SpaceX's Grasshopper Successfully Flew 325 Metres

Almost all of the cost of orbital launch is tied up in manufacturing cost and operational complexity. Fuel cost is about $200k for each Falcon 9.

The most important factor in reusability is making sure that the vehicle gets back to someplace convenient without any significant damage. If you crash-land using parachutes somewhere in the ocean or in a desert, that's not going to happen. So you need a controlled landing at a prepared site. The overhead necessary to return to the original landing site is generally more than made up by the fact that you get the rocket back exactly where you need it within a matter of hours and without any damage other than the normal wear and tear caused by every flight.

ICBM warheads have about a 90% chance of landing within a circle with a radius of about half a kilometer, at least according to declassified info, the actual accuracy is likely higher. Even so, that hardly compares to a precision landing on a small landing pad.

If you build your supply landers (and your supplies, of course) capable of withstanding high g's you can probably crash-land large things on airbags. NASA hasn't done it because the rovers are somewhat delicate, but if you are shipping frozen-solid food containers, I believe it would be fine.

No one quite knows, as no such system exists yet, though I imagine SpaceX has some internal numbers. You need enough to slow your mostly-empty (and thus fairly light) stage from its terminal velocity to 0, and however many seconds margin you need to be comfortable.

Let's do some very rough numbers to get a ballpark...

Say terminal velocity is 100 m/s, which is fairly fast, but a rocket is a skinny streamlined thing. (It could fall slower, maybe even 50 m/s, on its side or with parachutes, but parachutes are heavy and complicated, and falling sideways may be difficult to control, but seems quite possible.)

Say an F9 "1.1" first stage weighs 28,000 kg. The landing components mostly are already there: the engines are already gimballed and the additional software weighs nothing. Unless you're so awesome you can land in a cradle--which is not impossible--you need some landing gear, so let's add 3000kg, for a dry weight of 31,000 kg.

Amongst all its engines, our F9 has some 300,000kg of thrust. Just one engine is around 33,000kg thrust, so we'll probably only use one or two.

Two engines will get us easily about 10m/s slower per second, so we need to retro-fire for minimum 10 seconds. You probably want some margin over that; let's reserve 20 seconds fuel.

F9 1.1 stage 1 will have a burn time of 185s; our retro burn is only at about 25% capacity, so 20s of two engines is worth, oh, 5s of normal burn time. Say about 3% of the fuel.

Of course, it's not that simple, because (1) we've made the stage heavier with the landing gear, which requires more fuel and (2) to avoid losing performance we'll want to add enough fuel to add our extra burn time, which also requires more fuel, and (3) we need more fuel to lift all that fuel and mass we added. The rocket equation strikes again!

(Of course, it may not be that bad, because usually first stages have plenty of margin, and most payloads are not max. Maybe if you encounter a really heavy payload you just agree to throw away the rocket that time; use an old one or something.)

Most of my assumptions are pessimistic (I think) but in the real world they may turn out optimistic. I would guess you're looking at 3-5% of the fuel load for a recoverable VTOL stage. A lot depends on details, as with rockets weight scales very badly against performance.

I think that yes, it's the lack of a fixed pivot point.

The inverted pendulum case (as approximated when balancing a pencil on its tip) can be reduced to a single object resting on the surface of a frictionless sphere, under the influence of gravity. At the precise top of the sphere, it can stay stationary. Perturb it even slightly, and it begins to accelerate away.

With the rocket, the object is essentially resting on a flat frictionless plane. If you tilt it a bit, then it begins to slide away, but the acceleration is always proportional to the initial perturbation. This is because the rocket always remains in the same orientation, and so the force always comes from the same direction. That, in turn, is because the rocket engine moves along with the rest of the rocket, rather than being a fixed point in space.

Think about it another way: without external support, and ignoring the influence of the atmosphere, a rocket is in free fall. The fact that it's accelerating to counter the force of gravity doesn't change that fact. Again ignoring the atmosphere, a rocket firing with 1g of thrust counteracting gravity behaves the same as a rocket firing with 1g of thrust in empty space. The second case is obviously not unstable (what would it be unstable with respect to?) and so the first case isn't either.

Adding the atmosphere back in, I could be way off here, but I think what little effect it would have at low speeds would be beneficial. Since the goal is not to move, the drag from the atmosphere will help make sure that happens. Since the rocket is more or less of a uniform shape, that drag shouldn't end up applying much of a torque either.

Edit: forgot to mention, I'm afraid I don't have pointers to articles. I've collected this stuff through various readings and thought over the years and don't have anything specific to point to. I think I first learned about Goddard's pendulum mistake in high school physics, for example.