The Definitive Horse Sized Duck vs Duck Sized Horse Answer

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!

Engineering

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:

Mallard Duck          1-1.4kg
Horse                      380kg-550kg

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.

Physiology

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.

Military

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.

Miscillaneous Observations

– 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.

Conclusion

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?

What are the chances?

This post has nothing to do with bikes, wood or anything bike related but you might enjoy it if you’ve found any of the other posts interesting. I got married recently and the following post is essentially my grooms speech minus the usual thanks to friends and family. It went down better than I’d expected so I thought it might make entertaining reading and have reformatted it as a blog post.

Shortly after we got engaged, J turned to me, while we were having dinner one night, and said “We’re so lucky to have found each other. I mean, what are the chances of meeting THE person that you want to spend the rest of your life with?” I have to admit, I can’t remember how the rest of the conversation went after that but the question stuck with me and when it came down to writing me grooms speech I thought it might make an interesting subject. So, What are the chances of meeting your soul mate?

By soul mate I mean one person. The one. Your perfect match. With just over 7 billion people on the earth, finding your perfect match seems like an impossible task but a large proportion of married couples will tell you that they’re soul mates……at least they think that when they first get married! So, are they all deluded or is it easier than it sounds? The first thing I thought of when I started to think about this question was the 6 degrees of separation theory that you can connect any two people on the earth by an average of 6 social connections (a friend of a friend of a friend of a friend……). Interestingly this has more recently been calculated as 4.74 degrees of separation for the 721 million Facebook users. This means that, with the right chain of social connections we are all only 6 steps away from meeting our soul mate. Unfortunately, the probability of randomly finding the exact 6 people that will link you to your beloved is astronomically large (so large that I’m not even going to calculate it!) and I can’t see that it’s going to help me get any closer to answering the question.

Now, in my opinion, the group of people that you have to start looking for your soul mate in is fairly self selecting. You are only going to spend the rest of your life with someone that shares similar interests, views and philosophies as you, so they will almost certainly be in a vaguely similar social group to you. This means that we can start to reduce the population of people to chose from. I set up a set of social preferences that I thought vaguely described the groups that I hang out with with work, hobbies, family etc. I’ve tried to identify the total number of people in my extended social groups, that is if I met all the friends of friends of friends until I’d met everyone with similar interests to me. Here are some of my assumptions:

Location – I don’t travel much so I’m going to assume that my soul mate is UK based so that I could theoretically meet them. I mean there’s no use in having a soul mate that it is impossible to ever meet and I’m not about to start a world quest on the off-chance that my soul mate is living in a cave in Borneo. This brings my population down from 7 billion to 62,262,000 people (2011 Census). I’ve added a further 0.9 factor on this as, without being specific and without wanting to cause offence (!), there are some parts of the country that I can’t see my soul mate coming from.

– Age – My contact group through work and social activities is about 20-60 years old. Using the UK national census data shown in this graph, this cuts my population down by another 50%

– Interests – Like I said above, my soul mate is going to be from a similar sort of social/ interest group to me so i’ve put a 0.7 factor to rule out 30% of people that I won’t meet because they have completely different interests to me but it doesn’t matter because my soul mate wouldn’t be in those groups anyway.

– Intelligence – Now, you’ll already have realised, from reading my blog, that I’m a nerd. I can’t really deny that. The people I hang out with at work are similar (although they’ll try and deny it!) and the people I socialise with are generally well educated and I’d hope my soul mate would be too. So i’ve put a factor of 0.45 on intelligence. 15% of the UK population is university educated so I tripled that number for good measure.

Note that I haven’t included any phisical traits or male/ female as that isn’t something you chose in the people you meet on a day to day basis. Remember, we’re trying to calculate a total population of my extended social group. All of these factors leaves me with 7,354,699 people out of the original 62 million so the odds are heading in the right direction.

The next question I asked myself is how many of these people will I meet in my lifetime. Taking the UK average age of 77 for men I reckon I have about 62 years worth of meeting people that could turn out to be ‘the one’ or 22630 days. I think I probably meet around 7 new people per week through work and friends. By this I mean people that you actually speak to and get to know their name. There are plenty more that you speak to but never actually know their name and will almost certainly never see again. I’ve guessed that there are another 50 people per week that I talk to in this way but i’ve then factored that by 3% as I don’t count them as propper meetings. This gives me an average of 1.36 new people per day or 30,712 people in my 62 year period.

