Solving ODE45 with input vectors

I would like to elvaluate the speed of a float as it travels through the water column with depth/time dependent seawater density. The vector rho is the density profile of the sea water from 1000m to 0m. I would like o pass it through the function to evaluate speed.

clear;
close ;
%% m files for obtaining sea water denstiy profile
Sea_Water_Density_s_t_p; 
Float_Velocity_Calculation;

tspan=[0 1 1000];
y0 =[1000,0,1030];
rho=descending_rho_profile;
[t,y]=ode45(@(t,y) f(t,y,rho),tspan,y0);
hold off
plot(t,y(:,2))
grid on

Here is my function:
function speed=f(t,v,rho)
Cd=.16;
A=.0613;
m=82.25;
V=0.0809815;
%rho=1030;
g=9.81;
speed=[v(2); (rho*V*g/(m+V*rho))-(m*g/(m+V*rho))-((1/(2*m+V*rho)*Cd*A.*rho.*v(2).^2))];
end

Answer 1

[Alan Stevens]

I assume this represents some object sinking through changing desity sea water. The following shows how you might do it, though you will need to replace my simple linear density change with whatever you think is appropriate.

tspan=0:1000;
y0 =[0 0];
[t,y]=ode45(@f,tspan,y0);

subplot(2,1,1)
plot(t,y(:,1))
grid on
xlabel('time'),ylabel('depth')
subplot(2,1,2)
plot(t,y(:,2))
grid
xlabel('time'),ylabel('speed')

function speed=f(~,y)
Cd=.16;
A=.0613;
m=82.25;
V=0.0809815;
depth = y(1);
rho = interp1([0 10000],[1030 1070], depth); %This is just a linear interpolation between two values
                                             % Needs to be replaced with
                                             % your own relationship.
g=9.81;
speed=[y(2); (rho*V*g/(m+V*rho))-(m*g/(m+V*rho))-((1/(2*m+V*rho)*Cd*A.*rho.*y(2).^2))];
end

[Jestoni Orcejola]

How to use a nonlinear interpolation? I have a vector rho I want to pass through the equation.

[Alan Stevens]

Just replace my interp1 with spline(D,R,depth), where R is vector of known rho values at positions D, and depth is the value at which you want rho.

[Jestoni Orcejola]

I have a 1001x1 matrix that contains the volume change, but it only changes for certain amount of time until it stays constant. For example volume changes for 60 s then stays constan after that, how do I pass that through the equation? I'm assuming I can use a spline here. Unless my assumption here is wrong.

[Alan Stevens]

Yes, a spline should work.

[Jestoni Orcejola]

What if I want to count down whith the interpolation. Is that possible?

[Alan Stevens]

Interpolation is just that - it doesn't matter about direction! The interpolation routine will just take the depths you specify and spit out the corresponding densities.