For a spherical projectile traveling through air, a reasonable approximation to the drag force is F_drag =  ½C_DρAv² where A is the area πr² , ρ is the air density, v is the speed, and C_D is the drag coefficient, often taken to be 0.5, based on experiment. Our drag parameter is F_drag/m, where m is the mass.  C_D is difficult to determine theoretically—trajectory predictions with air resistance are at best semi-quantitative.   Experimentally, C_D increases dramatically at velocities near the speed of sound (the “sound barrier”) but then decreases again. So results at high velocities with a constant C_D are not too reliable.

For ping pong balls, tennis balls and golf balls, the lift force from a high spin will also affect the results, we haven‘t tried to include it here.

Putting in the numbers (you should check!), we find for a tennis ball b is about 0.002, for a golf ball about 0.001, for a ping pong ball about 0.014.

For a cannonball, a Napoleon 12-pounder as used in the Civil War, we get b = 0.000033, but the initial speed is about 440 m/sec, supersonic, so the drag will be increased to well above the naïve formula. For modern artillery shells, b will be lower since they are not spherical, so the mass is greater than for a sphere presenting the same cross-sectional area.

Exercises: Try plotting the range as a function of angle for a fixed initial speed. What angle gives the maximum range? Is this the same if there is air resistance?

Plot the maximum range as a function of initial speed.