Matlab: Finding the right angle

[Adel A]

Hi!

I have a question about an assignment.
It is about throwing a rock which should land 20 meters away from the "shooter". My job is to find the angle "a" which makes the rock land 20 meters away.

The motion is given by a multivariable, second order differential equation which I have already "converted" into a first order differential equation. I am only allowed to use the Runge-Kutta-method, and not allowed to use ODE45.

I have managed to write a code which works fine, but I need help with making a loop of some sort, in order to find a precise angle, a, which makes the rock land 20 meters away from the shooter.
I know that the angle should be approximately 1 radian.

Here is the code:

clear all
close all
clc

format long

%g = 9.81;
a = pi/4;
a = 1;

x0 = [0;1.5];
xPrim0 = 19.*[cos(a);sin(a)];

K = [0.02 0; 0 0.065];
f = [0; -9.81];

x = [];                         % Vi ska räkna ut X (positionen), men vad skickar vi till funktionen?
xPrim = 19.*[cos(a);sin(a)];    % Ska det multipliceras med 19?
                                % xPrim är väl inte en funktion av vinkeln
                                % efter kastet?
h = 0.3;
T = 100;
hojd = 1.5;

%Q = [0.1 1 1.5 3.1];

%c0 = [0.1;0];
u = [x0; xPrim0];
u0 = [x0;xPrim0];

%F = [xPrim; f - norm(xPrim).*K*xPrim];

W = [];

% for i = 1:4

% u0 = [u(i);0];

u = u0;
U = u;

tt = 0:9;

for t=tt(1:end-1)

    k1 = Fproj(u);
    k2 = Fproj(u+0.5*h*k1);
    k3 = Fproj((u+0.5*h*k2));
    k4 = Fproj((u+k3*h));

    u = u + (1/6)*(k1+2*k2+2*k3+k4)*h;

    U = [U u];

end

% W = [W U];

plot(U(1,1:end),U(2,1:end))
hold on
line([0 25], [0 0])
line([20 20], [-5 5])

 

[DrClaude]

I suggest that you encapsulate some of that code in a function that, given an angle a, calculates where the rock lands. It would be also easier if it returned the number as the distance to the desired value, such that it returns 0 when it is on target.

What you need then is to use some method for finding roots. You could use the Newton-Raphson method, but it is probably overkill for such a simple 1D problem. The simplest you could try is the bisection method.