Darryl Yong (he/him) Today at 9:15 PM Hint for everyone about ode45: ode45 numerically calculates solutions to a differential equation and it uses adaptive step-sizes for the independent variable

velsquared = [0];

for i = (2:length(xcoords))

dy = ycoords(i) - ycoords(i-1);

dx = xcoords(i) - xcoords(i-1);

dv2 = (M * 9.81 * sin(-atan(dy/dx)) - 1/2 * 1.2 * 0.05 * M^(2/3) * velsquared(end)) * 2 * sqrt(1 + (dy/dx)^2) / M * dx;

velsquared = [velsquared velsquared(end) + dv2];

end

with these new lines of code:

interpy = spline(xcoords, ycoords);

interpyder = fnder(interpy,1);

desolution = ode45(@(x,v2) (M * 9.81 * sin(-atan(ppval(interpyder,x))) - 1/2 * 1.2 * 0.05 * M^(2/3) * v2) * 2 * sqrt(1 + (ppval(interpyder,x))^2) / M, [0, 500], 0);

velsquared = deval(desolution, linspace(0, 500, length(xcoords)));

For some reason, running fmincon does not give the same result for the optimal parameters as the original code, even though I checked by hand that the resulting velsquared vector is the same for both methods up to differences of ~1%. I'm not sure if there's anything I can do about that, but if I think of something I will let you know.

Please view explanation and answer below.

After a bit of tinkering, I found a trick to make it work: In the minutility file, use this code instead:

options = optimoptions('fmincon', 'Algorithm','interior-point','DiffMinChange',1e-2);

myans = fmincon(@getUtility, [35 50000], [], [], [], [], [10 0], [40 50000], [], options);

The extra options will improve convergence. Hope that helps :)

Hey again, I had a chance to look into that extra credit problem - I found that constraining to 2.5 instead of 5.0 led to an optimal configuration of 31.4 m and 50,000 N. (The code itself didn't have to change much - just the numbers in defining exitVelocityErrorCount were altered from 5 to 2.5.