K^2

Unfortunately, there is a very big difference between what would be scientifically useful, and what people think NASA should do. There is no glory in a robotic lander. People want to see manned missions to other worlds, and honestly, Mars is the only thing within reach that isn't just a whole lot of gray rock. So NASA has to keep up at least an appearance of trying to send a manned mission to Mars if they hope to keep their funding. And to be fair, I think NASA is doing a very good job of combining scientific utility with satisfying public's thirst for space.

Keep in mind that you are looking at an area near the North Pole. Most of the features we have here on Earth in the extreme Northern/Southern regions is due to glaciers. If it weren't for them, we wouldn't have much in terms of rivers there either. Precipitation cycle is going to be much more powerful in near-equatorial regions, so that's where I'd expect to see rivers and canyons.

I'm sorry, but you are way off base here. These individuals had an intellect on the outside of the bell curve, as well as ability to think through problems both systematically and creatively. They wouldn't have achieved what they did without such abilities. As such, you cannot compare them to an average individual even in a context of a problem where any education or prior experience are useless. There can be people better qualified to make a judgement on these things, but we are still talking about a very small group of people.

In this particular case we should. The main design goal of the arrester system is to bring an aircraft down to a stop in a fixed distance under minimum stress, while allowing for maximum opportunity for safe abort. Linear force is not quite the worst thing you could do here, but it's very far from the best. Even using spring-and-pulley systems you can do much, much better. But hydraulics are used specifically because they give you the closest thing to constant, adjustable force.

That's if you have a constant pitch over long time. Which, by very definition, carries no information. If we were actually sending one out, it'd probably be detectable. Though, it could be mistaken for natural phenomenon. But our radio communication carries information. In that case, Shannon-Hartley applies, albeit in its corollary form, and given sufficiently strong noise, you cannot distinguish a channel with sufficiently high bit rate from the aforementioned noise. In other words, if we were sending a steady pitch signal, it'd be pretty detectable, but our radio chatter is not. It has peaks, but there are not any frequencies completely devoid of noise. And you can only push statistics so far. If you could just analyze the hell out of signal, we wouldn't be having spectrum crunch issues down here.

Nah. An arresting wire really is closer to a constant. At least, that is the goal. The wire is connected to a hydraulic system which is governed by a relieve valve set to a particular pressure. So deceleration builds up quickly to a target level and remains roughly constant.

When either your final or initial velocity is zero, the shortcut for relating velocity to distance and acceleration, assuming acceleration is constant, is v² = 2ad.

Doesn't matter how sensitive the receiver is. If transmission power is weaker than cosmic background, you cannot detect it with any receiver. There are simply no means, purely mathematically, of distinguishing it from noise. Of course, if they sent out probes, and one happens to pass close to Sol, then it's a different matter. As you said, we don't really know the distance from which transmission needs to be picked up, and something within a ly of Sol can certainly detect transmissions from Earth using sensitive enough equipment given long enough observation time.

If we are looking at very basic orbital motion, by analogy with circular orbit, there are formulae for periapsis and apoapss velocities of elliptic orbit. To get the notation consistent, you know: vc = Sqrt(GM/rc) Where vc is velocity of circular orbit, GM is gravitational parameter (Mass times gravitational constant), and rc is radius of circular orbit, which is planet's orbit + orbit's height. So lets say that instead you have an orbit with two different heights, yielding ra as the apoapsis distance and rp for periapsis. Just like circular orbit, that's planet's radius + corresponding height. In that case, the two velocities are given by the following. va = Sqrt(rp/ra) * Sqrt(GM/a) vp = Sqrt(ra/rp) * Sqrt(GM/a) Where a is the semimajor axis, a = (ra + rp)/2 Using these, you should be able to compute Hohmann transfer, which is elliptical orbit connecting two other orbits (often circular ones). So you can do the math, for example, on how much fuel it will take to transfer from Kerbin's orbit to Duna's. These tend to work pretty close in the real world too. They aren't exactly right, unfortunately, but any reasonable approximation can be made. One more useful thing to know is how much delta-V you need to go from surface to a low orbit. Since potential energy change is usually minimal, as a rough estimate, you can simply use velocity of circular orbit when there is no atmosphere. You'll need just a touch more to fight gravity, so have some reserves. If, however, there is atmosphere, you need one more formula to estimate extra velocity required to fight through it. This assumes that thrust of your engines is about 2x the weight of your ship. This is what you need to ascent at terminal velocity and what you should be aiming for. The extra delta-V is given by this formula. v = 4gH / vt. Here, g is surface gravity, H is scale height, and vt is the terminal velocity at whatever altitude you are taking off from. For planets in KSP, all 3 of these are available for every planet in the system. For real world applications, you have to be able to estimate vt, which will be very different for different rockets. This last one is for very rough estimates. So keep it in mind if you are going to use it in KSP to estimate fuel needed on a lander.

Actually, there are some developments that I can't get into detail of at the moment that might result in powerful radio broadcasts to make a re-emergence as a carrier for something that will act as point-to-point communication. Once there is a paper, I'll link it somewhere around here.

We have that. It's called a University.

And what did you expect? Anything that can be understood without spending a decade building up a proper background has been understood centuries ago. It has taken a score of people who were brilliant in mathematics, logic, and sciences, and who have spent their lives studying the world to build up the knowledge we have today. And it takes many years of study for us, common wizards, just to understand all of this and try to expand the foundation. Understanding isn't some sort of privilege you are granted as a birth right. It is something you gain through years of work. That makes it easy to forgive the ignorance of the subject, but not the lack of respect towards people who actually do something to advance sciences. It would be a dark day indeed when public money is spent on charlatans and snake oil.

