If we include turbulent drag, we get a good first approximation to the equation of motion of a 'real' baseball or golfball. Our equation is still only a first approximation because we have not attempted to deal with the effects of spin. Spin introduces a force transverse to the motion, and this can be good news or bad news. Golfers who hook or slice are well acquainted with the bad news. The good news is that spin provides lift. The lift enables a drive to be hit at a lower angle than otherwise, and this in turn increases the momentum transfer between the clubhead and the ball.
The example is too complex to act as an introduction to the Feynman algorithm in two dimensions - the examples in the next section are better for that purpose. The program finds the minimum initial velocity a baseball requires to travel 450 feet before returning to the elevation from which it starts. In order to write Newton's law, we must first resolve the drag force into x and y components. (A common error at this point is to use the squares of the x and y velocities in the equations for the x and y accelerations.) The acceleration equations are coupled (the equation for the x acceleration includes the y velocity and conversely), and have no analytic solution. The acceleration equations in the program are not written in terms of the terminal velocity of a baseball, but the terminal velocity can be calculated from the given data, and is 34.7 m/s.
The program is complicated by the fact that we are trying to find the minimum velocity required for a specified horizontal range. Had we been given the initial velocity (magnitude and direction) and asked for the horizontal range, we would have only needed the Feynman algorithm. In our problem we have to try different initial speeds and launch angles until we find the combination that has the lowest initial speed for a range of 450 feet. We do this in two stages. For every trial initial speed, we vary the launch angle to find the one that maximizes the range. The routine that does this steps the angle up and down across the angle for maximum range in progressively smaller steps. (The procedure is something like a one-dimensional version of the Simplex algorithm used in the section on the analysis of ellipsometric data). The other stage, varying the velocity to find the value which gives a range of 450 feet, uses a bisection algorithm. The bisection algorithm starts with velocity values that bracket the desired range, and progressively narrows the difference between them until they differ by less than a specified amount. Enough said - try the program.
One look at the trajectory the ball would have without air resistance shows how important air resistance is.
The program can be adapted to answer another question: why does newsreel footage from the second world war show battleships firing their guns at elevations above forty-five degrees? As our program stands, it will never give an optimum angle above forty-five degrees. The answer lies in the very long range of the large guns on battleships, and the fact that air gets thinner as we go up. The decrease in density is approximately exponential: the density falls by 1/e for an increase in elevation of 8 km. I leave this problem as an exercise for the reader. (The program is easy to modify, but you need to know the barrel diameter and range of large naval guns.)