# Advanced Mathematics and Mechanics Applications Using MATLAB, 3rd Edition

### Howard Wilson (view profile)

14 Oct 2002 (Updated )

Companion Software (amamhlib)

runtraj
function runtraj
% Example: runtraj
% ~~~~~~~~~~~~~~~~
%
% This program integrates the differential
% equations governing two-dimensional motion
% and atmospheric drag proportional to the
% velocity squared. The initial inclination
% angle needed to hit a distant target is
% computed repeatedly and function fmin is
% employed to minimize the square of the
% distance by which the target is missed. The
% optimal value of the miss distance is zero
% and the optimum angle will typically be found
% unless the initial velocity is too small
% and the horizontal velocity becomes zero
% before the target is passed. The initial
% velocity of the projectile must be large
% enough to make the problem well posed.
% Otherwise, the program will terminate with
% an error message.
%
% User m functions called: missdis, traject,
%                          projcteq

clear all; close
global Vinit Gravty Cdrag Xfinl Yfinl

vinit=600; gravty=32.2; cdrag=0.002;
xfinl=1000; yfinl=100;

disp(' ');
disp('SEARCH FOR INITIAL INCLINATION ANGLE ');
disp('TO MAKE A PROJECTILE STRIKE A DISTANT');
disp('OBJECT'); disp(' ');
disp(['Initial velocity = ',num2str(vinit)]);
disp(['Gravity constant = ',num2str(gravty)]);
disp(['Drag coefficient = ',num2str(cdrag)]);
disp(['Coordinates of target = (', ...
num2str(xfinl),',',...
num2str(yfinl),')']); disp(' ');

% Replicate input data as global variables
Vinit=vinit; Gravty=gravty; Cdrag=cdrag;
Xfinl=xfinl; Yfinl=yfinl;

% Perform the minimization search
fstart=180/pi*atan(yfinl/xfinl); fend=45;
disp('Please wait for completion of the')
disp('minimization search');
opts=optimset('TolX',1e-12,'Display','off');
bestang=fminbnd(@missdis,fstart,fend,opts);

% Display final results
[y,x,t]=traject ...
(bestang,vinit,gravty,cdrag,xfinl);
dmiss=abs(yfinl-y(length(y))); disp(' ')
disp(['Final miss distance is ', ...
num2str(dmiss),' when the']);
disp(['initial inclination angle is ', ...
num2str(bestang),...
' degrees']);

%=============================================

function [dsq,x,y]=missdis(angle)
%
% [dsq,x,y]=missdis(angle)
% ~~~~~~~~~~~~~~~~~~~~~~~~
%
% This function is used by fminbnd. It returns
% an error measure indicating how much the
% target is missed for a particular initial
% inclination angle of the projectile.
%
% angle - the initial inclination angle of
%         the projectile in degrees
%
% dsq   - the square of the difference between
%         Yfinal and the final value of y found
%         using function traject.
% x,y   - points on the trajectory.
%
% Several global parameters (Vinit, Gravty,
% Cdrag, Xfinl) are passed to missdis by the
% driver program runtraj.
%
% User m functions called: traject
%----------------------------------------------

global Vinit Gravty Cdrag Xfinl Yfinl
[y,x,t]=traject ...
(angle,Vinit,Gravty,Cdrag,Xfinl,1);
dsq=(y(length(y))-Yfinl)^2;

%=============================================

function [y,x,t]=traject ...
(angle,vinit,gravty,cdrag,xfinl,noplot)
%
% [y,x,t]=traject ...
%       (angle,vinit,gravty,cdrag,xfinl,noplot)
% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
%
% This function integrates the dynamical
% equations for a projectile subjected to
% proportional to the square of the velocity.
%
% angle  - initial inclination of the
%          projectile in degrees
% vinit  - initial velocity of the projectile
%          (muzzle velocity)
% gravty - the gravitational constant
% cdrag  - drag coefficient specifying the
%          drag force per unit mass which
%          equals cdrag*velocity^2.
% xfinl  - the projectile is fired toward the
%          right from x=0.  xfinl is the
%          largest x value for which the
%          solution is computed. The initial
%          velocity must be large enough that
%          atmospheric damping does not reduce
%          the horizontal velocity to zero
%          before xfinl is reached.  Otherwise
%          an error termination will occur.
% noplot - plotting of the trajectory is
%          omitted when this parameter is
%          given an input value
%
% y,x,t  - the y, x and time vectors produced
%          by integrating the equations of
%          motion
%
% Global variables:
%
% grav,  - two constants replicating gravty and
% dragc    cdrag, for use in function projcteq
% vtol   - equal to vinit/1e6, used in projcteq
%          to check whether the horizontal
%          velocity has been reduced to zero
%
% User m functions called: projcteq
%----------------------------------------------

global grav dragc vtol

% Default data case generated when input is null
if nargin ==0
angle=45; vinit=600; gravty=32.2;
cdrag=0.002; xfinl=1000;
end;

% Assign global variables and evaluate
% initial velocity
grav=gravty; dragc=cdrag; ang=pi/180*angle;
vtol=vinit/1e6;
z0=[vinit*cos(ang); vinit*sin(ang); 0; 0];

% Integrate the equations of motion defined
% in function projcteq
deoptn=odeset('RelTol',1e-6);
[x,z]=ode45(@projcteq,[0,xfinl],z0,deoptn);

y=z(:,3); t=z(:,4); n=length(x);
xf=x(n); yf=y(n);

% Plot the trajectory curve
if nargin < 6
plot(x,y,'k-',xf,yf,'ko');
xlabel('x axis'); ylabel('y axis');
title(['Projectile Trajectory for ', ...
'Velocity Squared Drag']);
axis('equal'); grid on; figure(gcf);
%print -deps trajplot
end

%=============================================

function zp=projcteq(x,z)
%
% zp=projcteq(x,z)
% ~~~~~~~~~~~~~~~~
%
% This function defines the equation of motion
% for a projectile loaded by gravity and
% atmospheric drag proportional to the square
% of the velocity.
%
% x     -  the horizontal spatial variable
% z     -  a vector containing [vx; vy; y; t];
%
% zp    -  the derivative dz/dx which equals
%          [vx'(x); vy'(x); y'(x); t'(x)];
%
% Global variables:
%
% grav  -  the gravity constant
% dragc -  the drag coefficient divided by
%          gravity
% vtol  -  a global variable used to check
%          whether vx is zero
%
% User m functions called:  none
%----------------------------------------------

global grav dragc vtol
vx=z(1); vy=z(2); v=sqrt(vx^2+vy^2);

% Check to see whether drag reduced the
% horizontal velocity to zero before the
% xfinl was reached.
if abs(vx) < vtol
disp(' ');
disp('*************************************');
disp('ERROR in function projcteq. The ');
disp('  initial velocity of the projectile');
disp('  was not large enough for xfinal to');
disp('  be reached.');
disp('EXECUTION IS TERMINATED.');
disp('*************************************');
disp(' '),error(' ');
end
zp=[-dragc*v; -(grav+dragc*v*vy)/vx; ...
vy/vx; 1/vx];