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So, for reasons, I'm trying to get a probe to Eve as quickly as I reasonably can, for certain definitions of "reasonable" -- and decided the way to figure this out was to start with the basic kinematics equation of .5at2+vt+d and do a lot of algebra starting with the conceit that we want the acceleration portion to equal d. This resulted in just deriving the brachistochrone equation like a doof: (total trip t = 2*sqrt(d/a), total dv = 2*sqrt(da)) Throw in the min distance from Kerbin to Eve being 3,668,900km, pick a comfortable acceleration, and there's the dv and burn time. However, that's assuming crazy things like instant re-orientation, and for a = 1m/s, leads to a 22 tonne ion craft I already have around but also a pair of 16.8hr burns. I could just make everything bigger to burn harder for less time, but that could make the lower stages get a bit out of hand. So, we come to realm of having a coasting phase, which highlights the fact that the brachistochrone equation is a special case of a more general equation, and we've just set coast time to zero. As such, instead of d being just a*t2/2, which was convenient, now it's a*ta2/2 + vmaxtcoast where ttotal = ta+tcoast and vmax= roughly(a*ta). I think I can maybe constrain it with a chosen ta -- say, 10hrs (I can handle a pair of 5 hour burns over a weekend), and a vaguely acceptable acceleration rate, say, 1m/s again, for ease. So to solve for the times, we substitute a*ta for vmax and ttotal-ta for tcoast a*ta2/2 + a*ta * (ttotal-ta) = d => (d-a*ta2/2)/a*ta = ttotal-ta => d/a*ta-ta/2 = ttotal - ta => ttotal = d/a*ta + ta/2 This gives us 1m/s2*(36ks)2/2 + 36km/s*(ttotal-36ks) = 648Mm + 36km/s*(ttotal-36ks) = 3668.9Mm => 3020.9Mm/36km/s = ttotal-36ks = 83913.89ks => ttotal = not much more...that kinda gets overwhelmed by coast time...of over 23 thousand hours. I might not've done that right. I blame it being after 2:30. Any thoughts/corrections?
