Methods for Modeling 2d Rocket Flight

[CSteiner]

I've formulated a method for modeling the flight of a rocket projectile. Can anyone read it over and point out any mistakes false assumptions, etc? thanks!

 

[mfb]

According to Newton’s Second Law, the product ma must remain constant.

For every reasonable definition of m and a (which is missing), this is wrong.

Since its acceleration is changing over time

Not necessarily, R does not have to be constant.

$\Delta m$ in the integral has a different meaning compared to above, but uses the same symbol.

Air resistance depends on the velocity and the direction of motion, you cannot integrate it like that (with both meanings for integrate).

Why does alpha depend on the position of the rocket?

Thus, this system leads to a seemingly circular and unresolvable issue.

You solved a differential equation before, where is the problem with setting up another?

The iteration does not work like that. It gives something like an arc, but not the correct results.

 

[CSteiner]

Thanks for taking the time to read and reply!

For every reasonable definition of m and a (which is missing), this is wrong.

hmm, you're right. I should have done the derivation purely in terms of momentum.

Not necessarily, R does not have to be constant.

Δm in the integral has a different meaning compared to above, but uses the same symbol.

Air resistance depends on the velocity and the direction of motion, you cannot integrate it like that (with both meanings for integrate).

Why does alpha depend on the position of the rocket?

Well, I was assuming a constant R.

There is no delta m in the integral.

Huh. I guess I just assumed that air resistance was always anti-parallel to direction of motion. I suppose I'll just set the air resistance to be negligible then.

Alpha can depend either on position or on time. Either way its an unknown function.

You solved a differential equation before, where is the problem with setting up another?
The iteration does not work like that. It gives something like an arc, but not the correct results.

I don't know any methods for solving an ODE containing an unknown function (the angle of attack). I tried many ways to get an explicit equation of velocity and angle of attack, but I didn't succeed. I don't believe there is a way to avoid the implicit nature of the system. Thus the approximation algorithm.

Is it too far off to even be an approximation? What if I modified the algorithm to take the average of angle n and angle n-1, like the RK2 method?

 

[mfb]

Well, I was assuming a constant R.

That assumption should appear somewhere then. For most rockets, it is not true.

There is no delta m in the integral.

In the equation with the integral, in the denominator.

I guess I just assumed that air resistance was always anti-parallel to direction of motion.

That is a reasonable approximation, but it is unrelated to my point. The acceleration from air resistance is not constant in time, so its contribution to velocity is not proportional to time.
Actually, real rockets have two different directions: direction of motion (relevant for drag) and direction of thrust (relevant for propulsion). Usually they are not so far apart from each other because that optimizes efficiency, but that is another approximation that should be mentioned somewhere.

Alpha can depend either on position or on time. Either way its an unknown function.

Sure, but alpha cannot be calculated based on position values relative to the starting point or some other fixed reference.

I don't know any methods for solving an ODE containing an unknown function (the angle of attack)

Every differential equation has at least one unknown function. Sure, alpha will couple the two equations for the directions (a realistic treatment of air resistance will do the same) and probably make an analytic solution impossible, but you can get the iteration steps out of this differential equation.

Is it too far off to even be an approximation?

In its current version, I would not use it for any predictions. To know "oh well, the rocket will go up and forwards and then fall down again" you don't need calculations, and I don't think it is more precise than that.

 

[CSteiner]

Every differential equation has at least one unknown function. Sure, alpha will couple the two equations for the directions (a realistic treatment of air resistance will do the same) and probably make an analytic solution impossible, but you can get the iteration steps out of this differential equation.

I meant that. What do you mean by the iteration steps though?

 

[mfb]

I meant that. What do you mean by the iteration steps though?

The same thing as you when you calculate the position in steps of 0.5 seconds.
You'll certainly need the velocity there.