LiquidSolid

Liquid fuel rockets are good because you can turn the engines on, and - more importantly - off.

Solid fuel rockets are good because they are a lot simpler and cheaper, plus fuel storage is not a problem.

Liquid fuel has a higher specific impulse than solid fuel (typically 500 seconds rather than 250), but this is - in part - because of the necessarily inefficient design of a solid-fuel rocket: the entire rocket body is the combustion chamber.

The usual solid rocket fuel is a mixture of ammonium perchlorate, aluminium, and polybutadiene acrylonitrile, which is a synthetic rubber.  The latter is the binder, but it burns along with the aluminium in the oxygen from the ammonium perchlorate, of course.

Thermites (like copper oxide mixed with aluminium) would make lousy rocket fuels on their own.  They are very stable until you set fire to them, and they create a lot of heat, and both these are good.  But they give off very little gas, and so there is nothing actually to eject stuff out the back end of the rocket.  You just end up melting its casing.

But suppose you were to mix thermite with a binder?  That would decompose to gas to give an exhaust, and the exhaust would carry the metals from the thermite along with it out of the nozzle.  And, because copper has a high atomic weight, the exhaust gas velocity needed to carry a given momentum would be low and easy to control.   Lead oxide and aluminium should work even better in this regard, though it might not be a good idea to get downwind of the exhaust...

But we still haven't solved the control problems for a solid fuel rocket.  How do you turn it off, and then on again?

Instead of making the entire rocket body one solid lump of fuel, lighting it at the bottom end, and then running away as fast as your legs can carry you, why not wind a filament of rubberised solid fuel into a reel?

The reel would fill the entire body of the rocket, except for a liquid-fuel-sized engine at the bottom.  You would unwind the reel and feed the filament into the engine, where it would be ignited.  If you match the filament feed rate to the rate of burn, the thing would run steadily.  To shut the motor down, simply guillotine the filament then stop the feed.  To start it again, start the feed and re-ignite.

It ought even to be possible to arrange things so that, as the filament unwinds, the reel moves slowly inside the rocket body to keep its centre of gravity in the same place, making flight control much simpler and the rocket more stable.