What's wrong withb my Runge-Kutta approximation?

[spraynard]

I've been playing this game to practice / learn classical mechanics. Last night, I wrote the Runge-Kutta algorithm in c++ in order to approximate the velocity of a rocket after a given period of time has passed, using a very basic model.

I'm only taking a few things into account:

- time

- constant thrust from the rocket

- constant gravity (not taking into account distance from the surface of kerbin)

- constant atmospheric density (not taking into account altitude)

Using these independent variables, I calculate:

- mass of the rocket given the time

- force of drag on the rocket, given the constant atmospheric density. I calculate the cross-sectional area of the rocket based off the total empty mass. Drag coeffecient is 0.2 Velocity is the velocity at the previous step of the algorithm. I just use the formula on the wiki for drag.

- acceleration based off the constant acceleration due to gravity, the thrust of the rocket, the force of drag, and the mass of the rocket after the given time.

- velocity using the runge-kutta algorithm.

I've built a very simple rocket to test the program. It is nothing more than the starting capsule, an RT-10 solid rocket booster, two modular girder segments, and two mk16 parachutes.

After 28 seconds the program estimates that the velocity is 362.807 m/s. In the game, the velocity is about 555 m/s.

What could be the source of this high error rate? Is using constant gravity and constant atmospheric density going to result in these extremely divergent values? I can post the code, but I figured I'd see if anything jumped out at anyone.

[diomedea]

Without doing a complete analysis (that may require to see the formulae you used to calculate the velocity), two things may have to be considered:

- ISP is variable, the speed of exhaust gases changes with atmosphere density;

- atmospheric density is decreasing with altitude, therefore drag (and that is pretty relevant itself).

[spraynard]

Without doing a complete analysis (that may require to see the formulae you used to calculate the velocity), two things may have to be considered:

- ISP is variable, the speed of exhaust gases changes with atmosphere density;

- atmospheric density is decreasing with altitude, therefore drag (and that is pretty relevant itself).

Thanks for the reply. I was basically guessing, but I figured that without accounting for these two forces (as well as the fact that the force of gravity grows weaker the higher you go), I would still get a reasonably accurate estimation. In your estimation, could these two factors really account for such a vast discrepancy (difference of about 200 m/s after 28 seconds) between the estimated velocity and the actual velocity?

[zarakon]

- ISP is variable, the speed of exhaust gases changes with atmosphere density;

Though in KSP, it's actually the rate of fuel consumption that changes, and the thrust remains constant (reverse of real-world)

That will definitely affect your results, but I think the changing drag with altitude is likely to be a bigger factor

[cantab]

I think assuming constant atmospheric density is the big mistake. Kerbin's atmosphere falls off surprisingly quickly, with a scale height of 5 km (compared to 8.5 for Earth) meaning that 5 km up the air pressure and thus the drag is just 37% of sea level.

[spraynard]

I think assuming constant atmospheric density is the big mistake. Kerbin's atmosphere falls off surprisingly quickly, with a scale height of 5 km (compared to 8.5 for Earth) meaning that 5 km up the air pressure and thus the drag is just 37% of sea level.

It seems like everyone agrees that this is probably what is throwing the results off. I will incorporate the diminishing atmospheric drag into the model and then come back with my results. Thanks everyone!