MATLAB: Numerically Integrating the differential equation below

integration

i know some boundary conditions such as, t0= 0 and v0=7000m/s (<===newly updated v0) and vf=0…. but do not know what, tf, i equal to.

Best Answer

You've got two 2nd order DE's, so that means you have a 4th order system (2x2=4) and thus your state vector will contain four elements. You could define them as follows:
y = your 4x1 state vector with the following definitions
y(1) = r_r
y(2) = v_r
y(3) = r_t
y(4) = v_t
Then your derivative function outline would be:
function dy = myeqn(t, y)
% put some constants here or pass them in, e.g. mu etc.
dy = zeros(size(y));
dy(1) = y(2); % derivative of r_r is v_r
dy(2) = ___; % you fill this in from your r_r doubledot equation
dy(3) = y(4); % derivative of r_t is v_t
dy(4) = ___; % you fill this in from your r_t doubledot equation
end
For the dy(2) and dy(4) code, you will need to calculate your gamma value from the y vector. You could either hardcode the other stuff (D, L, m, etc.) or pass them in as input arguments. To start with, you will need to define initial values for all four states, not just v0. Also, I would have expected to see a factor somewhere on the r_t double dot equation based on the atmospheric density (a function of altitude), but I don't see it ... and this seems suspicious to me.
Related Question