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Fix typo (from hitting z instead of back-apostrophe)
Fix (by integrating over time instead of only calculating momentary acceleration) and give equations in terms of payload mass also.
Correct and compact terms in x equation more
Correct x equation (I think)
Simplify x equation further (there are numbers that one part needs to be divided by that Mathematica decides to multiply other things by and put in the denominator)
Explain that solved equations are in terms of thrust, not impulse and consumption (which Mathematica's Newton-Raphson method realised on its own).
Simplify solved equations and standardise their orders (in products, the number goes first, then the unknowns, then the adjusteds, but in sums the terms are ordered by complexity in descending order) and no more x computation (z remains, otherwise more complicated) multiplying a sum by (10*m*x+z+10)^2 and dividing one of its values by it again (only multiplying the other things in the sums by it, the additional brackets take the same length as the exponentiation and less ugly),.
Fix disclaimer (not sure what happened initially)
Create
Update Desmos graph to new one (which also has defined functions for each operation)
Reduce cringe
Remove the cringe
Make clearer
Updated Optimum and Maximum Profitable Fuel Mass (markdown)
Make clearer, simplify equation for optimal x value in multistage rockets, remove unnecessary cringe part at the end of the Desmos links, I will will make a single link for both purposes and displaying with respect to altitude on planets with constraints of number of squares of engine surface area per tonne of fuel (changing reciprocally over altitude beginning at that of 'optimal stages' tutorial rocket).
Replace wrong previous simplification with correct one (1 in denominator still annoying)
Fix simplified upwards acceleration equation, the 1 in the denominator is very annoying.
Add simplified upwards acceleration equation
Simplify absolute optimal x and y values for maximum delta-v per payload without accounting for total mass's effect on prior stages (WolframAlpha said they could be simpler than previous ones)
Fix s equation also
Correct equation for optimal y for delta-v/total mass, was previously wrong and had mismatched brackets
Add continuousness disclaimer and simplified equation double-decker fraction replacing WolframAlpha link with quintuple-decker one, not sure if faster than substituting optimal x into y equation.
Change WolframAlpha link for given engine's optimal delta-v:mass ratio.
Fix brackets in previous edit
Simplify optimal x value equation, unfortunately equations of the form sqrt(x^2+y) can't have the x part expanded without using an infinite series.
Added link to [[Optimal engine:payload proportion for given gravity]].
Replace first Desmos graph link with new one (with all engines instead of just one and vertical bars representing planets).
Replace WolframAlpha equation with better one (using shorter x equation), gives shorter answer (if you replace its sqrt(5)*sqrt(g*m)'s with sqrt(5*g*m)'s).
Fixed equation formatting, made wording clearer.
Replaced part of optimal x equation with simpler one (that also works assuming g, m, and t are positive (which they are)).