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I've been trying to model the Saturn V's velocity using Tsiolkovsky's ideal rocket equation, and in the process, I think I may have made a mistake with regards to the specific impulse?

I've come up with the following equation, taking the change in gravity into account. (a(t) is the altitude function derived from actual values, with a very small error range, c(t) is the fuel consumption of the rocket per second)

(note: what appear to be powers after the end of most lines are footnote references, apologies for the confusion!)

And this a plot of the modelI created versus the actual values plotted against time-

The percentage error here between the two sets of values vary from ~80% to ~14%, and the graph shape is vastly different.

I'd like to ask if there is a change in specific impulse? Or have I done anything else wrong in modelling the equation above?

Ithinkspecific impulse does change, but is there a mathematical equation by which I can rewrite this equation to take that into account?

Also, I apologize if I've made a stupid mistake, I'm a HS student doing some fun research, quite new to this!

Thank you very much!

edit: title was shortened, "Saturn V specific impulse issue in velocity modelling with Tsiolkovsky's equation against actual values?" was the original title

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# Saturn V specific impulse issue in velocity modelling with Tsiolkovsky

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