3. An artillery officer needs to determine the angle at which to aim the cannon

Question

3. An artillery officer needs to determine the angle at which to aim the cannon to a target d meters away. He knows that the cannonball will leave the muzzle of the gun at an initial speed of vo m/s. Neglecting air resistance, the only force acting on the cannonball is then gravity, which imparts a downward acceleration of g = 9.8m/s2 The desired is a solution to the nonlinear equation 2 sin cos Given that to = 126 m/s and d = 1200 m. (a) Use Newton's method to find a proper to within 2 decimal places. As an initial guess, use 0-/6 (b) Use Secant method to find a proper to within 2 decimal places. As initial guesses, use 0 and ,-/6 present your results in the form of the following table Iteration 1 | | error ei where error is defined as e,-0,-1

Explanation / Answer

a)

%%% Matlab code %%%%%%%%

clc;
clear all;
close all;
format short
%%% restoredefaultpath %%%
syms x;
v0=126;
g=9.8;
d=1200;
f=sin(2*x)*v0^2/g-d;

%%% Newton raphson method
s(1)=pi/6; %%% initial guess
tol=10^(-2);
disp(' Iteration       Roots       error ');
for n=1:50;
    l1=subs(f,s(n));
    l2=subs(diff(f),s(n));
    s(n+1)=s(n)-l1/l2;
    err(n)=abs(s(n+1)-s(n));
     fprintf('    %d          %f    %f ',n,s(n+1),err(n));
    if err(n) < tol
       break;
    end
end

OUTPUT:

Iteration       Roots       error
    1          0.398314    0.125285
    2          0.416721    0.018407
    3          0.417086    0.000365

b)

%%%% Matlab code %%%%%%%

v0=126;
g=9.8;
d=1200;
f=@(x) sin(2*x)*v0^2/g-d;
% %%% Secant method
y(1)=0;
y(2)=pi/6;
tol=0.01;
disp(' Iteration       Roots       error ');
for n=2:4
    f1=feval(f,y(n));
    f2=feval(f,y(n-1));
    y(n+1)=y(n)-f1*(y(n)-y(n-1))/(f1-f2);
    err=abs(y(n+1)-y(n));
    fprintf('    %d          %f    %f ',n,y(n+1),err);
    if ( err < tol)
        break;
    end
end

OUTPUT:

Iteration       Roots       error
    1          0.447852    0.075747
    2          0.412445    0.035407
    3          0.417251    0.004806

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