I think I am on the right track...

Not sure what I am doing wrong with my plot below. Please help. Should be three different trajectories based on a ball being thrown at a wall, bouncing off to the ground and then bouncing back up to roughly the starting point.

% All distances and heights measured in meters, time in seconds, velocity in m/s, angles in degrees and
%%acceleration in m/s^2

clear all
clc

H0 = 8;
theta_0=26;
g=9.81;
dAC=16;
eR=.94 ;
H3=0;
v0=30;
err = 5;
counter=1;

while err>0.008
    %Iteration Control
    v0 = v0 - .01;
    counter=counter+1;
    
    %initial velocities in both directions
    v0x=v0*cosd(theta_0);    %Trig
    v0y=v0*sind(theta_0);   %Trig
    
    %Point B = Maximmum height along trajectory path 1
    H1=H0+v0y^2/(2*g);
    
    %trajectory path 1
    H2=H0+dAC*tand(theta_0)-g/(2*v0x^2)*dAC^2;
    
    %velocities in x and y direction at max height
    v1x=v0x;    %Zero Acceleration
    v1y=(sqrt(v0y^2-2*g*(H1-H0)));
    
    %velocity just prior to hitting wall
    v2x=v1x;  
    v2y=sqrt(v1y^2+2*g*((H0+H1)-(H0+H2)));
    v2=sqrt(v2x^2+v2y^2);
    
    %NEW NEW
    %H2 = H1 - v2y^2/(2*g);
    
    %velocity just after hitting the wall
    v3x=v2x*eR;
    v3y=v2y;
    v3=sqrt(v3x^2+v3y^2);
    theta_3_ref=atand(v3y/v3x);  % Trig: Reference Triangle
    theta_3_abs=theta_3_ref+180; % Absolute Angle (Quadrant "3")
    
    %velocity just prior to hitting ground
    v4x=v3x;
    v4y=(sqrt(v3y^2-2*g*(H3-H2)));
    v4=sqrt(v4x^2+v4y^2);
    theta_4_ref=atand(v4x/v4y);
    theta_4_abs=theta_4_ref+180;
    
    %Distance between wall and ground
    A=(-g/(2*v3x^2));
    B=tand(theta_3_abs);
    C=(1/(2*g)*(v4^2-v3^2));
    
    %Quadratic formula for distance between wall and ground
    %%based on energy Conservation 2-3 022& Trajectory Eq. 2-3 equations
    dCD=(-B+sqrt(B^2-4*A*C))/(2*A);
    
    %trajectory path 2
    H3=H2+dCD*tand(theta_3_abs)-g/(2*v3x^2)*dCD^2;
    
    %velocity just after hitting the ground
    v5x=v4x;
    v5y=v4y*eR;
    v5=sqrt(v5x^2+v5y^2);
    theta_5_ref=atand(v5x/v5y);
    theta_5_abs=theta_4_ref+90;
    
    %distance between ground and return height
    dDA=-dAC-dCD;
    
    %trajectory path 3
    H4=H3+dDA*tand(theta_5_abs)-g/(2*v5x^2)*dDA^2;
    
    %Flight time between point A & B
    time_AB=(H1-v0)/-g;
    %Flight time between point A & C
    time_AC=dAC/v0;
    
    %Flight time between point C & D
    time_CD=abs(dCD/v4);
    
    %Flight time between point D & return height
    time_D_return=abs(dDA/v5);
    
    %Update Error
    err=abs(H4-H0);
    
    x=[0:.01:16];
    H2=H0+x*tand(theta_0)-g/(2*v0x^2)*x.^2;
    
    x1=[16:-.01:9.05];
    H3=H2+x1*tand(theta_3_abs)-g/(2*v3x^2)*x1.^2;
    
    x2=[9.05:-.01:0];
    H4=H3+x2*tand(theta_5_abs)-g/(2*v5x^2)*x2.^2;
    
    plot(x,H2,x1,H3,x2,H4)
end

v0
v0y
v0x
v1y
v1x
v2
v2y
v4
dCD
dDA
H1
H2
H3
H4
time_AB
time_AC
time_CD
time_D_return
counter

Answer 1

[Voss]