So, I’ll meet 30,712 out of a potential 7,354,699 people with my soul mate being one of those people. That gives me a chance of meeting them of 0.4% and, assuming other people have similar odds to me, it means that only 1 in 219 people will ever meet their soul mate. That sounds better than the 1 in 7 billion that we started with but still rather small. Especially, because I gave this speech to a room with only 120 people in it and seeing as i’ve found my soul mate that means that the chances of there being a second pair of soul mates in the room is down to 1 in 525.

I didn’t want to end on this slightly depressing note as there were plenty of happily married couples in the room, so I thought: What if you could be very happy for the rest of your life with your 99.9% perfect partner? I went back to my list of factors and edited it as if I was writing a specification for my soul mate. Here’s how it went:

Location – I stuck with the previous assumptions (90% of the UK) as I don’t want to limit the possible options purely based on location

– Sex and sexual preference – Female and straight. There are currently 49% women in the UK with 2% of them homosexual. So a factor of 0.48 overall.

– Age – I think a 10 year spread would be acceptable (+/- 5 years) which, using the distribution above would only include 12% of the population in the age demographic above.

– Ethnicity – Looking at the UK ethnic demographic I estimate that 90% of the population would be on my preference list.

– Height – I’m not particularly fussy about height but taking a large range of 5′ to 6′ still knocks off 5% of women from my list

– Appearance – I know I said SOUL mate but, lets be honest, appearance makes a big difference! I’ve taken 15% for facial appearance and a further 20% for phisical appearance. This may sound picky but, hell, I’m happy to be picky about this one!

– Mind – I’d like my soul mate to be intelligent so i’ve used 15% as the proportion of university educated people in the UK. I’ve also added a fairly random 30% for emotional compatibility.

– Interests – I’ve taken a factor of 30% for people with shared interests.

– Religion – I’m not religious so this isn’t a priority for me. But if it had been then it would seriously sway the selection process.

All these factors get multiplied together with the original 62 million people in the UK to give a final number of 932 possible near pefect matches in the UK (I know they’re not necessarily all mutually exclusive but I’ve ignored that). That is one out of every 32,742 women which makes me sound pretty fussy! The 932 matches are all contained within the group of 7,354,699 that I calculated above and I’ve got 30,712 chances to pick the right one. I’m not going to bore you with the calculation but, using converse probability, this gives a 98% chance of meeting one of these ‘ones’ in your lifetime. So you will almost definitely meet at least one perfect match in your lifetime and the balance of meeting one swings from unlikely (less than 50%) to likely (more than 50%) at about 5000 meetings or just over 10 years. So that’s a more positive note to end on.

Interestingly, if these numbers actually mean anything at all, this would explain part of the reason that most people meet or decide to marry a long term partner at about 30 years old after 10-15 years of meeting potential ‘one’s’.

All these numbers are entirely dependent on the choice factors that you use in the first place and I think i’ve been fairly picky. Which is even better news for the less choosy among us! But that does err further away from the concept of ‘the one’. It also means that rather than turning to my wife and using the cliché of “You’re one in a million” I now know that she’s even more special than that and is in fact “One in seven million, three hundred and fifty four thousand, six hundred and ninety nine”. It’s not quite as catchy though. Conversely, rather than being “The luckiest man in the world” I’m only the luckiest man in 219 people!

There’s one more thing to take from this (note that this wasn’t part of my Grooms speech!). If we take ‘fairly likely’ as being over a 75% chance, then it becomes fairly likely to meet at least one perfect match after about 22 years. So, it is fairly likely that you’ll meet at least 3 of the 932 over your lifetime which could go some of the way to explain the 1.5% divorce rate in the UK!

Some of the idea behind this speech/ post came from the Soul mate calculator. I used some of the same variables as they have in their calculation but changed my method slightly to work out how likely it actually was to meet your perfect match and the timescales involved for the different situations.

Brakeless…….. Clueless

All the cool kids in London ride fixed gear or single speed bikes these days. Fact. Well, all the cool kids…..and me. However, there’s a trend for removing both front and back brakes on a fixed gear bike and using rear wheel braking only via the pedals. Riding a fixed gear bike allows you to control your speed and come to a stop easily without ever touching the brakes. If you lock your legs up when riding then you can even skid the rear wheel to stop when you want to. this is (incorrectly) considered to be a faster way of stopping a fixed gear bike and comes with the added down side that it wears out your tyres quickly. If you talk to any brakeless rider they’ll tell you that they can stop whenever they want in the same time/ distance as a normal bike. I guess, you’d have to think that before hitting busy streets on a bike with no brakes in the first place. So, my question is:

How safe/ stupid is it to ride a bike with no brakes and can you really stop in the same distance/ time as a normal bike in an emergency?

