Problem 5

%Problem 5
%Part A creates a chart illustrating free fall velocity calculated analytically and numerically
function[v_analytical,v_terminal,t] = freefalls(h,timespan)
t = (0:h:timespan) %creates time intervals size h between zero and timespan
c = 0.25 %define vars
m = 60 %define vars
g = 9.81 %define vars
v_terminal = sqrt(m*g/c);
v_analytical = v_terminal*tanh(g*t/v_terminal);

v_numerical = zeros(length(t),1);
deltat = diff(t)
for i = 1:length(t)-1
  v_numerical(i+1) = v_numerical(i)+(g-c/m*v_numerical(i)^2)*deltat(i);
end

indices = round(linspace(1,length(t),7));
fprintf('time (s)|vel analytical (m/s)|vel numerical(m/s)\n')
M = [t(indices),v_analytical(indices),v_numerical(indices)];
fprintf('%7.1f|%18.2f|%15.2f\n',M(:,1:3)');
plot(t,v_analytical,'-',t,v_numerical,'0-')

%Part B Runs the comparison
freefalls(.1,30); hold on
freefalls(1,30); hold on
freefalls(5,30; hold off
legend('h=.1 - Analytical','h=.1 - Numerical','h=1 - Analytical','h=1 - Numerical','h=5 - Analytical','h=5 - Numerical','Location','Northwest')