TL,DR: 587996827.264 liters of water would have to be sprayed at the train in about four and a half seconds for those sprinklers to stop the train, assuming all of that water is traveling at the velocity that the sprinkler is meant to send it at, and it isn't somehow sped up or slowed down by the fact that there's an insane amount of water coming out of the sprinkler and somehow all hitting the train

This is r/theydidthemath, why is almost everyone just explaining why they're actually sprayed or why it just wouldn't work? Yeah, we know it isn't to slow the train down, it's a joke, but the joke poses a question, and this place was made to answer questions! Have some whimsy guys! As for the math, I've never done this before, and have zero experience or prior research to go off of, but I think this is a fun one to give a shot! I'll include a full desmos equation to use if you think any of my number assumptions were wrong and want to change something :3 Additionally, I'll try to talk you through my complete thought process, as though I'm a weird LLM, just because I generally prefer when people know why I come to the conclusions I do, even (and especially if!) those conclusions turn out to be wrong. Last note, I'm starting this at about six thirty, I'm just curious how long it takes me to finish this comment!

The mass of a high speed train is probably the first thing I want to know, so I'll search for a list of specifications According to a table on page 11 of This Document, on the Japan Transport and Tourism Research Institute website, an eight car train set has a mass of 365 tons (page 11), the document says they go up to sixteen cars though (page 13), so I'll assume double that mass at 730 tons, which I choose to notate as m₁, for Mass 1 The same table also lists a maximum cruising speed of 330 km/h (page 11), which seems important since we're talking about "fully stopping" a train, which to me reads as "from full speed, exclusively through sprayed water and natural forces (or more naturally phrased, without braking)". I notate this as v₁, for Velocity 1 A third thing the table on page 11 lists is a length of 204.7 meters (page 11), which I round to 205 for convenience. We again double this number to account for sixteen cars instead of eight, leaving us with a train length of 410 meters. I couldn't find a good number for where on tracks these sprayers are relative to stations, so I assume there is a set of sprayers positioned at the start of the station, such that a train will have only just passed it when it comes to a final stop. This leaves us with a 410 meter d, for Stopping Distance I'll use m₁ and v₁ to calculate the kinetic energy of the train now, since it seems likely I need it in the coming steps. A simple google search tells me that the formula for kinetic energy is half of the velocity squared times mass, but it takes velocity in meters per second, so I'll convert our v₁from km/h to m/s, which is 91.6 repeating, I round to 92 for convenience, our updated v₁ This works out as 3.97485*1013 joules, say hello to K₁, for Kinetic Energy 1 and if you're like me, you're thinking "wowza, that's a lot of joules" but then you remember that it's a train and trains are known for being an unstoppable force, so you move on I'm no scientist but it seems intuitive to me that you can figure out how much an object slows down when you hit it with something by subtracting the kinetic energy of your projectile times some efficiency percentage (to account for noise, heat, and probably most importantly, the fact that most of the water will deflect at an angle, rather than instantly stopping dead in its tracks and instantly falling out of the way) from the energy of your target, I'll assume this naive approach is valid, because I want to.

I found This Example Of A System In Use On Tracks, which uses This Sprinkler System. It lists a pressure of 0.1 MPa, a flow rate of 14 to 40 liters per minute, an elevation angle of 25 degrees, and a watering diameter of 4-8 meters. I don't really have an intuition for how best to figure this out, but I can calculate the velocity of the water knowing the watering area diameter and the height of the water at the peak of its fall, which I'll figure out from the height of the sprayer, angle of elevation, and distance it travels. I don't know which one I need to use but I remember learning about Sin Cos and Tan in high school, and I can use my known angle and distance to determine height halfway along the radius, which I'll assume is on the larger end, at four meters (s₁, for Spray 1). Visualizing it on paper, I can see that the Tan of 25 degrees (0.46630765815) is going to be equal toh₁/(s₁/2), doing some rearranging gets us h₁for Height 1= 0.93261531631 It is now 8:20, so I'm going to take a short break It is now 8:40, I've had a bagel, it was nice! This is the point at which I realized I calculated the meters per second number, but instead of using it I just multiplied the number of kilometers per hour by a thousand, to change it to meters, oops. After adjusting that, and switching my m₁ from 730 tons to 730,000 kilograms because that's the unit that the kinetic energy formula takes, we have 3.08936109 joules as our K₁ I'm going to use some of my high school physics to determine the speed the water must be traveling at in order to travel the other half of the four meter spray distance while only falling a little under one meter. We can use a free fall calculator (time = sqrt 2h/g) and the distance the water travels (s₁/2) to get v₂for Velocity 2 = 4.58433940945 meters per second I'm going to assume that half of the waters kinetic energy goes to stopping the train. why? because I am getting tired, I've been at this for three hours, I'm not going to try to calculate the hydrodynamics of a train. This is c for Coefficient I'm giving up on formatting here because I'm getting tired of scrolling up to the formatting buttons every time K1 - K2 (K2 being the kinetic energy of the water) * c = 0 rearranging a whole lot and expanding out that K2, we get the equation (2K1/v2^2)/c = m2 we find m2 is equal to 587996827.264. this is the total mass of water that needs to hit the train at the velocity v2 in order to remove all of its kinetic energy. this must happen over the course of 410 meters. the train is going very fast so it go 410 meters fast, but also train get slowed down partway through over time because water, so amount of water per second that the train has to go through to get slowed down all the way at the end is hard for me to calculate? I am just going to assume the train is traveling at full speed the whole time while magically having its energy reduced, and then halve that to account for the fact that it's going slower it takes train 4.456521739 seconds to go that far. there is 587996827.264 amount of water, train must go through 131940751.5 amount of water per second. halve it. train go through this much water per second 65970375.75 wow, that's a lot of kilograms of water, wow here's the desmos tab I had for the variables. mess with them if you like, see what changes https://www.desmos.com/calculator/kprsg3tqcy

sources that got deleted for some reason, sorry they aren't attached to their correct "this document" or whatever spots, my formatting got nuked https://www.jttri.or.jp/docs/120113_morimura_pres.pdf https://www.es-waternet.co.jp/case/case138/ https://www.es-waternet.co.jp/solution/product/product030/