For every reasonable definition of m and a (which is missing), this is wrong.According to Newton’s Second Law, the product ma must remain constant.
Not necessarily, R does not have to be constant.Since its acceleration is changing over time
You solved a differential equation before, where is the problem with setting up another?Thus, this system leads to a seemingly circular and unresolvable issue.
For every reasonable definition of m and a (which is missing), this is wrong.
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?
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.
That assumption should appear somewhere then. For most rockets, it is not true.CSteiner said:Well, I was assuming a constant R.
In the equation with the integral, in the denominator.CSteiner said:There is no delta m in the integral.
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.CSteiner said:I guess I just assumed that air resistance was always anti-parallel to direction of motion.
Sure, but alpha cannot be calculated based on position values relative to the starting point or some other fixed reference.CSteiner said:Alpha can depend either on position or on time. Either way its an unknown function.
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.CSteiner said:I don't know any methods for solving an ODE containing an unknown function (the angle of attack)
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 said:Is it too far off to even be an approximation?
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.
The same thing as you when you calculate the position in steps of 0.5 seconds.CSteiner said:I meant that. What do you mean by the iteration steps though?
The main methods used for modeling 2d rocket flight include numerical methods such as Euler's method and Runge-Kutta methods, as well as analytical methods such as the Tsiolkovsky rocket equation and the two-body problem.
The accuracy of these methods depends on various factors such as the complexity of the rocket design, external forces such as air resistance, and the accuracy of input parameters. Generally, the more complex the model and the more accurate the input data, the more accurate the predictions will be.
While these methods are specifically designed for 2d rocket flight, they can be extended to 3d by adding an additional dimension to the equations and variables. However, this may significantly increase the complexity and computation time of the model.
These methods can take into account uncertainties and variations by using probabilistic methods and adding stochastic variables to the equations. This allows for a range of possible outcomes rather than a single precise prediction.
Some limitations of these methods include the assumption of a perfect and symmetrical rocket design, neglecting external forces such as wind and turbulence, and the need for accurate input parameters. Additionally, these methods may not accurately predict the behavior of the rocket during certain stages of flight, such as during separation events.