To answer this we’re only going to look at the mechanical efficiency of rear wheel braking vs front wheel braking. We will assume that a proficient fixie skidder can react as quickly with their legs as the rest of us can with a hand operated rim brake so reaction times are not an issue. I know I can’t react as quickly with my legs but I don’t regularly ride fixed. In fact, the brakeless bregade could argue that the friction and brake cable extension from the brake lever to the rear wheel and potential water on the brake pads on a rim brake causes a much bigger delay and would require faster reactions. But that is another topic altogether.

First we need a model of a bike and rider to base our analysis on. The figure below shows the combined centre of gravity (COG) of the rider and bike. This moves forwards, backwards, up and down as the rider changes position. A typical 58cm frame might have a wheelbase of 1020mm (L1 + L2) and with the rider in a normal riding position the combined COG will be located at around L1 = 600mm and L2 = 420mm at a height above ground of around H = 1100mm.

You can measure the position of your COG quite easily by standing on some scales holding the bike to get the overall mass (M). Then, put the front wheel on the scales whilst sitting in your preferred riding position and, balancing against a wall, read off the weight measurement (Mf). Do the same for rear wheel as a double check (you probably need someone else to read off the weight as it will alter your position if you try to read it). The front and rear weights should add up to give you the same as the total and the COG position can be calculated using only the front measurement:

L1 = Wheel Spacing x (1- Mf/M)

The vertical position is a bit trickier to measure but will generally be located around the level of your hip in a normal riding position.

Let’s start with the bike with front and back brakes. Braking is limited by 2 factors. Firstly by the friction that can be developed between the bike tyre and the road and secondly by the stability of the rider on the bike. We all know that if you pull the front brake too hard you’ll end up over the handlebars. Friction between the bike tyre and the road is calculated by FFr = μR. Where FFr is the friction force,  μ is the friction coefficient between the road and the tyre and R is the reaction force at the point of friction as shown by Rf and Rr on the sketch above.

The friction coefficient varies depending on the road surface, tyre surface and whether the two friction surfaces are static or sliding (kinetic). Here are some typical values for bike tyres on an asphalt road:

Dry (Static)     = 0.8
Wet (Static)    = 0.5
Ice (Static)     = 0.1
Dry (Kinetic)   = 0.65
Wet (Kinetic)  = 0.4
Ice (Kinetic)    = 0.08

The friction force causes the bike and rider to decelerate. The act of decelerating causes more weight to be applied to the front wheel then the back wheel due to the momentum of your body, which is why it is easy to go over the handle bars if you decelerate too sharply. Using Newtons 2nd Law (Force = Mass x Acceleration)  the deceleration force on the bike and rider is M x a. Using the front brake alone, at the point where the back wheel lifts off, all the weight is on the front wheel. If we take moments about the base of the front wheel at this split second then the overturning force from the horizontal deceleration is balanced by the weight acting downwards or: MaH = MgL1 which can be rearanged to give

amax = gL1/H

Using the numbers above amax = 9.81×600/1100 = 5.35m/s2. Which is equivalent to stopping from 20mph in 0.7s over a distance of 7.5m.

Note, that at this point the rear brake has no effect whatsoever as there is no weight on the back wheel. The friction force developed between the tyre and the road is Mamax or MgL1/H whilst the maximum friction force theoretically available would use our whole weight mg. So, the smaller the ratio L1/H gets the less stable the bike becomes during braking. Recumbent riders can actually skid the front wheel because L1/H > 1 so the wheel skids before the bike toples over.

Now, let’s do another calculation for braking, this time using the back brake only. We know that we can skid the back wheel without overbalancing so the maximum braking force we can get is FFr = μRr this also equals the deceleration force so balancing the horizontal forces:

Ma = μRr

We don’t know what the reaction on the back wheel is as it is a function of the deceleration force so taking moments about the front wheel:

MaH + Rr (L1 + L2) = MgL1

If we rearange the first equation and substitute Rr into the second then we get:

amax =          gL1             compared with amax = gL1/H for front wheel braking
.         .H+(L1+L2)/μ

Using the numbers as before this gives amax = 9.81×600/(1100+(1020/0.8)) = 2.48m/s2 Or a 20mph stopping distance of 16.1m in 1.5s.

So, using rear wheel braking is 2.2 times worse than front wheel braking without skidding or overbalancing. Let’s make this a bit more realistic because we don’t all brake so hard that we’re at the point of overbalancing even when you have to do an emergency stop. We’d be walking round with a lot more cycling injuries if that was the case, so let’s take 80% of the front wheel braking figure (4.28m/s2). In the same way that it’s very easy to overbalance when braking hard on the front wheel, it is very easy to skid the back wheel when you slam on the brakes. This changes your stopping force from static friction to kinetic friction which is about 80% of the static value (max deceleration is now 2.21m/s2). So, assuming that our front brake rider doesn’t want to head over the handlebars and our back brake rider either doesn’t want to skid or can’t help skidding the front brake is still 1.9 times more effective than the back brake.

