K^2

Yes. The problem treats water as incompressible and inviscid. In truth, it's not that horrible of an approximation. Air resistance as water goes up will cause a greater reduction in maximum height than viscosity of water.

Other way around. Zero seconds for them is an indeterminate amount of time for us.

That's what the thrusters on the rim wall are for.

The probability of absorption is trivially related to the angle it would have been scattered to from a point charge. Then you just integrate that over differential cross section to get total absorption probability. That's the easy way to solve it. How the hell were you planning to go about it? Unless you want proper QCD treatment, of course. In which case, you have to solve the relevant 4-point DSE. (Lepton vertex can be taken out without loss of precision.)

Dux, you divided by mass twice. Once you got energy per kg, then your kinetic energy formula just has 1kg in it. Or use the full 1MJ against these 500kg. Either way, the answer will be the same. Fair enough. But it's still just a matter of doing the math. I mean, if you're at the level where you know what a differential cross-section is, this is still a boring problem. And if not, then no amount of thinking is going to get it solved.

Problems that require precise evaluation of sin 6deg for an answer are not fun. Nor the ones that are ill posed. There is no classical cutoff for fusion, due to Boltzman. And what do you mean by "at least," when answer is so sensitive to available energy.

At the speeds you are quoting, it'd be closer to 70T. But there are significant losses in atmosphere. The impact is closer to 3km/s, which gives the 10T yield.

Some fire-fighting boats should be capable of something comparable. And sure, why not? We might even just make this into physics brain-teasers thread if it goes well. I have a few tricky ones.

I'm not sure you understand power or maybe just rates in general. Given rate of flow and power, you know final velocity. Both of the former need to be established to deduce the later.

It has practical applications, though. If you understand all of that and a few ideal gas formulas, you can easily do math on water rockets, keeping in mind that power is equal to pressure differential times flow rate.

There's a lot of gray area there, but the fundamental document on the topic is the Outer Space Treaty. Unfortunately, despite using the phrase nearly 40 times, the treaty never actually defines what constitutes "outer space," so interpretations vary from context to context. There is some discussion of that in the Karman Line article. For the most part, it comes down to two figures. 100km, which is roughly where centrifugal force exceeds lift for "typical" aircraft. And 150km, which is the lowest perigee a satellite can have and complete a full orbit. So in summary, anything bellow 100km and not over international waters is somebody's airspace, and everything above 150km isn't. The 50km in between are kind of a gray area. Whether or not an airspace is actually controlled is a separate question. In US, for example, everything above 60,000 feet is Class E airspace, which means that you don't need flight plan, ATC clearance, or any of that formal stuff that you'd need at lower flight levels. But these are still altitudes accessible to spy planes, so don't think you won't raise an alarm crossing borders at these altitudes. Come to think of it, simply climbing to 60k feet ought to put you on some sort of a watch list. With all of these complications and hazy areas in mind, if I was building a private space ship and didn't want various hassles with scheduling launches, I'd probably go for an aircraft launch over international waters. The carrier aircraft would take off from mainland and can cross the border using standard transceiver routines. The actual space ship would then only need clearance for a fly through the Class A.

That's what I got.

If you are going for the "It's not the fall that kills you," I would replace the formula for final velocity with vi + sqrt(2gh). That makes it more clear that you are talking about a fall from a fixed height. I'd also replace the final one with delta-p / delta-t. It's exactly the same thing (in Classical), but delta-p immediately tells me "impulse". And impulse makes me think of an impact. While the "mass times average acceleration" can be anything.

Gravity isn't constant. You have a central potential of the planet/star and the thrust of the ship, which may change direction as necessary during trajectory. Solution will not be anything like a brachistochrone.

In general, orbital intercepts are absurdly complex class of problems. They are inherently non-linear, unstable, and even your strategy changes entirely depending on your initial conditions. There isn't a formula. There isn't even a simple algorithm. Off the top of my head, I couldn't even tell you for sure that engines-always-on will guarantee shortest time. You have to bite the bullet and solve the numerical optimization problem for any given starting condition.

