I wouldn't say technology is uniform, the better you are the more likely you are to get access to sponsorships and therefore money which will enable access to better tech and training.

But you're right that things start to get very complicated, and this model is probably far too simple.

Do you mean stronger as in it's happening more nowadays than the past? If so wouldn't that be explained by the fact that as technology, training, etc. gets better our top athletes are getting and further into the tail of the distribution?

I guess in a very handwavey sense (i.e. I have no idea whether this holds in practice) you're basically sampling from the tail of a Gaussian and you're not taking many samples, so it doesn't seem unlikely to me that you would end up with the ability gap between 1 & 2 being much higher than 2 & 3.

Honestly feels like it's rarer for this not to happen at some point in any given competitive activity.

Micheal Phelps seems like an obvious example, Serena Williams as well maybe? In the competitive AoE2 community The Viper was leagues ahead of everyone for like a decade and now Hera is well on his way to doing the same. Gee Atherton in downhill mountain biking was like this at one point too.

Are you aware of the out-of-bounds issue with Stockfish 16, BTW? I don't see any issues about it on GitHub, but I'm also not sure whether it's a Lichess issue or a Stockfish issue 🤷‍♂️ Stockfish 17 works fine though, as a workaround for the time being.

It's difficult to explain in 300 words, but imagine what happens when you change x very slightly. The base is still negative but the exponent is no longer an even integer, meaning the value is now complex and won't be shown by Desmos. Be interesting to see what this looks like in the complex plane.

You can also be of the opinion that implicit multiplication should have higher precedence than explicit, that's better than fine.

But if you think they should have the same precedence you are wrong.

Sorry, I don't make the rules. We reach them by consensus.

I'm on the bus, so I couldn't do the maths on what realistic pre- and post-collision velocities would be, but hopefully the simpler example demonstrates why you generally wouldn't expect both to be conserved at the same time.

So changing the momentum of the individual bodies while still conserving overall momentum, even if no directions change, will not necessarily conserve the overall energy.

That but also the squared relationship means that the way the momentum is distributed affects the total energy.

Imagine two 1kg blocks, one at rest and the other with v=10m/s. Here p=10kgm/s and E=50J. If both blocks have v=5m/s in the same direction, p=10kgm/s still but E=25J.

I actually think there is benefit in having a section of high school physics which is essentially "here are the biggest topics in physics research right now, if you want to understand them go do a physics degree". Though I suspect if you cover them at school they stop being exciting by definition...

I'm not saying nobody can critique maths education, but you should at least try to understand it first...

What do you think is causing the hot spots at the half hours?