The Feynman algorithm needs to be modified slightly to deal with forces that depend on velocity. At the time acceleration is calculated, the only value available for velocity is the one from a half-step earlier. If we keep track of change in velocity as well as velocity itself, we can use the last change to project forward a half-step for the acceleration calculation. (This process is similar to the one we used when plotting the gravitational field.) In this section we test the accuracy of this procedure by looking at a one-dimensional problem that has an analytic solution.
When a sphere the size of a baseball or golfball moves through air at a normal speed, momentum is transferred to it from the air with which it collides, i.e. the air exerts a force on the sphere that opposes the motion. The volume of air the ball sweeps through in one second is the product its velocity and cross-sectional area, so the momentum transferred will be this volume multiplied by the density of air, the velocity of the air relative to the ball, and a 'drag coefficient'. (Sleek automobiles quote drag coefficients of around 0.3, and a sphere does a bit better than that.) For a sphere moving through air at normal elevations, the force of turbulent drag is proportional to the radius squared times the velocity squared, and the proportionality constant is about 0.87 in S.I. units. At higher elevations the constant decreases in direct proportion to the decrease in the density of the air.
The resistive force can be simplified by writing it in terms of the terminal velocity. The terminal velocity is the velocity at which the resistive force and the force of gravity balance: a ball falling at its terminal velocity accelerates no further. In terms of the terminal velocity, resistive force is weight multiplied by the square of the ratio of velocity to terminal velocity. Baseballs and golfballs travel at speeds well in excess of their terminal velocities, so air resistance is not something that can be ignored even to a first approximation.
We will begin by finding the speed at which a ball returns after being thrown upward at its terminal velocity. We can show using elementary calculus that it returns at the terminal velocity divided by the square root of two. We can also show that the maximum height it attains is the height with no air resistance multiplied by the natural logarithm of two. The program calculates these two quantities and their numeric/analytic ratios. It also plots displacement versus time with and without air resistance.
The close agreement between the numerical and analytic values shows that the Feynman algorithm can be extended to deal with velocity-dependent forces without adding complexity.