And it goes on to note that it exhibits CPT symmetry instead. For our purposes, that's the same thing as perfectly time-reversible. This is why typical explanations of entropy, these not amed at people who spent many years studying mechanics, chaos, thermodynamics, and field theory, well, frankly, suck. Loosly speaking, there are two types of entropy we consider. There is the fine grain entropy, which is defined for the universe as a whole, and it is a conserved quantity. It's boring, just like the universe as a whole is boring. It's an object that just is throughout all of the time, and any dynamics within it is fully reversible. What's far more interesting is considering a small corner of the universe, which, nonetheless, interacts with the rest of the universe. If I drop a glass on the floor, I do not need to know anything other than composition of the glass and maybe small region of the floor near impact to predict what happens. And yet, the details of the impact and how the glass shatters, will depend on everything that's happening. This is where irreversibility comes in. Precisely, this is described with coarse entropy. It's the quantitiy that doesn't just concern itself with total number of states, but rather the rough volume these states occupy. There is a way to state this with mathematical precision, but I don't think it will be useful. The important part is that if the system exhibits no chaos, and all trajectories flow together, then the coarse entropy never changes. The processes are reversible. A glass falling on the way to the floor is a good example. Once the impact happens, chaos. Small, tiny changes in original states of particles in the glass or floor make huge differences in where the cracks head and how the glass flies into shards. The trajectories diverge. As they do, the phase space volume occupied by these states increases. The corase entropy has increased, and the process is irreversible. Now, if you happen to have full information about all of the universe with precise state of every particle, you can still reverse it. The CPT symmetry still makes everything fundamentally reversible. But if you are existing within the universe, trying to gather the information you need would cause more irreversible processes to happen. Once coarse entropy has increased, you cannot make it decrease again. It is a strictly increasing quantity. But the actual amount of coarse entropy is strictly subjective. It depends on what you call time zero, what you consider as your sub-system, and so on. It's a useful quantitiy, but it's useful precisely for determining what can and cannot be reversed, as well as a few other things related to determining direction of certain processes. (Why things dissolve, for example.) But it's not a fundamental quantity. It's not inherent to the universe as a whole, but rather enters into our discription of small parts of it as means of accounting for the infinite vastness we have no access to.

Well, dividing by distance^2 + epsilon^2 makes sure that you are never dividing by a number smaller than epsilon^2. Without it, two bodies can pass really close to each other, resulting in huge forces, which will eject both bodies from the system. If you've ever seen galaxy simulations which spray "stars", that's basically what's happening. Adding epsilon^2 regulates the interaction. But it only works if typical distances are much, much greater than epsilon. If original simulation you've grabbed this from dealt with Solar System scales, and units were meters, than 3E4 is just 30km. You don't expect things to typically get within that range of each other. But your simulation has a totally similar scale, and so that epsilon started to interfere with dynamics. You can try simply tunning it down to see what happens. You should be able to get it low enough to where things still move in proper orbits, but where it prevents you from dividing by a value that's too small when two planets "collide".

Beale, why are you dividing by (dist*dist + EPS*EPS), instead of just (dist*dist)? Is this to protect from division by zero? If so, I'm pretty sure it's too large for your scaled-down system. That would both explain why orbits have funny shapes and why your initial velocity computations were wrong.

Mercury's orbit suggests error in your gravity equation or integration routine. It might help if you post how you get the forces.

Oculus definitely isn't there yet, but I wouldn't dismiss possibility of VR tech advancing sufficiently in the near-ish future.

On the money. Zuni, if you want to actually experiment with it, pay careful attention to what are termed the driving section and driven section of an autorotating rotor. If you mess that up, your rotor will stall, and it won't do anything.

Absolutely. What you are looking for is called the Collocation Method. Very frequently, the numerical solution will use a Collocation Method as a single step in an implicit RK scheme. But if you are working with compact space and you expect polynomial of sufficient degree to be a good approximation to true solution, you can just collocate the entire space and do this in one go. Unfortunately, general collocation method requires solving a non-linear optimization problem. However, in a special case where you are solving y'(x) = f(x), the collocation can be solved analytically. But then you are really just doing numerical integration and your analytical solution is the quadrature rule for your collocation points. For solutions approximated with polynomials, Gauss-Legendre quadrature points are a good choice for collocation points.

ZetaX, you are pulling at straws. I can see your point that one can approach mathematics as a science in certain contexts. One can think of conjectures as hypotheses. But certain conjectures can be proven true. A hypothesis can never receive that status. The fact that you are absolutely guaranteed no way of proving a hypothesis correct is a foundation of science. Any scientific knowledge is fundamentally statistical. That is an absolute goal in science. In mathematics, it just means that we have not found a proof or a proof that a proof does not exist. Or a proof that nothing conclusive can be said about the statement. In either case, what you call a final result in science is unsatisfactory to a mathematitian. And final result for a mathematical theorem is absolutely unachievable in science. Are there parallels? Sure. But one can find parallels between science and interpretative dance. That doesn't mean anything, other than broad scopes of subjects. Like I said, if you think it's possible to find a flaw in mathematics, you don't understand mathematics. The aliens might have a completely different mathematics. They can use a different algebra, which is not derived from counting. And it would be valid. But that wouldn't make any of our algebras less valid. Same for the rest of mathematics. You can have different sets of self-consistent axioms. And we might never discover all of them. And if we find other intelligence, it's likely that they have found things we haven't. But none of it would invalidate mathematics we have.

Yeah, if you have multi-engine, all you need is engine controls. If you have a single-engine aircraft, you can get some roll authority by shifting the weight. You'd be at a mercy of a lot of factors, not least of which is wind, but you can at least give landing a solid try. If you don't get thrown by a gust of wind at the worst possible moment, you should be able to perform a soft landing.

Yup. Just like most satellites, ISS is visible when it's just about to pass or has just passed the terminator. When it's dark on the surface where you are looking at it, but light at the altitude where the satellite is passing. The only real exception I'm aware of is satellites that do orbit correction burns. You can see their flares when they are in total shadow. But these are pretty rare.