Solving Rocket Launch Differential Equations

[mysticboon]

Hi I'm currently working on a project which involves solving the rocket launch differential equations to find the apogee of an orbit. I know the analytical model for the equations as:
Δu = Isp*g0*ln(mf/me), where Isp is fuel impusle, mf is mass of full tank and me is mass of empty tank, but for this project I need to solve the differential equations in MATLAB numerically.

I know the general equation as d(M*u)/dt = Fnet = Isp*g0*dmp/dt where dmp is the change in mass of the propellant. I'm just kind of stuck in working around the differential equations to get something I can solve for in matlab. I've tried substituting some differentials for each other in order to get one equation, but I ended up cancelling terms to get something that doesn't make sense to me. I've been using this site to work with the equations: http://exploration.grc.nasa.gov/education/rocket/rktpow.html

% Veq = Isp*g0
% mdot = F/(Isp*g0)
% mdot = dmp/dt
% Mdu = Veq*mdot
% dm = -dmp = -mdot = -F/(Isp*g0)
% dmp = fnet/veq = fnet/(Isp*go)
% du = -(Isp(g0)*fnet/(Isp*g0)/M

Can anyone please give me some help in how to get one or two differential equations to use in matlab? Thank you

 

[mmwave]

Hello,

Thank you for reaching out for help with your project. Solving differential equations numerically in MATLAB can be a challenging task, but with the right approach, it can be done effectively.

First, it is important to understand the physical meaning behind the equations you are working with. The differential equations you have mentioned are based on the principles of Newton's second law, which states that the net force on an object is equal to its mass times its acceleration. In the context of rocket launches, this means that the force generated by the rocket's engine (Fnet) is equal to the change in momentum (M*u) over time.

To simplify the equations and make them more manageable, it can be helpful to rewrite the equations in terms of the rocket's mass instead of its momentum. This can be done using the relationship between momentum and mass, which is M*u = m*v, where m is the mass of the rocket and v is its velocity.

With this in mind, the differential equations you are working with can be rewritten as d(m*v)/dt = Fnet. This can be further simplified by breaking down the force (Fnet) into its components, which include the force of gravity, the force of drag, and the force generated by the rocket's engine.

Once you have a simplified equation in terms of the rocket's mass, you can then use MATLAB to solve it numerically. This can be done using the "ode45" function, which is designed specifically for solving ordinary differential equations. You will need to define the initial conditions (such as the initial mass, velocity, and acceleration) and the time interval over which you want to solve the equation.

I recommend checking out the documentation and examples for the "ode45" function to get a better understanding of how to use it. Additionally, you can also reach out to your peers or a professor for further assistance in using MATLAB to solve differential equations.

I hope this helps and best of luck with your project!