5.4.calc-Indrani

First, I'm going to lay out a bit of math to ground things a bit more. As I previously mentioned, Archer is being used for scaling here. In particular, she can [fire arrows a mile and a half with ease](https://pastebin.com/Tem3q4b5). I previously said the initial velocity must have been at least 150 m/s, which comes from the fact that distance can be calculated from d = (v0)^2 sin(2θ)/g, where d is the total distance, v0 is the initial velocity, and θ is the initial angle. In this case, d is 1.5 miles, and the θ that produces the minimum v0 is 45 degrees. Math comes out to around 154 m/s. *However*, this is obviously ignoring a few things. First is that she's obviously not firing at the very edge of her range; she's very confident and nails them pretty casually. Second, this is discounting air resistance. Third, this doesn't match up at all with expectations of effective range of bows. Let's get to it.

Snagging a [random source](https://www.outdoorlife.com/story/gear/the-fastest-compound-bows-we-have-ever-tested/), it puts fast bows at ~350 fps—Google shows that this is decently consistent for the high-end of these kinds of bows. That's around 100 m/s—or about 2/3rds of the supposed v0 for Archer. You would expect these bows, based on scaling off of (v0)^2, to have a range of around 2/3rds of a mile. This is patently not the case—bows like this measure ranges of at most [90 meters, mostly around 30-60 meters](https://www.hattila.com/en/blog/at-what-distance-can-a-compound-bow-shoot-n289) (this is at least somewhat consistent with a quick Google search,[1](https://www.outdoorlife.com/blogs/big-buck-zone/2012/07/know-your-pin-range-deliver-lethal-shot/),[2](https://www.hunter-ed.com/nebraska/studyGuide/Speed-and-Range-of-Modern-Bows/20103002_82144/)), and regardless of the details you're not seeing claims of hitting targets at over 1000 meters. Even in terms of records, I found a claim of [600 fps and a maximum distance of around 1320 yards](https://www.archerytalk.com/threads/whats-the-bow-speed-record.1110244/?post_id=1056674130&nested_view=1&sortby=oldest#post-1056674130), or 3/4ths of a mile, which is [corroborated here](https://www.usarchery.org/resources/world-flight-records-150224192914.pdf) under Strother (top compound bow result). 600 fps is ~183 m/s, and yet the *maximum physical range* of the arrow couldn't even break a mile. And that's not even mentioning that the world record for the furthest *target hit* is a mere [330 meters](https://www.guinnessworldrecords.com/world-records/mens-archery-farthest-accurate-distance). And Archer can nail moving horses in the eye at that distance.

If we use (v0)^2 to scale Archer's arrow from the range we get from Strother or Brown, we get around 259 m/s (that's 183 m/s times the square root of 2, since 1.5 miles is double 3/4ths of a mile); that's 3/4ths the speed of sound. But then account for wind resistance and accuracy and that falls off even further—keep in mind that even *sniper bullets* firing .50 BMGs will only see a [maximum effective range of 2000m](https://en.wikipedia.org/wiki/Sniper_rifle#Maximum_effective_range), and that's with far easier firing procedures than a bow and more aerodynamicity. 1.5 miles, for reference, is 2414 meters. So Archer is outshooting anti-materiel rifles, and with bigger projectiles that are subject to greater air resistance.

What about the claim that Archer's bowstring is too tense for even an orc to pull back an inch? Let's assume that Nauk is merely peak human, despite being a superhuman orc—and the strongest orc that Cat knows, nonetheless. Really, the only thing that matters here is draw length—I'll explain in a second. A source I found states that the modern average draw length is [28 inches](https://bobleebows.com/measure-draw-length-traditional-bow/#:~:text=There%20is%20a%20correlation%20between,have%20a%2028%E2%80%9D%20draw%20length.); Archer's bow is noted to be comically large, but we'll stick with average for now as a lowball. This means that Nauk is unable to draw the bow to less than 1/28th of its full draw. Bows have been demonstrated to [obey Hooke's law](https://sciencedemonstrations.fas.harvard.edu/presentations/bow-and-arrow), at least approximately, so 1/28th of the draw of the bow means 1/28th of the *force* of the bow at full draw. Seems reasonable. Additionally, we can model elastic potential energy as 0.5kx^2, where k is the spring constant and x is the draw length—because Nauk's draw at 1/28th the distance is at least equal to full draw of a normal bow, we can say that the spring constant is at minimum 28 times greater than a normal bow. This also means that KE of the released arrow is 28 times greater, and KE is 0.5mv^2, so we can put the velocity at launch at sqrt(28) times greater, or about 5.3 times greater. Using the record of 600 fps, or 183 m/s, that comes out to 968 m/s, or Mach 2.8. (This is roughly consistent with outclassing .50 BMGs in distance shooting.)

All of this is to say that it's *very likely* that Archer's arrows are at least comparable to bullets in speed, particularly at close range.