To be fair, time symmetry really is broken.

Higgs Field is only responsible for a small amount of current mass. Most of the mass is dynamic, due to dynamic symmetry breaking, for example. Force isn't what you understand it to be. All of the fundamental forces are due to symmetries. Gravity is due to translational and rotational symmetries in space-time, while electroweak and strong forces are due to local structure. As a result, absolutely all known phenomena, all available observation, and every bit of knowledge we have of this universe is described by a set of 12 fermionic fields (6 leptons and 6 quarks) placed in R4 manifold and governed by a Lagrangian with Lorentz, U(1), SU(2), and SU(3) local symmetries, with all of the corresponding gauge fields and regularization terms. There is absolutely nothing to suggest that such description is in any way flawed or insufficient. This is very important to note. We aren't faced with any phenomena that we cannot place in this hierarchy. There is an issue of being able to actually model something within this formalism, which is practically resolved by considering separately a problem of particle fields in locally flat space-time (QCD) and a problem of space-time geometry with classical energy distribution (General Relativity). Attempts to come up with a mathematical formulation that allow to cover both aspects in relation to each other are at the leading age of modern physics. But any such theory has to yield QCD and General Relativity in corresponding limits. And while various string theory and holographic approaches to field theory have been explored, they have not given us anything new. Nor have we found any indication that additional degrees of freedom allowed by these theories are necessary. At best, they describe the same dynamics in a more convoluted ways. A standard 4-space with U(1)xSU(2)xSU(3) fields is enough as far as everything we observe tells us. If there is more to reality, we have not found any indication of it. So before you start making up a description different from Standard Model, I strongly suggest you learn more about the later. At least qualitative aspects of it. But it is very important to keep in mind that description is only valuable if it gives reliable quantitative predictions. A bunch of words are useless if you can't put some formulae with them and these formulae let you compute something you can compare to a measurement.

I might come off as a bit pedantic on this, but it's not the energy that's required. Energy in needs to be equal to energy out or the crew cooks. What you need is the entropy flow. Any living system is going to increase entropy, so you have to dump it somewhere. Technically, ship is also going to "age", so that's another contribution to entropy. Easiest way to dump entropy is with waste heat, however, which means that you have to replenish lost energy with low entropy source. But in principle, there can be other ways. Basically, any source of structured anything can be used in lieu of energy source.

Feel free to add me to the list of skeptics, but given the record of the Soyuz, backup parachute is probably a very good idea. If I was riding one of these, I'd also insist on manual controls for the descent engine, all of relevant valves, and manual release for the parachute. If the fuel isn't hypergolic, I'd probably want a match, too. I've driven Russian cars, and I just can't imagine trusting something like that falling out of space without having a score of redundant manual backups.

You assume it would be trouble. Besides, they'd probably just look at it as vacuuming up cobwebs in the cellar. You don't do this because you plan to live there. You do this because you don't like the idea of spiders dropping on you while you go to get something out.

That's just an environment bias. You are used to measurements that are a whole number of feet, so yeah, it's easy to divide by 3. What if I gave you a plank that's 120cm long. Is that any harder to divide by 3 than a 4-foot long one? But I can divide that by 5 as well. Or by 10. Working in metric, you aren't going to have things that are just 1m long even. If it's necessary to make something for construction that's easily divided by some number of pieces, lengths can be easily adjusted to have these numbers built in already.

And what would make it worth the trouble of going around?

Most of the common causes of common cold do have individual cures, however.

Bah. It's always best to start with something simple that you can really understand. The most basic way to compute Pi is a geometric one. Start with a regular hexagon inscribed in a circle of radius 1. Its perimeter is 6, which gives you first approximation Pi=3. Then drop a perpendicular bisector on each side to construct a regular dodecagon inscribed in the same circle. You should be able to use Pythagoras' Theorem to compute the new perimeter. Keep going. As the polygon gets rounder and rounder, your approximation will get closer to the value of Pi. The bad news is that it converges ridiculously slowly, so it's not something you'd use for thousands of digits, but it's a good start to see how the algorithm works. Then you can sit down and try to understand more advanced formulae and where they come from. Most of them are highly non-trivial, and programming them is much easier than understanding.

This wouldn't be fire-and-forget solution. If you are simply trying to break off some pieces on the cheap, dumb impactor makes sense. If you want to deflect an all-life-threatening asteroid, you'd get a bit more surgical about it. Arming would happen based on estimated approach, but triggering would be purely radar based. Furthermore, the warhead's approach would be corrected based on visual data collected from the cameras on the probe. We have means of doing all that. It's just a bit more 'spensive, that's all. And I don't think anybody is going to complain about the bill when there is a frigin' asteroid heading for frigin' Earth. So that's not the hard part. The hard part is spotting danger with enough time for ablation due to proximate detonation would result in sufficient deflection to avoid collision. Odds are, we won't have luxury of such a warning when we find asteroid that's actually heading here. That doesn't mean we shouldn't be preparing anyways. We might just get lucky and have info in advance.

I'm pretty sure K-9 Units have been around before the K9 companion was introduced in the show.

Are you joking? NERVA had an empty mass nearly that of the entire Apollo CSM. Fueled up it was more than 3x as heavy. Saturn V simply could not lift NERVA-based craft to LEO in one go. It would take multiple launches and an orbital assembly.

Insertion, as OP noted, could be done with aerobraking. Then you only need enough fuel to circularize, and that's definitely manageable. It's the burn for the return trip that's completely out of the question with the S-V for your lifter.