In reality the difference probably comes down to a knats hind leg sand papered on both sides.

Strictly speaking, air drag increases linearly with speed as long as the air flow remains laminar. As the speed increases, the flow eventually detaches and becomes turbulent, and the air drag becomes proportional to the square of speed. In real life it is usually somewhere in between, and for faster speeds it moves closer to v^2.

For every car, for example, there's a more-or-less well defined threshold speed, over which the turbulent component starts to dominate (the exact value depends on the aerodynamic properties of a specific car). Under that threshold the car's fuel consumption (MPG) depends very little on the actual speed. Over that threshold, as v^2 component begins to become more prominent, the car's MPG is dropping noticeably as the speed increases. It is often assumed that the threshold value is 55 mph, while in reality the number makes little sense, since for each car model it is different (sometimes significantly).

A typical bike with a cyclist on it is not a very efficient aerodynamic shape, so it is not a surprise that the relationship will largely depend on v^2 even for low speeds. A trailer can be built in a much more efficient aerodynamic shape, thus extending the range of speeds in which the linear air resistance component dominates over the squared component. This is splitting hairs, of course, especially if the trailer is pulled by a bike (which immediately negates any aerodynamic benefits).

In any case, to call the relationship "exponential" (as previous poster did) is certainly a major error.

So basically, for bicycles, aerodynamic drag increases as the square of speed. Where have I read that before?

Have fun "cipherin".....

where
FD is the force of drag, which is by definition the force component in the direction of the flow velocity,[1] ρ is the mass density of the fluid, [2] v is the velocity of the object relative to the fluid, A is the reference area, and CD is the drag coefficient — a dimensionless constant related to the object's geometry and taking into account both skin friction and form drag.

Short of having a wind tunnel and some way of measuring resistance(like a spring scale....LOL!)I'm not sure what good this will do.Your going to have a few problems coming up with some numbers.

Look mom...I have a wind tunnel and a spring scale.....I'm ready to do some math.....should I get panniers or a trailer....let's find out.....

If things were that simple...

Cd in the above formula can only be assumed constant under certain set of circumstances. In general though Cd is a function of so called "Reynolds number" (https://en.wikipedia.org/wiki/Reynolds_number_)

"At a low Reynolds number, the flow around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. At very low Reynolds numbers, without flow separation, the drag force is proportional to v instead of v^2; for a sphere this is known as Stokes law."

Have you bothered to consider the magnitude of Re at typical cycling speeds, and by how much Re and Cd change over the range at which an average cyclist rides?

I guess you haven't beed following what people have been doing with power meters (do a search on Coggan & regression or Chung method). It's possible to measure CdA with high levels of accuracy and precision; better than the power meters themselves.

This guy answered best.

A third wheel is that much more weight to expend energy in order to accelerate it rotationally.