You’ll also remember that this article started off with a discussion of the ultra cool, brakeless, fixie skidders. In the process of starting a skid, most riders shift their weight forwards to take weight off the rear wheel and make it easier to initiate the skid. This reduces the dimension L1 and makes the rear wheel even less effective. This is taken to the extreme in a skid competition where the winning skidder will be the one to get their weight the furthest forward over the handlebars. Following the calculations above, if you shift your centre of gravity by only 100mm forwards to help initiate a skid then your maximum deceleration drops to 1.83m/s2 which is now 2.3 times worse than a front brake.

So, the brakes (well the front one for sure) are staying firmly on my bike and I will continue to despair at the brakeless (clueless) riders out on the streets. I’m also sure that my suggestion that Newtons Second Law shows that you shouldn’t ride brakeless will fall on deaf hipster ears!

Another interesting thing to note from this is that because bike riders (with a front brake!) are limited by trying not to overbalance, rather than being limited the friction coefficient between the road and the front tyre, you can NEVER brake as quickly as a car because bbecause a car’s L1/H ratio is much higher than that of a bike. I guess we’ve all experienced a close shave (actually a couple of crashes for me) when following a car too closely when they brake suddenly but now you have the proof…. and I still don’t learn.

One gear…….. one idea

As i’ve mentioned in other posts, I ride a Condor Pista singlespeed (untill the Hardwood Single Speed is finished!). It’s my daily commuter into London, my weekend sportive ride, my tourer with paniers for long trips….. I think you get the drift. Like most other single speed and fixed gear riders, I always get asked by my geared friends why I make life so hard for myself by riding with only one gear. The short answer is that I don’t think I do. There are very few rides around London, including the North and South Downs and the Chilterns that are out of the range of one gear. I usually get dropped on the downhills as my legs don’t spin fast enough but uphill is never as bad as I think it should be. For most rides I ride 44t chainring with a 16t sprocket, or 73.2 gear inches, changing to a 15t or 17t sprocket very occasionally for particular rides. I reckon I can get up anything up to about 15% on 44×16 which means Ditchling Beacon in the South downs is just on the limit for me on my standard gearing. The monstrous Yorks Hill (http://www.catfordcc.co.uk/hillclimb/about.aspx?sm=21_1) in the North Downs at 24% remains well out of reach though, and certainly as the 9th hill at the end of the annual King of the Downs!

So, that brings me on to the point of this post. I was out on a ride in the Chilterns in February last year and Charlie had plotted a route that had a hill, at about 50 miles in, that he was confident would defeat me (unfortunately I can’t remember where it was). I was certainly up for the challenge. The sign at the bottom of the hill said 20% but I got out of the saddle and sprinted ahead of him. It wasn’t that bad and I was within metres of the top when it ramped up to 20%…… and defeated me. It was February though so I hadn’t done much training and wasn’t particularly fit. I’d get up it next time. Right? The next time I met the hill was in September. By this stage I had done a lot of cycling over the summer, several centuries, lots of hills and a tough triathlon in the Lake District. I was the fittest I’d been all year so I should have no problem with the last few metres……but I only made it about a metre further than before. Why?

This bothered me for a while as I simply couldn’t work out why I was so much fitter but didn’t get any further (and because Charlie teased me about it).I thought it must have something to do with another funny thing i’d noticed before. That is, when I get defeated by Yorks Hill at the KoTD every year, I have to get off and push my bike at about the same point as all the slightly less fit guys riding carbon fibre compact chainset bikes (2 chain rings rather than a triple chainset). Why do they have to get off when they’ve got sooooo many more gears than me?

So, when no one was looking, I turned to trusty old Excel to give me a hand! As with most engineering problems we’re going to have to make some assumptions and simplifications to demonstrate our point in an easy way and you’re going to have to remember a bit of GCSE maths as well : (  Here’s my model for me going up a hill on my bike. I’ve ignored fricton in the bike parts and air resistance (both minimal in this example)The total weight of me and the bike is W, there’s a reaction force between the road and each of the wheels and there’s a force driving me up the hill. At the point that I fall off because I can’t push any harder, I’m stationary, so all these forces balance as shown in the little force triangle I’ve drawn in the corner. Now, as the hill gets steeper, to keep the forces balanced R gets smaller and F gets bigger. How do we work out F?

