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in the classical theory Tsiolkowsky is normally explained as:

$\Delta v=v_{eff}*\text{ln}\frac{m_0}{m_b} $

Then the books say, for staying in LEO one need approx. $\Delta v=7.8$ km/s

But what about the drag resistance the rocket encounters and the potential energy it has to overcome? When I want to include this it should be:

$\Delta v_{requried}=7.8 km/s + t*9.81 m/s^2 + t*\frac{F_{Drag}}{m_0} = v_{eff}*\text{ln}\frac{m_0}{m_b} $

To make it more correct, one needs to integrate this formula since the mass for the aerodynamic drag is changing. Is this formula correct? So this is true for a vertical launch. Otherwise one needs to include some angles.

Is it correct that the term $t*9.81 m/s^2$ stands for the "potential energy". So if the rocket climbs 100 km up, the time t which is needed for this times gravity constant stands for the potential energy?

Thank you for your feedback Best regards

Lucas M.

Max H.
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    The term $t\cdot \mathrm{9.8}$ doesn't have units of energy. So that part is not correct IMHO. So, it wouldn't be true for vertical launches either. For 100 km up the potential energy should not depend on the time taken to reach there IMHO. Yes, you need integration; but AFAIK, there won't exist an analytic solution for the result of that integration. – AJN Jan 25 '22 at 11:55
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    I usually see it spelled Tsiolkovsky but perhaps this is an alternate English spelling. Not a duplicate but related: Derivation of a variant of the Tsiolkovsky rocket equation which includes gravity – uhoh Jan 25 '22 at 12:03
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    To my mind, you only have to use for gravity : -gsin(angle)time – Max H. Jan 25 '22 at 14:39
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    how did you come up with that formula? Can you explain from first principles? With a diagram and well defined variable definitions if possible. – AJN Jan 25 '22 at 14:59
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    Please also mention what your final application for this result would be. There might be easier methods of solving the original problem without resorting to finding the corrective terms for the Tsiolkovsky equation. – AJN Jan 25 '22 at 15:02
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    The goal is: I want to calculate the optimal staging of a rocket, where I can include gravity and drag. But the original staging formula does always have a delta_v calculation: see here: https://spacecraft.ssl.umd.edu/academics/791S16/791S16L03.rocket_perf.pdf – Max H. Jan 25 '22 at 17:31

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