Section I-B: Air Density

Air density, r, is calculated as follows:

r = (p * M) / (R * T)

where M is the molecular weight of air, or 0.028964 ^kg / _mol

where R is the universal gas constant of 8.31447 ^J / _{(mol * K)}

where T is altitude temperature in Degrees C, calculated as follows

T = T₀ + L * h

where T₀ is the standard temperature at sea level, or 288.15 K (converting 15 C to Kelvin)

where L is the adiabatic lapse rate for dry air of -0.0065 ^K / _m

and h is the altitude in meters above sea level

and where p is calculated as follows

p = p₀ * { 1 + [ (L * h) / T ₀ ] } ^{[ (g * M) / (R * -L) ]}

where g is the average gravitational acceleration at the Earth's surface of 9.80655 ^m / _s 2

and p₀ is standard pressure at sea level, or 101,325 ^kg / _{m * s} 2

As temperatures increase or as altitude increases, air density decreases.  A ballpark rule of thumb is that for a 10 F (or roughly 5.5 C) increase in temperature, air density will decrease by 0.02 ^kg / _m 3.  Also, as altitude increases by 500 ft (or 328m), air density will again decrease by 0.02 ^kg / _m 3.  (And again, these are ballpark figures, for an example.) 

It should come as no surprise that as air density decreases, the trajectories of non-spinning BB's increase.  For BB's with backspin (i.e., hop-up applied), a decrease in air density contributes to a greater trajectory, which is further explained in Section V-B: Effect of Altitude on Trajectory and Section V-C: Effect of Temperature on Trajectory.

Note that my calculations do not take into account the amount of water vapor in the air (relative humidity). This was done initially however the effects of humidity on density were so negligible that it was removed from the equation. Even temperature and altitude -- both of which have a much more pronounced effect on air density than humidity -- still account for very little variation in performance (though it is still observable and thus I have included charts detailing the effects of altitude and temperature on trajectory).

As humidity increases, it causes the air to become less dense.  This is certainly counter-intuitive.  One would think that adding water vapor would add to the air density simply because water is much more dense in comparison to air.  But we're not adding water; we're adding water vapor.  Water vapor is less dense than air.  The reason for this is the molecular weight of water vapor is lower than that of standard air.  Think of it this way: air is, for all practical purposes, primarily comprised of nitrogen and oxygen with nitrogen constituting 78% and oxygen 21% of total air volume (with Argon and CO₂ making up almost all of the remaining 1%).  The molecular weight of oxygen and nitrogen combined is, simply, heavier than that of the combination of hydrogen and oxygen.  Therefore, adding water vapor to air causes the combined medium to become a less dense than dry air. 

The amount of water vapor that can be held in air is related to the temperature of the air.  Simply put, warmer air can hold more water vapor.  If you compared the density of air that is 40 F with 0% relative humidity to that of 40 F air with 100% humidity, you'd find that the density difference is very, very small.  Even at higher temperatures (and again, warmer air can hold more water vapor, affecting density to a greater degree) it's still a negligible factor.  The change in density of air that is 90 F with 0% relative humidity to that of 90 F with 100% humidity is about 0.002 ^kg / _m 3.  Taking humidity into account for calculating air density wouldn't give us any benefit since the subtle change of 1 degree Fahrenheit will have more of an effect on the air density.  And in reality, a BB's trajectory will more than likely take it through several regions of air sporting different temperatures (as in passing over a patch of sand, or a patch of grass, over asphalt, through a region of air that is in the shade, etc.).