Originally Posted by AEO

I haven't learnt fluid dynamics, but basically put, it is: air resistance vs. rolling resistance.
Factors that cancel out each other: weight, gradient, friction (of bike), air resistance (of bike and rider)
Although velocity is constant, because you are going up a hill, you are constantly accelerating to beat force of gravity and force of displacing air

Then you're left with:
air resistance (of panniers) vs. air resistance (of trailer) + rolling resistance (of trailer)

Now, since energy is conserved, rolling resistance does not change whether you go up or down the hill and air resistance increases exponentially the faster you go. By using potential energy, that is the energy stored by being at a higher altitude, one can measure how efficient each bike will conserve its energy by going DOWN the hill without pedalling and maintaining the same riding posture as going up. The bike with the longer run out or faster speed at the base will require LESS energy while going up the hill.

Potential energy is measured by Eg=mgh (Energy = mass x gravity x height). At the base of the hill, you have zero potential energy, because h=0. You will however, now have kinetic energy, which is Ek=0.5mv^2 (Energy = 0.5 x mass x velocity^2). Both bikes will have the SAME amount of energy when they are at the same height and weight. The less energy efficient bike will lose its energy to the surrounding system faster and end up slower at the base.