I ran into an error in the calculation of H3 at the bottom of the loop. Looking into it, it was suspicious to me that H2, H3 and H4 are overwritten there having been calculated previously, and I suspected that the H2 and H3 which are used there to calculate H3 and H4, respectively, should actually refer to the previously calculated H2 and H3. Changing those bottom H2, H3 and H4 to other names, I was able to generate a plot with 3 distinct trajectories (no idea if they are correct). Here is the code with my changes:

clear all
clc
H0 = 8;
theta_0=26;
g=9.81;
dAC=16;
eR=.94 ;
H3=0;
v0=30;
err = 5;
counter=1;
while err>0.008
    %Iteration Control
    v0 = v0 - .01;
    counter=counter+1;
    
    %initial velocities in both directions
    v0x=v0*cosd(theta_0);    %Trig
    v0y=v0*sind(theta_0);   %Trig
    
    %Point B = Maximmum height along trajectory path 1
    H1=H0+v0y^2/(2*g);
    
    %trajectory path 1
    H2=H0+dAC*tand(theta_0)-g/(2*v0x^2)*dAC^2;
    
    %velocities in x and y direction at max height
    v1x=v0x;    %Zero Acceleration
    v1y=(sqrt(v0y^2-2*g*(H1-H0)));
    
    %velocity just prior to hitting wall
    v2x=v1x;  
    v2y=sqrt(v1y^2+2*g*((H0+H1)-(H0+H2)));
    v2=sqrt(v2x^2+v2y^2);
    
    %NEW NEW
    %H2 = H1 - v2y^2/(2*g);
    
    %velocity just after hitting the wall
    v3x=v2x*eR;
    v3y=v2y;
    v3=sqrt(v3x^2+v3y^2);
    theta_3_ref=atand(v3y/v3x);  % Trig: Reference Triangle
    theta_3_abs=theta_3_ref+180; % Absolute Angle (Quadrant "3")
    
    %velocity just prior to hitting ground
    v4x=v3x;
    v4y=(sqrt(v3y^2-2*g*(H3-H2)));
    v4=sqrt(v4x^2+v4y^2);
    theta_4_ref=atand(v4x/v4y);
    theta_4_abs=theta_4_ref+180;
    
    %Distance between wall and ground
    A=(-g/(2*v3x^2));
    B=tand(theta_3_abs);
    C=(1/(2*g)*(v4^2-v3^2));
    
    %Quadratic formula for distance between wall and ground
    %%based on energy Conservation 2-3 022& Trajectory Eq. 2-3 equations
    dCD=(-B+sqrt(B^2-4*A*C))/(2*A);
    
    %trajectory path 2
    H3=H2+dCD*tand(theta_3_abs)-g/(2*v3x^2)*dCD^2;
    
    %velocity just after hitting the ground
    v5x=v4x;
    v5y=v4y*eR;
    v5=sqrt(v5x^2+v5y^2);
    theta_5_ref=atand(v5x/v5y);
    theta_5_abs=theta_4_ref+90;
    
    %distance between ground and return height
    dDA=-dAC-dCD;
    
    %trajectory path 3
    H4=H3+dDA*tand(theta_5_abs)-g/(2*v5x^2)*dDA^2;
    
    %Flight time between point A & B
    time_AB=(H1-v0)/-g;
    %Flight time between point A & C
    time_AC=dAC/v0;
    
    %Flight time between point C & D
    time_CD=abs(dCD/v4);
    
    %Flight time between point D & return height
    time_D_return=abs(dDA/v5);
    
    %Update Error
    err=abs(H4-H0);
    
    x=[0:.01:16];
    H22=H0+x*tand(theta_0)-g/(2*v0x^2)*x.^2;
    
    x1=[16:-.01:9.05];
    H33=H2+x1*tand(theta_3_abs)-g/(2*v3x^2)*x1.^2;
    
    x2=[9.05:-.01:0];
    H44=H3+x2*tand(theta_5_abs)-g/(2*v5x^2)*x2.^2;
    
    plot(x,H22,x1,H33,x2,H44)
end
v0
v0y
v0x
v1y
v1x
v2
v2y
v4
dCD
dDA
H1
H2
H3
H4
time_AB
time_AC
time_CD
time_D_return
counter

Answer 2

[Peter Denardo]

That's very helpful. Any idea how I get the second trajectory to start where the last one left off and the 3rd one to pick up from where the 2nd one left off?

