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Rachel made me do this.Also showed me how to do most of it.
I have recently watched the entire The Hunger Games series. That wasn’t Rachel’s fault. I take full responsibility. They’re OK but they shouldn’t have split the last one into two movies blah blah blah Jennifer Lawrence etc.
I asked my friend Rachel if she watched it, even though I assumed she hadn’t. She said she was pretty much pressured into watching it and that she “hated the train”. Which I found surprising because who doesn’t love Maglevs? She said that the thing took a sharp turn going too fast and the g-force would be too strong and would squish people. I said it didn’t look that bad to me.
“Do the math,” Rachel said.And, ironically, here’s a tangent: I used to not like math, or “get” math, at all. But since going back to school and taking a required math class… I’m kind of enjoying it. It is confusing.
I knew how to do some of the math, and Google showed me a bit more, but there was a lot I didn’t know how to do, so I got Rachel to tell me how to do that.
When we were done, Rachel said I need to publish the findings.
So, here we go.
Disclaimer: while we’re using real math, physics and geometry, I had to guesstimate some stuff. This is noted in the text.
The formula for calculating g-force is Velocity Squared divided by Radius:
Which means we need to figure out the velocity and the radius of the turn.
Velocity is easy – we are told that the train travels at 200MPH.Which is, frankly, not that fast. It’s the normal speed of the famous Japanese bullet train, which is not even a magnetic levitation train, and it’s top speed is actually higher. And we’ve had a maglev train actually hit 500KPH, which is about 300MPH. We need that in meters per second. We multiply by 1.6 to convert miles to kilometers, then multiply by 1000 to convert kilometers to meters, and then divide by the number of seconds in an hour (3600):
I’m just going to round that up to 90 because I want to.
Now for the fun part. We need the radius. Rachel said that’s easy, too, because for some reason she thought I’d know basic geometry. I do not,I think my math class starts geometry next week. despite Rachel saying PI at me repeatedly. So she gave me the formulas. First, we know (now) that Circumference equals two PI times Radius:
But since we don’t know the circumference, we need to calculate that. Luckily there’s a formula that can give us that based on stuff we do know: Length of the Arc equals Circumference times Arc’s Angle divided by 360:
We can combine these two formulasWhich Rachel made me do before I was allowed to put numbers in anywhere. by replacing C in the latter with it’s equivalent from the former:
Since we want the radius, we need to get that r all on it’s own. So we divide by 2π:
And yes I could display that a little nicer, but since we still need to isolate that r, we now divide by (A/360):Also we regret using MathML.
I personally think that’s good enough but Rachel said to simplify it even more, plus we may as well switch sides a bit, so we cross-multiply or some math voodoo:
Now we’re allowed to put numbers in, which means we need to figure out the arc’s length and angle.
This is where it gets fun and I had to start guessing things.
First, I had to estimate the length of the train. We’re shown the inside of the cars, so I estimate them to be 15 meters in length. This makes the entire train (5 cars plus some stubby things at the ends) 85 meters long. Here, visual aid:
We’re shown that train for a bit more than 3 seconds, which is enough for 3 one-second screenshots:
We see that the train moves slightly more than one train-length each second. Since we know the thing is going about 90 meters per second, that validates my size estimation. We can use this to figure out the length of the arc.
This is here things get a bit more guessy – we’re not shown the train actually entering the arc (it’s not curving yet), and based on how much smaller it is, I’m estimating how long it’d take to get all the way to the end there at about 12 seconds. But it straightens back up by then, so I’m thinking time in the arc itself is about 7 seconds. Again, guessing.
We know that Distance equals Rate times Time, and we have the Rate (90m/s) and Time (7 seconds), so:
But what about the arc’s angle? Well… I’m not sure this counts as guessing. I think it’s more extrapolation.But I’m not going to claim it’s any more valid or any less nonsense than the straight up guessing. I basically drew the rest of the circle based on the arc, then straightened it up, and then put a protractor on it:
So we have a 40° angle.
Which means that, at long last, we can figure out the radius!And be thankful to the gods of Cut and Paste. Which is actually a thing. It’s called The Church of Kopimism.
Which gives us:
This is the point where I think I’m done, and then remember I did all that to figure out the g-force. So now that we have our velocity (90m/s) and our radius (902.4), we plug that back in and get:
Now, if you were strapped down correctly, that’d probably not kill you. You might not even black out, assuming you happen to be a trained fighter pilot and are wearing a specialised flight suit.
But since Jennifer Lawrence doesn’t seem to have one of those, we need to slow that train down. Unless we want to issue “Return to seats, buckle in and hope for the best” warnings before each turn, we should probably keep it at 2.5g.
And because I’m tired of using MathML,I know what you’re thinking. “There’s no MathML in here!” Well there was, but I couldn’t get it to work right so I replaced it with images… I’m going to just tell you that the train needs to slow down to a relative crawl of about 170 KPH (which is about 105 MPH)
I should also point out that the level of technology exhibited in the movie is pretty much at the “Magic” level.In fact, it pretty much invalidates the premise for the entire movie. We could very easily hand-wave this whole thing and say they have inertial dampers.
But where’s the fun in that?