A major obstacle against space exploration is the difficulty of carrying large payloads from the Earth's surface up to space. The problem is that we need large amounts of propellant to give the payload enough speed to get free of the Earth's gravity, but we need even more propellant to carry the propellant, etc. Chemical engines allow a limited speed of the ejected propellant and thus a limited efficiency, that is why nuclear power becomes so attractive for space applications.
Without nuclear power, the only solution I can see would be to have a powerful energy storage system, like a giant battery. My idea is to pump up energy into the system before launch, and then use it to accelerate the propellant (and, in low atmosphere, why not the air?). There is much progress being done today in manufacturing relatively light and efficient flywheels (where a rotating system is speeded up and the rotational kinetic energy can be reused). Is it realistic to think about a big flywheel accelerated at high speeds at ground, and then used to power the rocket up to space?
What mechanism do you propose to transfer the stored gyroscopic kinetic energy to linear kinetic energy of the propellant? Just about any energy conversion system you choose will be far less efficient than simply burning chemical propellants to generate the necessary jet velocity.
Also, the sheer magnitude of energy released during even a modest rocket flight is light-years away from the small amount you are liable to store in a gyroscope. It is simply a strength of materials issure--no gyroscope could ever store as much energy as the chemical equivalent volume of gasoline. A modest sized gyro would have to spin at hundreds of thousands of RPM. While it is possible for dental drills to achieve nearly 500,000 RPM, a 50kg gyroscope made of even the stongest materials on the planet would catastrophically fail at not much greater that 100,000 RPM.
High field superconducting Titanium-Niobium magnets, cooled with liquid helium--have achieved magnetic field energy densities approaching the energy density of TNT (which is only about 1/2 to 1/4 of the energy density of gasoline.) However, this is very close to the critical field strength at which the magnet material transitions from superconduction to regular conduction. The result is called a 'quench' which belies the violence of such an event. When a quench happens, the enormous currents flowing within the magnet coils collapses the magnetic field and deposits almost all of the field energy as heat into the coils which instantly vaporize--the whole magnet will explode as if it were made of high-explosives! I think the Russians had a superconducting "Quench" accident which destroyed a laboratory and probably killed a few people back in the 1980's.
Anyways, it is a very difficult thing to store energy with reasonable efficiency in a compact volume. Chemical energy is a relatively high 'density' energy that has reasonably low entropy--it can be extracted and converted more or less directly in a rocket engine into kinetic energy of exhaust. The efficieny comes from the fact the energy conversion is one step. Typically rocket engines can convert almost 80% of the available chemical energy into kinetic energy of the exhaust. Nuclear power obviously has a much higher energy 'density' and thus (in an NTR) can achieve slightly better results than chemical alone. Without some radical shifts in energy conversion technology, then it is physically difficult to do much better than this.
Some have proposed leaving the energy production and conversion system on the ground. A high powered laser beam propulsion system, 500MW to 1 GW could deliver enough power to a vehicle to utilize air as an efficient propellant. Such a system could, theoretically, launch a couple of hundred kilograms of payload into orbit in only a few minutes, and do it again and again, for just a few hundred dollars per kilogram of payload (possibly.) Others have proposed using beamed microwave energy from a solar power sattelite to power this process. (However I am very skeptical of the magnitude of the microwave power interception/conversion that is frequently stated in this research, so I reserve judgement on this one. I'll believe it when I see it!)
A Space Elevator is theoretically just about the most efficient means of space transportation, because the actual energy cost of transportation versus delivered kinetic energy of payload approaches unity--that is, all of the energy expended is used to move payload and nothing else! Delivered payload costs would approach perhaps $1 per kilogram (basically the price of electricity to run the elevator.) Of course the real trick is building the thing in the first place--but that I will leave for a clever engineer and another post to figure out!
I have always felt that anyone who had the capability to build a space elevator wouldnt actually need one!
Having said that, Building a space elevator using a single Orion launch would make good sense!
Even the most enthusiasitic Orion programme is not going to see them popping off like 747's at Heathrow airport! and something for "Day to Day" travel is still going to be needed, either some sort of spaceplane or Space elevator.
Or would waiting a little bit longer after the first flight be more worthwhile?
Instead of launching the skyhook on your Orion, launch the (basic) equipment needed to begin asteroid mining (or even lunar) and orbital construction. It would pr'y take a few years longer than your way to get to the same stage launch wise, but you would be in a better overall position as you wouldn't then have to launch the more massive and large parts of the mining equipment on the tether.
And also that way you are better set if the Earth decides to quit supporting you, tho I must admit it might make that more likely. At least you can do more with the mining equipment and an Orion that you can move where'er needed, than with a tether that is unused and an Orion that must stay close enough for the hopeless task of guarding it. (or at least raining down death and destruction in retribution, lol)