[Voss]

After the H33 and H44 are calculated, subtract off their first element and add in the last element of the previous trajectory. See:

clear all
clc
H0 = 8;
theta_0=26;
g=9.81;
dAC=16;
eR=.94 ;
H3=0;
v0=30;
err = 5;
counter=1;
while err>0.008
    %Iteration Control
    v0 = v0 - .01;
    counter=counter+1;
    
    %initial velocities in both directions
    v0x=v0*cosd(theta_0);    %Trig
    v0y=v0*sind(theta_0);   %Trig
    
    %Point B = Maximmum height along trajectory path 1
    H1=H0+v0y^2/(2*g);
    
    %trajectory path 1
    H2=H0+dAC*tand(theta_0)-g/(2*v0x^2)*dAC^2;
    
    %velocities in x and y direction at max height
    v1x=v0x;    %Zero Acceleration
    v1y=(sqrt(v0y^2-2*g*(H1-H0)));
    
    %velocity just prior to hitting wall
    v2x=v1x;  
    v2y=sqrt(v1y^2+2*g*((H0+H1)-(H0+H2)));
    v2=sqrt(v2x^2+v2y^2);
    
    %NEW NEW
    %H2 = H1 - v2y^2/(2*g);
    
    %velocity just after hitting the wall
    v3x=v2x*eR;
    v3y=v2y;
    v3=sqrt(v3x^2+v3y^2);
    theta_3_ref=atand(v3y/v3x);  % Trig: Reference Triangle
    theta_3_abs=theta_3_ref+180; % Absolute Angle (Quadrant "3")
    
    %velocity just prior to hitting ground
    v4x=v3x;
    v4y=(sqrt(v3y^2-2*g*(H3-H2)));
    v4=sqrt(v4x^2+v4y^2);
    theta_4_ref=atand(v4x/v4y);
    theta_4_abs=theta_4_ref+180;
    
    %Distance between wall and ground
    A=(-g/(2*v3x^2));
    B=tand(theta_3_abs);
    C=(1/(2*g)*(v4^2-v3^2));
    
    %Quadratic formula for distance between wall and ground
    %%based on energy Conservation 2-3 022& Trajectory Eq. 2-3 equations
    dCD=(-B+sqrt(B^2-4*A*C))/(2*A);
    
    %trajectory path 2
    H3=H2+dCD*tand(theta_3_abs)-g/(2*v3x^2)*dCD^2;
    
    %velocity just after hitting the ground
    v5x=v4x;
    v5y=v4y*eR;
    v5=sqrt(v5x^2+v5y^2);
    theta_5_ref=atand(v5x/v5y);
    theta_5_abs=theta_4_ref+90;
    
    %distance between ground and return height
    dDA=-dAC-dCD;
    
    %trajectory path 3
    H4=H3+dDA*tand(theta_5_abs)-g/(2*v5x^2)*dDA^2;
    
    %Flight time between point A & B
    time_AB=(H1-v0)/-g;
    %Flight time between point A & C
    time_AC=dAC/v0;
    
    %Flight time between point C & D
    time_CD=abs(dCD/v4);
    
    %Flight time between point D & return height
    time_D_return=abs(dDA/v5);
    
    %Update Error
    err=abs(H4-H0);
    
    x=[0:.01:16];
    H22=H0+x*tand(theta_0)-g/(2*v0x^2)*x.^2;
    
    x1=[16:-.01:9.05];
    H33=H2+x1*tand(theta_3_abs)-g/(2*v3x^2)*x1.^2;
    H33 = H33-H33(1)+H22(end); % shift to start at H22(end)
    
    x2=[9.05:-.01:0];
    H44=H3+x2*tand(theta_5_abs)-g/(2*v5x^2)*x2.^2;
    H44 = H44-H44(1)+H33(end); % shift to start at H33(end)
    
    plot(x,H22,x1,H33,x2,H44)
end
v0
v0y
v0x
v1y
v1x
v2
v2y
v4
dCD
dDA
H1
H2
H3
H4
time_AB
time_AC
time_CD
time_D_return
counter
 

[Peter Denardo]

Not sure what I am doing wrong somewhere in my code but trajectory 2 should start where trajectory 1 ends, reach 0 on the y axis and then make its way back up to the starting point. The trajectories go in the correct directions but trajectory 2 goes all the way down to -5 and trajectory 3 is off too. I am so stumped

[Voss]

My advice would be to disable the while loop for now and focus on getting one set of trajectories to work. Step through the code and make sure all the variables appear to have correct values, especially their signs (that things moving up have positive y velocity and things moving down have negative y velocity, etc.).

One thing to be aware of is it may be a good idea to use atan2d for the arctangents rather than atand so that you don't have to know which quadrant the angle is in and add the appropriate multiple of 90 degrees. atan2d is four-quadrant arctan, so it returns an angle between -180 and 180 degrees.