Readouts of the last projectile motion:

Time taken: s

Range: m

Height: m

Final velocity: m/s

For a spherical projectile traveling through air, a reasonable approximation to the drag force is

*F*_{drag} = ½*C*_{D}ρ*Av*^{2}=*bv*^{2},

where *A* is the area π*r*^{2} , 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. The
*b* is standard notation.

Our drag *parameter* is *F*_{drag}/*mv*^{2}=*b*/*m*, where *m*
is the mass.

The coefficient *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. (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.