption, then: When upscaled from a normal bear, an Ursa Minor's weight increases in proportion to its cross-sectional area, rather than to its volume.
Which type of bear to use, though? For the purposes of this analysis, I will use the grizzly bear, Ursus arctos horribilis, also known as the North American brown bear. The typical adult male weighs around 300kg, depending on the season (they weigh the most just before hibernation and the least just after—the Ursa was woken up from hibernation, but it didn't appear to be winter, so it must have been early in the cycle and thus closer to its highest weight). I know the Ursa Minor is only a baby, but its proportions are comparable to those of an adult grizzly, so that is what I will use.
The grizzly bear is, on average, about 100 cm high at the shoulder. How tall is our Ursa?
Did you have to use that picture?
Hey, it's the best one I could find with a pony to use as a height reference. Anyway, Trixie, how tall are you?
Trixie is seven hands at the withers.
...Right. Thanks. Well, a hand is four inches, making Trixie 2'4" at the withers. To the top of the head, or what we humans would define as height, she is near as makes no difference four feet tall. Or, in SI units, 0.7112 m.
Trixie would like to point out that we ponies use height at the withers because our necks are much more flexible and would change the measurement a lot.
So, going back to our previous picture, we can compare Trixie's height to that of the Ursa and get an approximate size for it. I'm getting a height of about 3.67 m at the shoulder, or more than twelve feet. Sound about right?
Well, Trixie doesn't know the conversions...
That's about 36 hands tall. Twice my height.
Is that all? That seems too small.
For reference, that's more than three times the height of an average grizzly bear.
So, based on assumption #2 above, the Ursa Minor weighs 4041 kg. This is 2 metric tons, or 4.5 short (US) tons. The Ursa isn't the only thing Twilight lifted, however; she was also holding a water tower full of milk. Because water towers vary in design, capacity, and construction, I'm going to have to make another assumption: The water tower can be treated as a right cylinder with diameter equal to its height. Additionally, the water tower itself can be neglected in relation to the weight of the milk within.
That's quite a big assumption you're making there.
I'm aware of this, but there's really no way around it that doesn't involve maths far more complicated than I'm willing to do. Anyway, using similar size comparisons (length of Ursa to length of water tower), we can get a volume for the water tower of about 17.3 cubic meters, or 4500 US gallons. This is much, much smaller than any water tower in typical use on Earth today (they tend to hold on the order of hundreds of cubic meters), but might make sense for a small town like Ponyville. Also, it was filled up by a barn's worth of cows in the space of a few seconds. Make of that what you will.
Oh, well, that's because our cows are magic. Or Twilight just left half the water in the tower and diluted the milk with it. That stuff's expensive, you know.
Sure, why not. I'll treat it as just milk, though. Milk weighs 1035 kg/m^3, so our milk-tower weighs approximately 18,000 kg. This is 9 tonnes, or almost 20 tons. In short, much, much more than the Ursa Minor weighs.
So far, I have been using the term "weight" in place of mass, because it is easier to say "the Ursa weighs 4041 kg" than to say "the Ursa has a mass of 4041 kg". From here on, however, I will use weight properly; that is, to refer to a force. Also, assumption #4: Equestria's gravity is the same as Earth's.
Trixie can confirm this. She certainly isn't any heavier here on Earth than she is in Equestria.
Combined, the milk-tower and the Ursa have a mass of 22000 kg. The force exerted on them by gravity is thus 215,600 N. So, Twilight can exert a force of 215,600 N through her magic. Impressive, but this is force, not power. Power is work over time, and work is force times distance. Work is also change in energy, whether kinetic, potential, or internal. I'm going to ignore kinetic energy, because the nature of the shots involving the Ursa+tower system moving make finding their speed annoyingly difficult. So, potential energy.
Using yet more size-comparisons (this time, the Ursa vs. the buildings), I find that Twilight lifted the Ursa at least 4.5 m, and probably closer to 5 m. PE=mgh is the equation to use here, giving the work to lift the Ursa+tower against gravity as about 1,000,000 N*m.
Ah, no, here Trixie must step in. Twilight only gave the milk to the Ursa when it was already in the air, and the tower was actually higher at the start than when it ended.
You're right. Thank you Trixie, seriously this time. So, we cannot treat the Ursa and the tower as a single object for energy calculation purposes. Let's take the Ursa first, then: lifting it five meters into the air is (4041 kg)(9.8 m/s^2)(5 m) = 198,000 N*m. Now, for the water tower, I will concede that it ended up lower than it started. However, it was filled with milk at its lowest point, then raised back up the height of the Ursa. So, (18000 kg)(9.8 m/s^2)(3 m) = 529,200 N*m. This also changes our power calculations, as we can no longer combine the work values.
Twilight lifted the Ursa the full height in six seconds, making her power in this case 33,000 W. The water tower she raised in around ten seconds (it happens off-screen), so her power is 52,900 W. As these happen largely simultaneously, Twilight is putting out a total of close to 86 kW of power. For comparison, this is more than 114 horsepower, and is more than 1,000 times the power output of a typical day laborer.
Well, uh... Huh. Trixie doesn't feel quite as bad about losing to her now.
Yeah. Puts things in perspective a bit. So, to recap:
Force: 215,600 N
Total Work: 727,200 N*m
Power: 86 kW
And, according to S3E05 "Magic Duel", she's only been getting stronger. If I wanted to be really thorough, I would analyze her other feats of magic in different episodes to compare them. For example, when she bursts into flame in S1E15 "Feeling Pinkie Keen", I could have used blackbody radiation to figure out how much heat she was giving off and used that to gauge her power. That's analysis that's beyond even me, unfortunately (just wait until I finish Thermal Physics). I think this gives a pretty good sense of the scale of her magic, anyway.
897302
If the weight scales up cubically, rather than quadratically, then we're talking closer to 174 kW. Again, I had to make a lot of guesses and assumptions, because we're never given any concrete numbers.
For truly ridiculous power levels, you want to look at the time she materialized a doorframe for Spike to walk through (slam in his face? I forget exactly what she did with it.) That's direct conversion of energy to mass!
Science!