I would think an open cylinder type rocket where the rocket fuel gets injected into the "hollow" cavity that air is going through would be a more effective than the rocket fuel just burning itself. Does anyone know of any links showing data from when they tested these types of rockets?
[Physics] Why doesn’t NASA use RAMJET rockets to get into space
rocket-science
Related Solutions
Specific impulse is usually defined as $I_{sp} = \frac{F_T}{\dot m ~ g_0} $
That's true only if you use standard metric units. With force expressed in pounds-force and mass flow rate expressed in pounds, one simply divides the force (in pounds-force) by the mass flow rate (in pounds/second) and voila! you have specific impulse in lbf·s/lb. For example, the first stage of the Saturn V produced 7,715,150 pounds-force of thrust at launch while consuming fuel at a rate of 29,157.58 pounds/second. Divide 7,715,150 by 29,157.58 and you get 264.6, the specific impulse at launch. Properly, this value of 264.6 is in units of lbf·s/lb. If you do the math, it is also numerically equal to the specific impulse in seconds.
Alternatively, one could convert that force to newtons and the mass to kilograms, yielding 34.3817 meganewtons of thrust and 13.22565 metric tons per second of fuel consumption. Now the division yields 2594.9 m/s. Divide by g0=9.80665 m/s2 and you get 264.6 seconds.
A second alternative is to convert that force in pounds-force to kilogram-force. This conversion yields a force of 3,499,530 kilograms-force. Now we're back to using the trick of simply dividing the force (in kg-f) by the mass flow rate (in kg/s): 3499530/13225.65 = 264.6.
Germany was the leading European developer of rocketry up until 1945. They used kilogram-force to express thrust rather than newtons. The Americans and Russians took over after that point. The Americans stuck to using customary units, pounds-force and pounds-mass. The Russians followed the German tradition of expressing thrust in kilograms-force. In both cases, simply dividing force by mass yields specific impulse in seconds.
In fact, despite being a banned unit, most European aerospace engineers tended to express thrust in kilograms-force rather than newtons up until the 1980s, and some still do. It is a very convenient unit for spacecraft and aircraft that operate near the Earth.
A major problem with your idea is the following: If you're looking to eliminate the need for a supply of propellant with the railgun, it won't happen. The impulse imparted to the rocket is, by Newton III, imparted o on the railgun and so each launch will de-orbit the latter. It will need a constant supply of fuel to keep its orbit stable. A given quantity of fuel imparts a given impulse, so whether impulse is imparted directly to the rocket through chemical engines or whether to the launcher, you won't save on the need for propellant.
The other engineering issues aside (like rail wear cited by Dmckee):
I know railguns are cool and all, but the rail wear problems are still insufficiently resolved for them to move out of the laboratory. The current needed to get high performance just chews through the material of the rails.
another major reason that if you planned to boost humans in this way, or delicate instruments, the rails would need to be very long. Not impossibly so, but clearly a project hugely beyond the ISS, for example. Suppose we are in LEO and wish to boost to Earth escape speed: we need a delta-V of about $3.5{\rm km\,s^{-1}}$ ($11.2{\rm km\,s^{-1}}$ from LEO orbital speed of $7.8{\rm km\,s^{-1}}$). If you did this at constant acceleration $a$, then (calculating in a frame comoving with the LEO orbiting railgun):
$$s=\frac{(\Delta V)^2}{2\,a}$$
and limiting $a$ to, say, $5\,g$, I get a rail length of about $123{\rm km}$.
Best Answer
There are several issues with using ramjets on rockets:
The startup problem. Ramjets rely on the supersonic velocity of the vehicle to create the compression needed for the combustion chamber to operate. So you'll need something (traditional motor stage, air breathing engine, etc) to get the craft moving to the point the ramjets can actually function.
Ramjets have relatively low thrusts. They don't weigh much because they have no moving parts, but they don't produce a whole lot of thrust. So a rocket would need a large number of the engines to function.
Ramjets will eventually stop working as the speed gets higher. At some point, the duct upstream of the combustion chamber will not slow down the incoming air enough. This will make the air in the combustion chamber supersonic instead of subsonic. This is what a SCRAMJET engine is used for, but that's a slightly different configuration. So either the ramjet would need to have moving parts to change its geometry to handle subsonic and supersonic flow in the combustion chamber, or you would need to shut down the ramjets and open up some scramjets (which means the rocket would now have 3 different types of engines).
Ramjets, and scramjets, are air-breathing engines. For anything that gets very high in to atmosphere, there isn't much air left. So the engines won't work very well anymore. Traditional rockets carry both their fuel and oxidizer so they work anywhere.
All of that said, there is considerable research going on in the area. The USAF is actively funding research into them, and the Russian military has a pretty cool design for some self-guiding bombs that drop off of an ICBM and fly to their targets using scramjet engines.