%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%                       Computational Physics
%                             FALL 2008
%
%                     Instructor: Kevin H. Knuth
%
%
% HW 3
% Numeric Integration
%
% The purpose of this homework set is to introduce you to the subtleties of 
% numeric integration while being conscious of both numerical errors and 
% speed.  In addition, you will learn about function handles, which allow
% you to write functions that take functions as arguments.
%
%
% TURN IN:
% m-files titled myTRAP.m and myQUAD.m
% Report on methods and accuracy and errors as a function of spacing
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Problem by: Kevin H. Knuth
% Created on: 09 February 2008
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% PART 1
% Write a function myTRAP that integrates a function based on the
% trapezoidal approximation.  You may compare to the MATLAB function TRAPZ,
% but cannot use TRAPZ.m.
%
% Usage:    integral = myTRAP(funHANDLE, x)
% Inputs:   funHANDLE = function handle
%           x = array of x-coordinates
% Outputs:  integral = Integral over limits imposed by x
%
% 
% PART 2
% Write a function myQUAD that integrates a function based on the
% Simpson approximation.  You may compare to the MATLAB function QUAD, but
% cannot use QUAD.m
% Keep in mind that the number of evaluation points must be odd and that 
% there must at least be seven points.  You may want to test for that!
%
% Usage:    integral = myQUAD(funHANDLE, x)
% Inputs:   funHANDLE = function handle
%           x = array of x-coordinates
% Outputs:  integral = Integral over limits imposed by x
%
% 
% PART 3
% Write a report analyzing the motion of a falling raindrop.  
%
% 1. Start with Newton's Laws and assume that the droplet experiences a 
% linear drag due to air resistance.  That is:
% F = -cv
% where c is a constant.
% Derive the equation for the velocity as a function of time.
%
% 2. Next derive the equation for the position as a function of time.
%
% 3. Then write a function velocity.m that takes three arguments: 
% the the current time t, initial time time to, the mass of the particle,
% and the drag coefficient c and returns the velocity.
%
% 4. Now we will solve a more realistic problem.  For a raindrop with 
% diameter d = 10e-4 meter, 
% and a drag coefficient of c = d * 1.55 * 10e-4 Ns/m^2,
% use your solution to step 2 to compute how far the drop will fall 
% in 100 seconds after starting from rest.
%
% 5. Determine the appropriate number of time steps for the integration
% and run the two integration algorithms myTRAP.m and myQUAD.m to 
% numerically compute the distance the raindrop will fall in 10 seconds 
% after starting from rest.  Compare these results to your answer in 
% Step 4.  
% HINT: You will need to pass a function handle referring to velocity.m
% however, you will want to fix the values of to and c.  Consider
% defining the anonymous function:
% raindrop = @(t) velocity(t, to, m, c);
% and pass it to myTRAP and myQUAD.
%
% 6. Make a plot of the difference between your numeric estimate and the 
% analytic estimate over an interesting range of time intervals.
%
% OPTIONAL. Make plots of the droplet's velocity and position over the 
% first 10 secs. How does the distance the droplet falls compare to a 
% particle in a vacuum?
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%