Modeling a rocket using Tsiolkovsky's equation and ordinary differential equations

Question

So I have this uni assignment to make a model out of ODEs, and my idea was to use rockets. After some research I found about Tsiolkovsky's equation and tried using it on a Falcon 9 (disconsidering gravity and drag for now). However, my calculations always return a speed smaller than the real one, which means the gravity/drag thing isn't the problem. For reference, I'm using this for data (the Falcon 9 v1.1 bit, page 9), which gives me, at first stage separation: $$ \Delta v = 282\times9.81\times ln(\frac{439+78+6}{28+78+6})\rightarrow4383m/s $$ Which is nowhere near the speed the rocket has here near the stage separation. I know the first stage mass isn't dry by the end, but considering that, the results would be even farther. I've also considered the dragon shuttle's mass.

What I would like to know is if my mistake is in the model I'm using or the data. All help is welcome.

Answer 1

Which is nowhere near the speed the rocket has here near the stage separation.

The video is reporting velocity in km/hour, not in m/s. 4383 m/s is 15780 km/hour -- over twice as fast as the Falcon 9 is moving at stage separation.

In general, the rocket equation isn't going to give you directly usable results for the initial ascent phase of a rocket. Drag varies with velocity and air density (dependent on altitude); the flight path of the rocket is continuously changing, so the effect of gravity on velocity isn't straightforward, etc.

Answer 2

I would like to thank everyone for the immense help, and inform that I've found a study on modeling rockets using the forces involved:

https://pages.vassar.edu/magnes/2019/05/12/computational-simulation-of-rocket-trajectories/

It works like a charm, but for some reason I still don't understand it gives out half the actual speed and half the actual altitude for the Falcon 9 (I compared the model to the CRS 10 launch). And yes, I changed the rocket parameters in the code (it's originally the Falcon 1). If I multiply the end result by 2, it ends up with a quite accurate prediction, which is amazing (less than 5% error up to the end). Also, it accounts for Max Q, (but not for multiple stages (that's only a matter of adding some lines of code I guess)).

Thanks to you guys now I actually understand what my mistake was (unit conversion) and what Tsiolkovsky's equation is about (an ideal estimate of the maximum speed a rocket can achieve with given mass of propelant).

As for the new model (using the forces instead of Tsiolkovsky's equation), I got to a ODE as intended and it is quite... big. I'll probably stick to using the computer.

Thanks again!