I have been wondering as I load up my gear into my panniers if there is any work advantage [amount of watts] it takes to move say 50 lbs of gear [pannier weight included] up a 6% grade at 5 mph vs. say a BOB trailer pulling 50lbs of gear [weight of the trailer included] up the same grade at the same speed. I wonder if the wattage output would be the same ? any ideas ... I'm not interested in the old debate panniers vs. trailers .. I'm strickly interested in actual wattage output .. however it would be interesting to know if you were on the flats and keep the loads the same and kept your speed at 12 mph I wonder if your actual wattage output would be different because of the difference in wind resistance .. I appologize if this test has already been posted and if it has I would sure like to review it ... Glenn

I would bet that panniers were lighter and with less wind resistance than a trailer, so they'd take less power. However, people that care about wind resistance aren't using trailers and panniers, so there's not a good way to prove it.

More wheels= more resistance.

Do State which law of Physics you have in mind, Inertia? Gravity?, Thermodynamics?
conservation of angular momentum?

TOE

https://en.wikipedia.org/wiki/Theory_of_everything

You say total weight is the same so the work against gravity will be the same. What will increase is friction due to the additional wheel bearings and tire and perhaps some increased wind resistance because the low slung BOB is below in the draft of your legs,whereas the paniers would be right behind them.

This.

If the wheel on the BOB trailer was the same as the wheels on the bike then it probably wouldn't make much difference and the rolling resistance is proportional to the load carried by the tire.

But a BOB trailer has a 16" wheel with a fat tire which will have higher rolling resistance that won't be offset by the lowered resistance of the bike tires. I would guess the difference would be less than 10W.

Well, if you're towing a 15 pound trailer with 35 pounds of gear vs. 50 pounds of gear on the bike, that's 15 pounds of stuff you don't have if you absolutely need it. On the other hand, if all you need is 35 pounds of gear, then that's all you need and you don't need the trailer.

IMO: if you're touring, it doesn't matter one way or the other.

I would suspect that the trailer would be easier if the total weight was equal.

Yes, more wheels equals more resistance but it doesn't take a whole lot of power to spin a wheel. I think this would be outweighed (no pun intended) by the extra effort involved with the side-to-side motion of the frame and the extra weight in the panniers. The trailer would be mostly isolated from this side-to-side movement.

+ a cheap Hub on BoB trailers upps the profit, so they use a cheap one.

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.

It's not quite as simple as this. Air resistance will increase exponentially with speed but rolling resistance will increase linearly. Since speed will probably be much higher going downhill than up, the air resistance will exert a much higher relative force on the bike versus rolling resistance on the downhill run. To do this experiment correctly, you will need to find a very shallow hill which will give you a downhill coasting speed that approximates the uphill speed of the rider.

Doesn't matter......90% of touring effort,trailer or panniers,is in your head.....

When I bought a trailer to haul my kids around, the first thing I did was build new wheels for it. Nice aluminum BMX rims and sealed bearing hubs. 25 years later my brother-in-law is still using it to haul his kids around.

The second statement makes the first obvious.

Air resistance increases as the square of speed. That's a second order linear relationship, not an exponential one.

To the subject at hand: from experience, the trailer will be harder work than the panniers if your bike is built properly.

I used to compete in triathlons and since I wasn't a good swimmer (23:30 for 1500 m) I would spend most of the bike leg overtaking the better swimmers, especially on hills. I rode the the only bike I had at the time which had pannier racks and since I like annoying triathletes, I would occasionally ride races with panniers attached (with empty cardboard boxes in them). I take from thsi that the added air resistance of panniers is quite small.

I rode the same bike with a kiddy trailer for a while (until we found they were illegal where I lived). I never tried competing in the tris with it attached but the drag effect was very obvious.

If something is squared, the exponent is 2, so I'd say that's exponential.

Same weight, the difference in rolling resistance will be trivial. If any, because the rolling resistance is proportional to weight (and speed) so while more wheels can give more resistance each of those wheels has less resistance in proportion to the lesser weight on each. The mechanical losses in the extra sets of bearings is almost nothing.

At the speeds mentioned, 6-12 mph, air resistance is not significant enough to be concerned about any delta. So the power requirements will be very close either way.

Handling and balance may be affected, which may make the rider less efficient on one setup or the other. My "physics" guess is that this is where the only real difference in wattage is found. Unless you want to consider higher speeds.

I got a 28 spoke WTB Grease Guard Front Hub with intention of doing that,
but the cargo practicality of the BoB trailer, got low marks after having it a while,
My typical use, ... so I sold it, instead.
local Hunters Use them for carcass hauling, Elk season, Mountain bike into the woods.

I still Have the Grease Guard sealed bearing Hub brand new
if some one reading this wants to buy it.

Usually we say f(x)=a^x is exponential. But since cyclists tend to say "exponential with speed" and we all know they really mean v^2, or v^3 talking about power, it doesn't merit arguing about.

All the theoretical debates aside, there is a reason that the scientific method includes experimentation. In this case, get a bike with a power meter. Set-up pannier and trailer experiments. Repeat each a few times (or more). Then see that the actual data says.

As with most real world situations; the theoretical makes assumptions (and simplifications) that may or may not be realistic. The only definitive way to determine the answer to a question like this is to MEASURE the real world results.

It is semantics, but when the exponent is a constant it is usually referred to as polynomial growth. Exponential growth (where the exponent is a function) will always surpass both linear and polynomial growth over time.

Linear growth:
f(x) = a * x +b

Polynomial growth:
f(x) = a * x^b

Exponential growth:
f(x) = a * b^x

You physics wonks crack me up, but I respect you nontheless. At 5 m.p.h. we can ignore wind resistance. A bike with only 2 wheels will always be more efficient than the same bike with a third wheel in contact with the ground.

Question is answered, clearly and concisely.

You're welcome.

Sorry, but no. Simply add in head winds, tail winds, incline, pavement types, etc... Such answers are never simple and frequently counter intuitive.

I like the power meter approach as long as the rider must try to keep the same pace going up that hill. Then come to a dead stop at the top and just roll down to a finish line for timing.

Adding wind conditions messes up everything. Isn't there a ski slope in the Middle East that's completely encapsulated? Abu Dabi or Dubai, or something like that?

Myrridin is right, the scientific method is empirical, systematic observation.