Last week I recieved the following email from Chris, a good friend of mine :
“OK dude, you’re my expert on all things to do with engineering and I’m carrying a huge bet on from the pub yesterday which concerns all sorts of nonsense and I’m hoping you can help me resolve it.
Dave asked me if I’d rather fight ten duck-sized horses or one horse-sized duck and my response was based upon what I assumed to be the fact that if you grew a duck to the size of a horse it would be unable to function as a duck. It would be utterly flightless and, I ventured, there was a good chance its legs would break under its own weight. Its heart might even give way.
Everybody mocked me for this assumption and the long and the short of it is that if I can back it up, my old flate mate Jez has to send me a trophy with an apology engraved on it. If I can’t back it up, I have to send him a trophy with a KFC Zinger Tower Burger in it, at a time of his choosing, wherever he happens to be in in the world at that point. Burkina Faso was mentioned.
Any chance you can help me out on this?”
I’d never heard this conundrum before but a quick Google search brings up a whole host of hits for the topic which interestingly range from between 5 to 100 duck sized horses versus the mighty horse sized duck. Common consensus seems to be that people would rather fight the duck sized horses, however, these seem to be highly qualitative personal opinions from internet speculators and there’s no conclusive case to substantiate this view other than that they are the smaller beast.
Rather than jumping straight on the case of forming a defence for my friend Chris, this clearly required an independant review of the question to formulate a difinitive answer to settle the debate. So, I consulted experts in several fields for their take on the situation. The following response is a compilation of the various replies that I recieved. I’ve also written it up here so Chris can’t edit the answer when he responds!
First we need to quantify a horse sized duck and duck sized horse. As they differ in proportion between length and height the easiest way to scale the 2 beasts is by weight:
Taking the average of each of these, a horse has a mass that is 388 times larger than a duck. Mass is directly proportional to volume (ignoring the variation in lung volume and fluffyness of the duck’s feathers which would add volume without significantly increasing mass) so the horse has 388 times the volume of a duck. Volume is equal to length x length x length so to find the increase in size (length) between the horse and the duck we have to take the cube root of 388. So a horse is an average of 7.3 times longer/ taller/ wider than a duck.
Bearing this in mind lets look at how this affects the duck and horse when you scale them. The duck gets 7.3 times bigger in length but 388 times heavier. It’s muscles and bones get 7.3 times wider in all directions so the cross sectional area of the muscles and bones gets 7.3 x 7.3 = 53 times larger. The stress on the muscles and bones = Force/Area so the increase in stress as you scale the duck up is 388/53 = 7.3 times. To understand what effect that would have, that is equivalent to the duck having to carry 7.3 times the weight it was designed for (it’s own weight) or me (85kg) having to carry 620kg! I’d definitely fall over under this weight, probably pull some muscles and most likely break some bones. So, the duck definitely can’t walk and it goes without saying that it can’t fly.
The miniture horses have the opposite effect. They would suddenly find that they could leap incredibly high relative to their height and would be very nimble. Their performance would be similar to that of a small dog like a terrier.
Fluid Dynamics and Aerodynamics
The next question is can the duck still swim? The answer is yes. The duck floats by displacing an equal mass of water to it’s own mass. The displaced water will be scaled to the same scale as the duck and so the monster duck will float in exactly the same way as the normal duck. The ducks feet have been scaled to the area scale factor (53) which is the same as the ducks muscles so it should find that it can still pull its feet through the water. The primary drag when swimming is from the frontal area facing the direction of travel. Again, this has been scaled by 53 so will not significantly impede our feathered friend. The monster duck does, however, have a proportionally much higher mass than before so has a significant inertia to overcome as it sets off, stops or changes direction. In effect it will turn into the oil tanker of water based birds.
Bearing this in mind – Would this fight be taking place on land or in water? The duck clearly stands more chance of survival if conducting a naval battle.
I rather briefly glossed over the fact that the duck wouldn’t be able to fly in the last section. That was on the assumption that since it couldn’t stand up then it wouldn’t be able to achieve the take off speed required for flight. However, what if the duck managed to find a hill or cliff to roll off in order to get airborne? Well, it’s still not going to get very far which we can see if we look at the scale factors again.
