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I've been researching this for some time now and I got stuck.

The best resource I've found is this book. The "optimising a multistage rocket" part talks about a three stage rocket but it seems applicable here. The part where I got stuck is at the page 29, where author represents fractional mass of the three stages. Could someome please explain the steps after that to me ? His "simple manipulations" don't seem as simple to me :D

Other solution I wanted to try was to modify the Tsiolkovsky rocket equation for my two-stage rocket into a function with a variable being the ratio between the stage 1 and 2. This function's maximum would be the desired ratio. But my high school mathematics obviously weren't enough for that, because I got nowhere.

Could you please nudge me in the right direction ? Thank you very much.

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These are long explanations in the book and you effectively ask for even longer ones because the author seems too fast to you. It's a bad template for a question to ask on this server. One can't write doubled books for free and just for you.... The Tsiolkovsky equation doesn't deal with any stages and to answer questions about optimal separation to stages, one needs to consider the mass of the payload and the rest of the rocket separately, payload ratios, and many other things described in the book. If all the mass of the rocket were just fuel, the separation to stages would be immaterial. –  Luboš Motl May 26 '13 at 6:00
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