Time taken: s
Final velocity: m/s
For a spherical projectile traveling through air, a reasonable approximation to the drag force is
Fdrag = ½CDρAv2=bv2,
where A is the area πr2 , ρ is the air density, v is the speed, and CD is the drag coefficient, often taken to be 0.5, based on experiment. The b is standard notation.
Our drag parameter is Fdrag/mv2=b/m, where m is the mass.
The coefficient CD is difficult to determine theoretically—trajectory predictions with air resistance are at best semi-quantitative. Experimentally, CD increases dramatically at velocities near the speed of sound (the "sound barrier") but then decreases again. So results at high velocities with a constant CD are not too reliable. (Our code could be modified to include a velocity-dependent drag, though.)
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 that here.
Putting in the numbers (you should check!), we find for a tennis ball b/m is about 0.019, for a golf ball about 0.009, for a ping pong ball about 0.14.
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 perpendicular to the direction of flight.
Thanks to John Welch of Cabrillo College for pointing out some errors in earlier figures.
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.