How do I implement friction into my script?

clear;
close all;
clc;

tic

aT = 0;
ad = 0;

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 = 3000;
    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];
    distance = (alpha/360) * 2 * pi * r;
    ad = [ad, distance];
    
    fprintf('Checked for degree: %.0f\n', i)
    
end

toc

This is a model that I've created for calculating the distance a missile travels under the following conditions:

velocity = 2000 m / s

gravity = dependent on height

distance = influenced by curvature of the earth

friction = not present

Now, I am trying to implement friction into my script and I know that the friction force is related to the speed by a coefficient:

Ff = gamma * | v | * v

With Ff and v being vectors in the direction of the missile.

Could anyone please give me a hint to what I could do to implement this relation into the script.

I will then alternate gamma and compare the results of the script to get the best value for gamma.

Answer 1

[Thiago Henrique Gomes Lobato]

Friction is a force that produces acceleration which changes the velocity, so if you add an additional term to the velocity update you may get the effect that you want to. Theoretically you would need to define the mass of the missile, but this can be controlled by the parameter gamma (both are constant, costant*constant = constant), in the end you will have something like:

gamma = 1e-5;
vy = vy - (gy * dt) - gamma*abs(vy)*vy*dt;
vx = vx - (gx * dt) - gamma*abs(vx)*vx*dt;

[The Legend]

Thank you so much, this will definetely help me out!

[Jim Riggs]

This equation is not the correct form.

it should be

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

the product gamma*Vtot*vx is a FORCE in the x direction.

Force/mass is an accleration in the x direction.

(although, I suppose that is is possible that the 1/mass factor is already included in gamma, so if this is the case, then the equation should be:

vy = vy - dt * (gy + gamma*Vtot*vy);
vx = vx - dt * (gx + gamma*Vtot*vx);

For more detail on the Vtot * vx factor see the discussion here

[Thiago Henrique Gomes Lobato]

You're right, it should be Vtot. I didn't reflect enough time at first and gave a too quick response. Thanks for coming back to clear this even though it was already in the other question.