Derivation of the terminal velocity optimal ascent

[Borogove]

It's common knowledge that the fuel-optimum ascent speed is terminal velocity; someone over at Stack Exchange did the math to explain it. The calculus is just beyond my attention span, but I thought I'd link it:

http://space.stackexchange.com/questions/5419/climbing-at-terminal-velocity-minimizes-losses-but-why-and-of-what

[Vanamonde]

We like to keep the more nuts-and-bolts/math kinds of discussion in Gameplay, so people know where to find it consistently. And so, thread moved.

[Mesons]

I've wondered the same thing for a long time.

If only the math was as simple as so many claiming terminal velocity seem to think. The big problem that the post you linked misses is the complication of Isp varying with pressure. Using F = dm/dt * v_e would only work if dm/dt wasn't also a function of height z. F is the quantity that's fixed, not v_e as the author of that answer assumed.

I worked on this problem for a good couple hours before I decided I have better things to do. Suffice it to say that only for engines like the aerospike, where Isp is essentially fixed, is terminal velocity the best. For anything with a substantial Isp change, I'll bet that hauling ass through the thick lower atmosphere is the best, so that you can get up to where the drag is negligible and your Isp is maxed ASAP. Besides, you can only surpass terminal velocity if your TWR is 2.0 or more. For most rockets, your TWR won't be more than 3 or so, so there isn't really a lot to be lost by gunning it the whole way. You definitely minimize gravity losses once you are getting to thinner air that way.

[Jimbimbibble]

My intuition tells me that gunning it is not best because in order to gun it you need to be burning a lot of fuel in the lower atmosphere where it's less efficient. Sure, you get out in less time, but time doesn't matter here because we care about the amount of fuel consumed for a given change in height. To test this, build a rocket with a TWR of about 3:1 and about 3km/s deltaV. Launch it strait up at full throttle and again at terminal velocity using Mechjeb. The one that achieves a higher apogee is more efficient.

[Red Iron Crown]

Careful with the terminology, the linked derivation minimizes delta-V expended, not fuel used. Minimizing fuel used results in a much lower initial TWR (1.2-1.3) and never approaches terminal velocity.

It seems like a contradiction to consume more delta-V but less fuel, but it happens because of the lower engine mass in the fuel-efficient craft.