increasing the launch angle of projectile so it reaches a certain distance

I am currently doing a school project where i have to plot the trajectory of a projectile launched from the ground with initial speed u, and angle theta above the horizontal. The projectile has to hit a target of a horizontal distance D=10000m away once it has reached the ground. I have used an initial guess of theta=pi/12 as the angle, so that the projectile does not reach D. The program should then automatically pick a new value of theta by increasing theta from pi/30 in steps dtheta=theta/100 until the target is overshot. A parachute is deployed at t_pchute=15 seconds.

How do I increase the angle iteratively such that the projectile reaches the target? I was told to use theta(i+1)=atan(vy(i)/vx(i)), but i'm not sure why. If you have time, please help me.

 %Constants

 D=10000;%m
 u=600;%m/s
 m=50;%kg
 t_pchute=15;%s
 g=9.81;%m/s^2
 a_proj=0.01;%m^2
 a_pchute=0.05;%m^2
 C_proj=0.4;
 C_pchute=1.2;
 p0=1.207;%kg/m^3

 theta=pi/12;
 dtheta=theta/100;

 %initial conditions
 x=0;
 i=0;
 t=0;
 y=0;
 dt=0.1;
 p=p0;

 vx=u*cos(theta);
 vy=u*sin(theta);
 v=sqrt(vx.^2+vy.^2);

 for i=1:1000

    Fa(i+1)=0.5*p(i)*C_proj*a_proj*v(i);
    Fp(i+1)=0.5*p(i)*C_pchute*a_pchute*v(i);

    if t<t_pchute
     ax(i+1)=-((Fa(i)*cos(theta(i)))/m)*vx(i); 
     ay(i+1)=-g-(((Fa(i)*sin(theta(i)))/m)*vy(i)); 
    else
     ax(i+1)=-((Fp(i)*cos(theta(i)))/m)*vx(i);   
     ay(i+1)=-g-(((Fp(i)*sin(theta(i)))/m)*vy(i));
    end

    vx(i+1)=vx(i)+(dt.*ax(i));
    vy(i+1)=vy(i)+(dt.*ay(i));
    v(i+1)=sqrt(vx(i).^2+vy(i).^2);

    x(i+1)=x(i)+(dt.*vx(i));
    y(i+1)=y(i)+(dt.*vy(i));

    p(i+1)=p0*(1-2.333e-5*y(i+1)).^5;

    theta(i+1)=atan(vy(i)/vx(i));
    t=t+dt;

    if y(i+1)<0
        break
    end
end

 plot(x,y),grid
 xlabel('Distance/[m]')
 ylabel('Height/[m]')
 title('Projectile Trajectory')

[Walter Roberson]

How does this question differ from http://www.mathworks.co.uk/matlabcentral/answers/71006-how-do-i-increase-the-angle-so-that-the-projectile-reaches-the-target and http://www.mathworks.co.uk/matlabcentral/answers/71143-so-stuck-please-help-how-do-i-find-the-correct-angle-for-the-graph-trajectory-of-my-projectile-mot ?

[Amit]

sorry, my apologies. it was inconsiderate of me. I have now deleted the other two questions.

Answer 1

[Jürgen]

Hi, I admint I did not check all your code, but

if you want something like:

theta= Value;
thetaStep= StepValue;
for i= 1:someValue
     Theta= Theta+ StepValues*(i-0);
     do something with theta
end

then I find it strange that you use Theta/100 as step value because your step is then a function of Theta, or is that done deliberately?

R,J

[Amit]

hi thanks for the help.

basically, dtheta, which I put as theta/100, can be any small step value (eg. 0.01). I would just like theta to iteratively increase by the step value so that the projectile reaches D=10000 metres. In other words, whilst my projectile lands at a horizontal distance which is less than D=10000m, I would like the program to increase the angle theta and restart the loop. I think I have to use a while loop, but I'm unsure how? Thanks again.

[Jürgen]

while horFlightDist < 1000
       do something
end

[Amit]

Thanks. I tried doing that but I get a funny looking graph. I'm also not sure where to put the while loop?

Answer 2

[Amit]

can anyone help? thanks.