FOURTH ORDER RUNGE-KUTTA METHOD- MOTION OF A SPHERICAL MASS WITH AIR RESISTANCE

## Motion of a spherical mass with air resistance
## Fourth order Runge-Kutta Method
## Very Important!!! The positive velocity direction
## is the direction of the gravitational acceleration
m=1E-2; ## mass of the object(kg/m)
g=9.8; ## Acceleration due to the gravity (m/sec^2)
v0=0; ## initial velocity of the object(m/sec)
k=1E-4; ## Air drag coefficient(kg/m)
t=[]; ## Empty time vector(sec)
t(1)=0; ## Released time(sec)
dt=1E-1; ## Increment in time(sec)
v=[]; ## Empty velocity vector(m/s)
v(1)=v0; ## Initial velocity
v_nodrag=[]; ## Velocity by ignoring air drag
## Analytic solution by zeroth order approximation.
v_nodrag(1)=0; ## Velocity vector with no drag.
n=1; ## Initialize the loop index
f0=[];
f1=[]; ##The 1st order derivatives by
f2=[]; ##by Runge-Kutta. Eqn 5.33 pg217
f3=[];
## Run the loop until the time reaches the value 10sec.
while (t(n)<=10);
f0(n)=g-(k/m)*v(n)*v(n);
v_f0(n)=v(n)+(dt/2)*f0(n);
f1(n)=g-(k/m)*v_f0(n)*v_f0(n);
v_f1(n)=v(n)+(dt/2)*f1(n);
f2(n)=g-(k/m)*v_f1(n)*v_f1(n);
v_f2(n)=v(n)+dt*f2(n);
f3(n)=g-(k/m)*v_f2(n)*v_f2(n);
v(n+1)=v(n)+(dt/6)*(f0(n)+2*f1(n)+2*f2(n)+f3(n));
t(n+1)=t(n)+dt;
v_nodrag(n+1)=v_nodrag(n)+g*dt; ##analytic solution
n++;
endwhile
plot(t,v,'r-',t,v_nodrag,'b-');
title('Velocity vs Time');
xlabel('time(sec)');
ylabel('velocity(m/sec)');
legend('v(Runge-Kutta)','v(no drag)')
axis([0,13]);
save -text RUNGEKUTTA.dat
print('-dpsc','RUNGEKUTTA.ps ')