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%                       Computational Physics
%                             SPRING 2008
%
%                     Instructor: Kevin H. Knuth
%
%
% HW 5
% Baseball: Air Resistance and Spin
%
% We often get to solve projectile motion in two-dimensions with the
% effects of air-resistance proportional to the velocity and velocity
% squared.  Here we consider a more complete example that take into account  
% the effects of both spin and drag.  We will find that the effect of spin 
% takes us to the regime where we must consider motion in 3-dimensions.
%
% We consider the effect of three forces: gravitation, drag, and the Magnus
% force, which is due to the effects of the differential velocity between
% the ball's spinning surface and the air.
%
% The drag force will be assumed to be proportional to v^2, so that
% Fdrag = m*Cd*v^2 where Cd = 6e-3 m^(-1), with a direction of v_hat.
%
% The Magnus Force is proportional to both the velocity of the ball v, and
% the difference in velocity on the extreme surfaces of the ball due to its
% rotation (wr), so that
% Fmag = m*Cm*(w x v), where w is the angular velocity vector, v is the
% center of mass velocity vector, and Cm = 4e-4 m^(-1).
%
% The mass of the ball is 0.149 kg.
%
% References:
% Watts, R.G. and Ferrer, R. (1987).
% "The lateral force on a spinning sphere: Aerodynamics of a curveball"
% American Journal of Physics" 55 (1): 40.
%
%
% IN YOUR REPORT:
% -Include force diagram for the ball.
% -Include a description of the forces involved (justify/verify units).
% -Rewrite the vector form of the equation of motion into three ordinary
% second-order differential equations.
% -Write a function that takes the intital conditions and solves these
% differential equations.
% -Write a function that plots (in two subplots; side-by-side) the vertical 
% and horizontal deflection of the ball. 
% -Add a function that plots the trajectory of the ball in 3D space.
% -USE YOUR FUNCTIONS TO ANSWER THE FOLLOWING:
%
% a. Set the drag and Magnus forces to zero and verify that you can
% reproduce classical projectile motion.  For a 40 m/s fastball, what
% initial angle is required to have the ball pass 1.5 m above Home Plate.
%
% b. Turn on the drag, and compare the trajectories with those of
% projectile motion without drag.  Does it behave as expected?  How could
% you check?
%
% c. Assume that the ball is released z=1.8m and x=18m from Home Plate 
% (origin). Select initial angle above the horizontal and the ball
% speed.
%
% d. Relying on the initial conditions set up in Part a, turn on the
% Magnues force, and add a topspin to the ball w_y = 200 rad/s.  How does
% the topspin change the height of the ball as it flies over Home Plate?
% What does a backspin do?
%
% e. How does a 35m/s curveball deflect if it is given a topspin of
% 200 rad/s?
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% Problem by: Kevin H. Knuth
% Created on: 18 January 2008
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