If we know the angle of the hill and we know the weight of the rider and bike then:
F = Wsinθ
and R1 + R2 = Wcosθ The force is applied by me standing on the pedals with a proportion of my weight (xW) causing a tension in the chain. The chain then pulls on the rear sprocket which then turns the rear wheel causing a force F against the road surface.

There’s one more slightly trickier aspect in that you get maximum leverage on the cranks when they are horizontal and you get nothing when they are vertical. We’re going to use an average horizontal distance from the centre of the pedal to the centre of the bottom bracket as below.

I’m not going to bore you with the equation so you’ll have to take my word that the answer is that the average horizontal crank length = 0.637C or 63.7% of the full crank length.

Lets try an example on a 10% hill (θ = 5.7degrees) with my 44t (88.9mm radius) x16t (32.3mm radius) gearing, 700×23 tyres (334mm radius) and 170mm long cranks. That means the average crank lever arm is 0.637 x 170 = 108mm

The Force F = 0.0995W (using F = Wsinθ)

So the chain tension is 0.0995W x 334mm/32.3mm = 1.029W (the ratio of wheel radius to sprocket radius)

This force goes into the chain ring to be resisted by my weight on the pedals (xW). So, the force on the pedals to exactly balance F is:

1.029W x 88.9mm/108mm = 0.85W (ratio of chain ring to the average crank lever arm)

That is, 85% of my body weight is required on one pedal to balance on a 10% hill. More than this and i’ll go forwards, less than this and i’ll most likely fall on my arse! To speed this up I made a graph.

This shows that, using all the sizes and gearing above I can almost get up a 12% hill by standing up with all my weight on one pedal. Now, I know that I can actually get up about a 15% hill so the graph tells us this requires 126% weight on the pedals. I ride with cleats so I must be pulling upwards with my other leg by 26% of my bodyweight + the weight of the bike. That’s 26% of 85kg + 10kg which is 25kg per pedal stroke. Sounds like hard work and explains why I can’t keep it up for very long before my legs give in.

Ok, you’ve made it this far and you’re still waiting for the punch line. Right? How does this answer any of the questions that I started out with? I know that I can apply about 126% of my body weight down on the pedals as an absolute maximum to get up the last bit of a hill. But, using the graph again, to get up the elusive final part of that 20% hill I’d have to go up to 167% of my body weight. Let’s say everything up to 100% is possible for everyone, as that is essentially just like walking up stairs with all your bodyweight on one foot for every step. I call it the bit you get for free as you are just using your weight to it’s full extent. Anything above 100% requires pulling up with your other foot which requires cleats, technique and most importantly a very different set of muscles that you don’t use very much when you’re sat in the saddle for 95% of the ride. So, even though I got a lot fitter over the season I certainly wasn’t 2.6 times stronger/ fitter (I’m saying that going from 26% extra to 67% extra is an increase of 2.6x) and this shows me that realisticallyI’m never going to get up that 20% hill in my normal cruising gear. Once, I go past that 12% boundary where I need more than just my body weight, each small increase in steepness requires a significant increase in effort and strength.

Also, knowing my 126% limit, I can now work out a gear ratio that will let me get up my 20% nemisis. Using the calculation above with the same 44t chainring, I’d have to switch to a 21t sprocket from my usual 16t (and even that wouldn’t quite get me there but it’s the closest without changing the chainring). That’s a change of 31% on the gear ratio meaning that I’ll be spinning my legs 31% faster for the rest of the ride, just to get up the final part of one hill. No thanks! It also shows that when I switch between my 15/16/17t sprockets it wont really change the maximum gradient of hill I can get up by very much. It’ll just change my cadence over the ride and make each of the hills very slightly easier/ harder.

Finally, back to the guys on their carbon fibre compacts, walking up Yorks Hill. To get up a 24% hill in a 34t chainring with a 26t sprocket means 94% body weight on the pedals. That certainly means that you’ve got to get out of the saddle and, assuming they’re already pretty tired from the 90miles of hills that preceeded Yorks Hill in KoTD, that could well mean a premature walk just a few metres ahead of me whilst the smug triple chainsetted riders (on 30×26) are still ploughing on with a mere 83% body weight, which is still quite a lot of effort considering they have a gear ratio that’s more than twice as easy as mine.

So, the point is that as long as I’m happy to get out of the saddle from time to time when I get to hills, the 44×16 gear ratio that I like using really doesn’t limit me until I get to the steepest of hills, at which point geared riders also start to struggle because small increases in steepness past your 100% point require you to shift through the gears very quickly to maintain the same pressure on the pedals.