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How much higher is a rocket following a gravity turn going to be able to reach compared to a rocket following a vertical flight ?
I understand that it depends on which type of rocket and which type of gravity turn but just a rough estimate for a real rocket (Saturn V type) not necessarily based on any calculations just from your empirical knowledge ? 50% higher ? Twice as high ?
I'm just trying to get an idea of how important the change is.
A rocket flying vertically will reach a much higher altitude than one flying a gravity turn; it just won’t stay at that altitude.
A vehicle flying sideways at sufficient speed will continuously “miss” the Earth as it falls, yielding a circular or elliptical closed path. If instead all the velocity the rocket can produce is applied vertically, the result will be like a baseball thrown straight up instead of sideways; it will reach a quite high altitude before falling straight back to earth and crashing.
The exact altitude reached will vary a lot with the design of the rocket; a low-thrust upper stage as used on Ariane 5 will not reach as high an altitude going straight up as would something like Falcon 9.
As an example, an Atlas V 401 might be capable of putting a given payload into a circular orbit at 250km altitude via a gravity turn trajectory. The same stack going straight up would reach an apogee of perhaps 4000km according to my crude simulations.
As Russell Borogove says, if you just go straight up you fall back. Thus I'm looking at this from a standpoint of going straight up and then burning horizontal.
I don't know what the numbers look like in the real world but I have played with trajectories almost like this in Kerbal Space Program when hauling drag monsters. (The game has no good answer for lifting large rovers, thus some very ugly rockets go up at times.) Going nearly vertically until I'm out of the atmosphere costs about 10% more delta-v than a proper flight profile. Note that this is with a thrust to weight ratio well above what is normally used for real world space launches and thus understates the penalty.
Alternately, you're looking at an interplanetary trajectory. That way you will not fall back, it's simply a matter of efficiency. My memory of the one time I tried it was that the penalty was over 20%--and that with an even higher thrust to weight ratio as I was trying to approximate a vertical cannon launch + minimal fuel to go into orbit. (Go out to the edge of the sphere of influence, a small burn to put my periapsis in the upper atmosphere, aerobrake until my apoapsis was in low orbit, the a final burn to circularize.) Note that this is even more sensitive to the thrust to weight ratio as you're spending even longer without horizontal velocity.
