I am not one for minmaxing my playing, but the underlying math of KSP has lit a fire on my already rabid appetite for mathematics. I am curious as to how one would calculate the ideal escape from Kerbins atmosphere with a one stage rigid body. (One stage, since more stages just complicate the final equation.) As far as I know, there are three forces (At least three significant ones) acting on the body while in flight; the constant thrust (constant for each stage, as long as there is fuel left), the downward force due to gravity (which decreases linearly with the mass) and the drag force, which appears to be the most arbitrarily simulated phenomena in KSP. I am assuming the drag force is proportional to the square of the velocity, just like in textbook examples of differential equations involving drag. Also, seeing as the drag is affected by the atmospheric density, which in game is simulated with an exponention function, the final equation seems to become something way beyond my current abilities - a second order non linear differential equation. Therefore, I call on the help of those with a sound knowledge of such matters, and the magicks of Laplace and Fourier! (Of which I know next to nothing) This is my premature attempt at translating the problem into an equation I am unable to solve. Would this be a good start for answeing my problem? I see that this equation would only account for vertical motion, but someone more able that this here iHorse might be able to translate the problem into vector calculus to account for circular orbits and stuff like that. In shorter words, what I am asking is this: Is there a way to calculate an ideal escape trajectory out of Kerbins atmosphere? What variables needs to be accounted for, and how should this problem be translated into calculus? I would love to see someone take a shot at this, both for my own learning purposes and for the collective good of the OCD's among us.
