Skip to main content

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 ')

Comments

Popular posts from this blog

PHYSICS MACHINE

Physics Machine  Ümit Alkuş  Abstract Physics machine is a software which does physics like a physicist. First, all the things human being has developed so far, for doing physics, will be available to this machine. Secondly, all the consistent theories, successful experiments, and published articles will be included into this machine in the form of traced and readable knowledge, in other words, this machine can read and understand these all. Finally, as the last target, this machine can observe the universe and physical events with the aim of creating theories and physical laws.  METU, Physics Department, 06800, Ankara, Turkey   Keywords: Artificial Intelligence, Machine Learning, Data Mining, Artificial Physicist   Introduction There are approximately millions of articles over physics, huge collection of very successful theories, and physics books. In the earth, no physicist could have attempted to read and understand these accumulations since it re...

NEWTON-RAPSON METHOD-8th degree Legendre polynomial

## Newton-Rapson Method to the smallest non negative root ## of the 8th degree Legendre Polynomial ## P8(x)=(1/128)(6435x^8-12012x^6+6930x^4-1260x^2+35) ## where -1<=x<=1. ## for the smallest non negative root, we can ignore ## all the terms except the last two by truncated ## the function to be zero and find ## x=0.167 as the initial smallest non negative ## root. ##Constants and initializations x=[]; ## Empty array for the iterated x roots x(1)=0.16700000; ## Initial guess to begin the iteration for the ## smallest non-negative root. L8=[]; ## Empty array for the Legendre polynomial L8p=[]; ## Empty array for the derivative of the Legendre polynomial for i=1:100 ##The value of the function at x L8(i)=(1/128)*(6435*x(i)^8-12012*x(i)^6+6930*x(i)^4-1260*x(i)^2+35); ##The value of the derivative of the function at x L8p(i)=(1/128)*(6435*8*x(i)^7-12012*6*x(i)^5+6930*4*x(i)^3-1260*2*x(i)); x(i+1)=x(i)-L8(i)/L8p(i); ## the iteration endfor ## For plot let's ...