The Chair Force Engineer doesn't think much of a catapult start for rockets. His point is that if your catapult gets the rocket going fast horizontally, it just makes the max Q problem and drag losses worse.
As the rocket accelerates, the dynamic pressure on the front of the craft gets larger, until it gains enough altitude and vertical velocity that the atmospheric density drops faster than the velocity component rises. The maximum pressure experienced is max Q. Max Q is a problem because that drag turns the rocket's forward velocity into heat, which the structure has to deal with somehow, usually with a thin layer of ablatives on the nose.
A study done at UC Davis, A Study of Air Launch Methods for RLVs looked at dropping the rocket from an airplane, as is done for the Pegasus and SpaceShipOne rockets. They found that the high starting altitude helped, as did releasing the rocket with some significant vertical velocity component.
The thing that air launch and (vertical) catapults have in common is they both allow the rocket to delay fighting gravity.
As I pointed out in an earlier post, a rocket coming vertically off a launch pad is losing nearly all of its delta-V to gravity losses. Later on in the flight, once a good bit of propellant has been burned off and the acceleration has improved, a much smaller portion of the delta-V is lost to gravity. Note that the total vertical impulse required is a function of the time spent getting to orbital velocity, and is insensitive to the actual flight path.
The situation is much different for a rocket firing horizontally. Delta-V expended horizontally adds to the rocket's final velocity regardless of what the current acceleration is. Unfortunately, horizontal velocity added early in the flight adds a lot of drag loss.
So to minimize gravity losses, the rocket should start out firing at least somewhat horizontally, losing vertical velocity, and only later recover that vertical velocity. There is some limit to this pattern, since too much horizontal velocity too low in the atmosphere will exacerbate drag losses, and since we don't want the rocket to smack into the ground. Starting at high altitude helps, as does starting with some vertical velocity.
This leads to a concept I'll call "catapult gain", because I haven't read about it and don't know what it's really called. If I start the rocket off with a couple hundred m/s of vertical velocity, I increase the velocity at first stage burnout by more than the initial velocity boost, for two reasons. First, the gravity losses are lower, since the rocket can postpone some of its vertical impulse to when it is more efficient. Second, the first stage rocket can start out with more gas in the tanks, burn longer, and deliver more delta-V, because it doesn't have to have positive vertical acceleration right from the start.
Catapult gain is limited to the first few hundred m/s of velocity. The first effect is limited because we're reducing gravity losses, and increasing drag losses. There is only so much gravity loss to be mined, and drag losses rachet up quickly. The second effect is limited for the same reason that upper stage minimum accelerations are limited: eventually the extra delta-V is stretched over so much time that the gravity losses start to overcome all the extra delta-V. More starting velocity is always good, of course, but the point is that it reverts to a gain of one. To give a rough idea of catapult gain, some spreadsheet calculations indicate that a 250 m/s jump reduces the upper stage delta-V requirements of a SpaceX Falcon 9 by 500 m/s, for a catapult gain of two. Lest that seem small, note that it would allow the dead weight of the upper stage to increase by around 18%, and I'd guess the payload would increase by at least twice that.
So now that I've established a motivation for a vertical catapult, let me suggest one: a steam rocket. This is a very big bottle of very hot water at very high pressure, which partially flashes to steam as it exits. It has really crappy specific impulse (about 50 seconds), but can produce really large amounts of thrust very cheaply. For a Falcon 9 stage 0, it would be an 80 tonne steel tank (HY 100, safety factor 1.5) holding 400 tonnes of water, starting at 300 degrees C and 86 bar. The thing would produce around 26 MN of thrust for about 7.5 seconds, boosting the Falcon to 280 m/s. A 25 meter pipe, inserted up the throat of the nozzle, would add another 40 m/s by acting as the piston in a cylinder, allowing the Falcon to react against the Earth instead of mere propellants.
Let me point out how simple this system is. It has no moving parts on the vehicle, no flight-operational valves, no sequencers. The nozzle is bolted down onto a seal while the water in the tank is heated (pumped through an external heat exchanger). Fire the explosive bolts and it goes. There are no gimbals, and no guidance system. The LOX-Kero first stage is started while on the pad, and fires directly onto the top of the tank, which being solid steel backed by water is unaffected. The interstage is a truss which the engines fire through. There is no recovery system: the thing sails through the air for about a minute, coasting up to 4 kilometers high, and then comes crashing down into the sea, where it eventually resurfaces, nozzle down, until it's dragged back to shore. If we wanted to be sure it doesn't sink, we could inflate a balloon inside the casing, which would passively inflate as the internal pressure dropped during boost, and would require no pyrotechnics or interface of any kind. The exhaust is hot water and steam, has mild overpressure, requires no water suppression, nor cleaning of nasty chemicals afterwards. I will guess that the tank and nozzle can be built for $500k. Fuel costs for heating it up are about $10k, so an extensive program of test-firings with dummy rockets atop would be cheap.
I recommend tilting the launch pad slightly to ensure that the thing comes down in the ocean and not back onto the pad.
The thing vents 53 tonnes of propellant per second, which is orders of magnitude more than any chemical rocket ever built, and will produce a singular launch spectacle. So long as pictures of parboiled parrots can be avoided, the PR value alone should be immense. As a side benefit it should allow the Falcon 9S9 to put more payload into orbit than a Delta IV Heavy, and maybe enough more than the Shuttle to allow it to lift Shuttle ISS cargoes with an added strongback.
EDIT NOTE: In an earlier version of this post I claimed a doubling of throw weight. Math error. That's what I get for posting after midnight.