Gliding – The ducks wings have increased by an area scale factor of 53 and the aerodynamic lift provided by the wings is proportional to wing area. However, the duck’s weight, hence the lift required has increased by 388 so the duck will fall exceptionally quickly and break whatever remaining bones were left in it’s bloated body.
Flapping flight – To counteract the 7.3 times extra weight it’s going to have to flap like mad to create the extra lift required to keep it in the air. At full extension the ducks wings are required to support it’s weight to keep it up in the air. The force exerted on the ducks pectoral muscles is proportional to the square of the length of its wings. This combined with the 7.3x relative increase in weight compared with the muscle power means that the duck is sorely underpowered for flapping flight.
Unfortunately there’s not a whole lot to say about the horses on this one apart from the fact that they may have been inferior to the ducks in their original form as they were solely land based compared with the full terrestrial, aerial and marine assault that could be undertaken by normal ducks. This difference has clearly been put in the horses favour by the change in size of the duck.
Birds, like mammals, have a 4-chambered heart (2 atria & 2 ventricles), with complete separation of oxygenated and de-oxygenated blood. Birds tend to have larger hearts than mammals (relative to body size and mass) which is thought to be necessary to meet the high metabolic demands of flight. Among birds, smaller birds have relatively larger hearts (again relative to body mass) than larger birds. Hummingbirds have the largest hearts (relative to body mass) of all birds, probably because hovering takes so much energy.
The duck’s heart will be increased in size by the volume scale factor of 388 which will adequately match the increase in blood volume that it is required to pump around the body. The large heart relative to the size of the bird will mean that, now in it’s sedentary state, the ducks heart will be larger than physically required but would most likely function normally.
The most important stage to winning a battle is locating the enemy (not as easy as it sounds – see any army/news report from the herrick and telic era) and it’d be a darn sight easier to locate one big duck rather than 10, small, fast moving horses.
Once locating the enemy has been done, you then proceed to prevent it from being able to attack you before you attack it. In conventional warfare this is done by dividing your force into 3 (assault, suppress, reserve). The suppress team engages the enemy to ‘fix’ them – they cannot move, nor fire back because they are so busy taking cover to keep themselves safe. All the while the assault force is moving in cover to attack – the enemy won’t see them as they are busy cowering in fear of the onslaught. Job done, regardless of how dangerous a foe you are facing.
10 horses would be difficult to locate, and they could well employ teamwork to attack and defeat you – imagine fighting three off from the front, and then suddenly a further three attack from nowhere and get you in the back.
One fat duck is easily located, and relatively easily distracted by some Battle Ready Edible Animal Distractor (BREAD), so you can attack it from the flank and win a glorious victory.
Weaponry – A feature of birds is their lightweight but strong beaks. Whilst the beak would also suffer from the issue of scalability discussed earlier this may not render it unusable as the duck no longer has to worry about flying with a heavy beak. This could make for a formidable weapon on the end of a long (but rather weak) neck. The duck also has a new wingspan of nearly 7m which it could use to swipe at the enemy. The effectiveness of the wings will be severely impaired by their increased mass so could only be used like a pair of heavy clubs and would be useless for more than a few hits so they better be accurate!
The horses by comparison are fairly unarmed aside from a kick in the ankle from a small hoofed foot.
– If you apply the theory that “the sum is greater that the parts” then in affect you wouldn’t be fighting 10 duck sized horses but 12, or possibly more.
– The power of Rubberduckzilla has been recently demonstrated in the following TV advert. Although, once you know it can be tamed with a regular sized bottle of Oasis it becomes a rather less terrifying opponent.
Most of this response has focused on the negative impact of scaling up a duck. This is not just to appease Chris with an answer to his original theory but is a common issue of scalability that I come across regularly at work. Making things smaller is generally much easier than making them bigger (until you get very small) as the mass of whatever you’re dealing with quickly overrides other design issues. The fact that we can build a scale model that works well in the office often tells us little about how it will actually perform when scaled up by 500 times unless the model has been constructed with careful thought to the dimensional and scale sizes stiffnesses and weights of the parts.
The duck doesn’t seem to have faired very well in the comparison and it does look like the duck sized horses would make for a more formidable opponent in a fight so I hope Jez has a suitable apology to engrave onto the trophy.
I think my work here is done, although I’m more than happy for additional contributions as very little of this was my own work in the first place. Now where can we find an oven big enough for that duck?