Implement friction into a model, related to the velocity. (Missile Launch)

Does anyone know how I could implement friction into my code located down below?

The friction should be defined as: Ff_vector = - gamma * | v_vector | * v_vector, and should be substracted from the different velocity components of the missile.

The current distance and time outputs I get from running my code are (without friction implemented):

Also important to note is that there is no thrust, just a starting velocity of 2000 m/s.

The rocket also has a mass now of 500 kg.

clear;
close all;
clc;

tic
tic
aT = 0;
ad = 0;
format long

for i = [37:2:53]
    r = 6371 * 10^3;
    G = 6.674 * 10^-11;
    M = 5.972 * 10^24;
    g = (G * M)/(r^2);
    theta0 = i;
    ax = 0;
    ay = r;
    v0 = 2000;
    vx0 = v0*cosd(theta0);
    vy0 = v0*sind(theta0);
    x = 0;
    y = r;
    vx = vx0;
    vy = vy0;
    T = 0;
    dt = 0.01;
    at = 0;
    landed = 0;
    z = 1;
    while landed == 0
        z = z + 1;
        T = T + dt;
        xo = x;
        yo = y;
        x = x + vx * dt;
        y = y + vy * dt;
        d = sqrt(x^2 + y^2);
        alpha = atand(x/y);
        g = (G*M)/(d^2);
        gy = cosd(alpha) * g;
        gx = sind(alpha) * g;
        vy = vy - (gy * dt);
        vx = vx - (gx * dt);
        v = vx/sin(alpha);
        ax = [ax, x];
        ay = [ay, y];
        if d < r
            landed = 1;
        end
    end
    
    aT = [aT, T];                            % This is the list of times it took for each missile to land
    distance = (alpha/360) * 2 * pi * r;
    ad = [ad, distance];                     % This is the list of distances each missile travelled
    
    figure(1)
    
    th = 2.8*pi/6:pi/500:3.1*pi/6;
    xunit = r * cos(th) + 0;
    yunit = r * sin(th) + 0;
    
    figure(2)
    hold on
    plot(ax, ay)
    toc
end

th = 2.8*pi/6:pi/500:3.1*pi/6; 
xunit = r * cos(th) + 0;
yunit = r * sin(th) + 0;

plot(xunit, yunit)
hold off

legend('37 degrees', '39 degrees', '41 degrees', '43 degrees', '45 degrees', '47 degrees', '49 degrees', '51 degrees', '53 degrees', 'earth')
title('The trajectory of the different missiles')
xlabel('x(m)')
ylabel('y(m)')
toc

The desired output, with friction included should be:

Any help is immensly appreciated!

[The Legend]

bump

[Walter Roberson]

should be substracted from the different velocity components of the missile.

Would that be before or after

        x = x + vx * dt;
        y = y + vy * dt;

? Are you calculating the friction coefficients before moving, or are you calculating them after moving ?

[The Legend]

After the missile is launched with initial velocity 2000 m/s

Answer 1

[Jim Riggs]

This question is familiar.

look here.

[The Legend]

This unfortunately did not help, I am unable to get the distance values that I should get, thank's for the help.

[Jim Riggs]

Assuming that the code gives you the correct answer without the friction term, to add friction do the following;

change this:

vy = vy - gy * dt;
vx = vx - gx * dt;

to this:

Vtot = sqrt(vx^2 + vy^2);
Fy = -gamma * Vtot * vy;
Fx = -gamma * Vtot * vx;
vy = vy + dt*(Fy/mass - gy);
vx = vx + dt*(Fx/mass - gx);

mass is 500 kg.

EDIT: after some thought, it ocurrs to me that the 1/mass factor could be included in gamma. There is no way for me to know. If this is the case, then omit the division by mass "/mass"