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Can you derive the Tsiolkovsky rocket equation without calculus?

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    $\begingroup$ This sort of questions without showing any progress or without any detail are considered to be trivial and are prone to be withheld and to be downvoted. Show your progress under the question asked with detailed pictures of what you have come through. These type of trivial questions are not encouraged here. Also, first of all, calculus is the study of variations and in a rocket everything varies during its take off for example, the fuel hence the mass, position, velocity everything varies. So, you can't actually solve it without calculus. $\endgroup$ – John Brookfields Jun 19 at 6:23
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No. You want to get to the total change in velocity. To do this, you have to add up (integrate) all the tiny contributions from each little lump of fuel as it is ejected. Each contribution is different because as each "lump" is ejected, the rocket gets lighter.

You could get an estimate if you work out the $\Delta{v}$ from the first ton of fuel, reduce the rocket mass by 1 ton, get the next $\Delta{v}$ from the next ton, reduce by another ton, and so on until you use up all the fuel, then add all the $\Delta{v}$s - but this would give only a crude estimate of the total. You could repeat the calculation and do it for each kg of fuel (1000 $\times$ as many steps). This would get you a better result, but still not precise.

Finally, you'd do it for an infinite number of infinitesimal quantities of fuel at each step - which is exactly how integral calculus works.

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