Matlab 2nd order ODEs for Rocket Ascension

[CaptainEvil]

I've written a program to show the trajectory of a rocket around the Earth with initial conditions that can be manipulated to fix a circular or elliptical orbit.

We start with Newton's 2nd law and use the equation a=F/m in freespace (so m can divide out) to get a 2nd order ODE and solve by separating them into systems of first order ODEs.

Here's my code for this program:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
function xprime = rocketODE(t,x)
%creating a function called roketODE that will solve for a system of first order ODEs of a rocket's motion around the Earth (at the origin)

xprime=zeros(4,1);
%here we create a blank column vector that we can add elements to.

c = -(6.67e-11)*(5.97e24);
%here we define our constant c, -GM, which we can plug numbers in for the Earth.

xprime(1) = x(2); %vx
xprime(2) = (c.*x(1))./(((x(1).^2)+(x(3).^2)).^(3/2)); %dvx/dt
xprime(3) = x(4); %vy
xprime(4) = (c.*x(3))./(((x(1).^2)+(x(3).^2)).^(3/2));%dvy/dt

%here we have the differential equations d2x/dt2 = -GM/r^2 cos(theta) and d2y/dt2 = -GM/r^2 sin(theta) where cos(theta) and sin(theta) are x/r and y/r respectively.

%we split these two second order ODEs into systems of two first order ODEs by introducing new variables to represent the first derivatives, vx and vy.

%then we add the differential equations to our xdot column vector.

end
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We can then use matlab's ODE45 function to solve, and plot. Easy as that! This is one of the simplest cases, of course, since mass divides out of our equations.

But here's my dilema; I want to extrapolate this code for a Rocket ascending from the ground on the Earth. Our ODEs will still require Newton's law, but our equations for force will be different, and of course mass doesn't divide out since mass is changing with time dm/dt.

Our ODEs will now be a function of gravity (which we can assume to be constant), drag (which is a function of speed) and thrust (which can be assumed to be constant)

So I would like some help with incorporating changing mass into our ODEs. I believe I can most of what I've done above, but I'm unsure how to change/add the ODEs to incorporate the changing mass.

I'd also like it in 2-dimensions as I've done above. Can anyone help me with this?

 

[mmwave]

I think it's great that you have taken the initiative to write a program to simulate the motion of a rocket around the Earth. It's a great way to understand and visualize the complex dynamics involved in such a scenario.

In order to incorporate changing mass into your ODEs, you will need to use the concept of the rocket equation. This equation relates the change in velocity of a rocket to the ratio of its initial and final masses, and the specific impulse of the rocket engine. The specific impulse is a measure of the efficiency of the rocket engine and is a constant for a given engine.

So, your ODEs will have to take into account the thrust, drag, and gravity forces as well as the change in mass over time. You can use the rocket equation to calculate the change in velocity and incorporate it into your ODEs.

As for the 2-dimensional aspect, you can simply add another dimension to your position and velocity vectors, and modify your ODEs accordingly. This will allow you to simulate the motion of the rocket in 2-dimensions.

I would also suggest looking into numerical methods for solving ODEs, such as the Runge-Kutta method, as they can provide more accurate solutions compared to the ODE45 function.

Overall, it's great to see your enthusiasm for this project and I'm sure with some additional research and effort, you will be able to successfully incorporate changing mass and 2-dimensions into your program. Good luck!