There is an art, or rather, a knack, to flying…

The aerofoil is one of the greatest inventions mankind has come up with in the last 150 years; in the late 19th century, aristocratic Yorkshireman (as well as inventor, philanthropist, engineer and generally quite cool dude) George Cayley identified the way bird wings generated lift merely by moving through the air (rather than just by flapping), and set about trying to replicate this lift force. To this end, he built a ‘whirling arm’ to test wings and measure the upwards lift force they generated, and found that a cambered wing shape (as in modern aerofoils) similar to that of birds was more efficient at generating lift than one with flat surfaces. This was enough for him to engineer the first manned, sustained flight, sending his coachman across Brompton Dale in 1863 in a homemade glider (the coachman reportedly handed in his notice upon landing with the immortal line “I was hired to drive, not fly”), but he still didn’t really have a proper understanding of how his wing worked.

Nowadays, lift is understood better by both science and the general population; but many people who think they know how a wing works don’t quite understand the full principle. There are two incomplete/incorrect theories that people commonly believe in; the ‘skipping stone’ theory and the ‘equal transit time’ theory.

The ‘equal transit time’ theory is popular because it sounds very sciency and realistic; because a wing is a cambered shape, the tip-tail distance following the wing shape is longer over the top of the wing than it is when following the bottom surface. Therefore, air travelling over the top of the wing has to travel further than the air going underneath. Now, since the aircraft is travelling at a constant speed, all the air must surely be travelling past the aircraft at the same rate; so, regardless of what path the air takes, it must take the same time to travel the same lateral distance. Since speed=distance/time, and air going over the top of the wing has to cover a greater distance, it will be travelling faster than the air going underneath the wing. Bernoulli’s principle tells us that if air travels faster, the air pressure is lower; this means the air on top of the wing is at a lower pressure than the air underneath it, and this difference in pressure generates an upwards force. This force is lift.

The key flaw in this theory is the completely wrong assumption that the air over the top and bottom of the wing must take the same time to travel across it. If we analyse the airspeed at various points over a wing we find that air going over the top does, in fact, travel faster than air going underneath it (the reason for this comes from Euler’s fluid dynamics equations, which can be used to derive the Navier-Stokes equations for aerofoil behaviour. Please don’t ask me to explain them). However, this doesn’t mean that the two airflows necessarily coincide at the same point when we reach the trailing edge of the wing, so the theory doesn’t correctly calculate the amount of lift generated by the wing. This is compounded by the theory not explaining any of the lift generated from the bottom face of the wing, or why the angle wing  is set at (the angle of attack) affects the lift it generates, or how one is able to generate some lift from just a flat sheet set at an angle (or any other symmetrical wing profile), or how aircraft fly upside-down.

Then we have the (somewhat simpler) ‘skipping stone’ theory, which attempts to explain the lift generated from the bottom surface of the wing. Its basic postulate concerns the angle of attack; with an angled wing, the bottom face of the wing strikes some of the incoming air, causing air molecules to bounce off it. This is like the bottom of the wing being continually struck by lots of tiny ball bearings, sort of the same thing that happens when a skimming stone bounces off the surface of the water, and it generates a net force; lift. Not only that, but this theory claims to explain the lower pressure found on top of the wing; since air is blocked by the tilted wing, not so much gets to the area immediately above/behind it. This means there are less air molecules in a given space, giving rise to a lower pressure; another way of explaining the lift generated.

There isn’t much fundamentally wrong with this theory, but once again the mathematics don’t check out; it also does not accurately predict the amount of lift generated by a wing. It also fails to explain why a cambered wing set at a zero angle of attack is still able to generate lift; but actually it provides a surprisingly good model when we consider supersonic flight.

Lift can be explained as a combination of these two effects, but to do so is complex and unnecessary  we can find a far better explanation just by considering the shape the airflow makes when travelling over the wing. Air when passing over an aerofoil tends to follow the shape of its surface (Euler again), meaning it deviates from its initially straight path to follow a curved trajectory. This curve-shaped motion means the direction of the airflow must be changing; and since velocity is a vector quantity, any change in the direction of the air’s movement represents a change in its overall velocity, regardless of any change in airspeed (which contributes separately). Any change in velocity corresponds to the air being accelerated, and since Force = mass x acceleration this acceleration generates a net force; this force is what corresponds to lift. This ‘turning’ theory not only describes lift generation on both the top and bottom wing surfaces, since air is turned upon meeting both, but also why changing the angle off attack affects lift; a steeper angle means the air has to turn more when following the wing’s shape, meaning more lift is generated. Go too steep however, and the airflow breaks away from the wing and undergoes a process called flow separation… but I’m getting ahead of myself.

This explanation works fine so long as our aircraft is travelling at less than the speed of sound. However, as we approach Mach 1, strange things start to happen, as we shall find out next time…


Hitting the hay

OK, so it was history last time, so I’m feeling like a bit of science today. So, here is your random question for today; are the ‘leaps of faith’ in the Assassin’s Creed games survivable?

Between them, the characters of Altair, Ezio and Connor* jump off a wide variety of famous buildings and monuments across the five current games, but the jump that springs most readily to mind is Ezio’s leap from the Campanile di San Marco, in St Mark’s Square, Venice, at the end of Assassin’s Creed II. It’s not the highest jump made, but it is one of the most interesting and it occurs as part of the main story campaign, meaning everyone who’s played the game through will have made the jump and it has some significance attached to it. It’s also a well-known building with plenty of information on it.

[*Interesting fact; apparently, both Altair and Ezio translate as ‘Eagle’ in some form in English, as does Connor’s Mohawk name (Ratonhnhaké;ton, according to Wikipedia) and the name of his ship, the Aquila. Connor itself translates as ‘lover of wolves’ from the original Gaelic]

The Campanile as it stands today is not the same one as in Ezio’s day; in 1902 the original building collapsed and took ten years to rebuild. However, the new Campanile was made to be cosmetically (if not quite structurally) identical to the original, so current data should still be accurate. Wikipedia again tells me the brick shaft making up the bulk of the structure accounts for (apparently only) 50m of the tower’s 98.6m total height, with Ezio’s leap (made from the belfry just above) coming in at around 55m. With this information we can calculate Ezio’s total gravitational potential energy lost during his fall; GPE lost = mgΔh, and presuming a 70kg bloke this comes to GPE lost= 33730J (Δ is, by the way, the mathematical way of expressing a change in something- in this case, Δh represents a change in height). If his fall were made with no air resistance, then all this GPE would be converted to kinetic energy, where KE = mv²/2. Solving to make v (his velocity upon hitting the ground) the subject gives v = sqrt(2*KE/m), and replacing KE with our value of the GPE lost, we get v = 31.04m/s. This tells us two things; firstly that the fall should take Ezio at least three seconds, and secondly that, without air resistance, he’d be in rather a lot of trouble.

But, we must of course factor air resistance into our calculations, but to do so to begin with we must make another assumption; that Ezio reaches terminal velocity before reaching the ground. Whether this statement is valid or not we will find out later. The terminal velocity is just a rearranged form of the drag equation: Vt=sqrt(2mg/pACd), where m= Ezio’s mass (70kg, as presumed earlier), g= gravitational field strength (on Earth, 9.8m/s²), p= air density (on a warm Venetian evening at around 15 degrees Celcius, this comes out as 1.225kg/m3), A= the cross-sectional area of Ezio’s falling body (call it 0.85m², presuming he’s around the same size as me) and Cd= his body’s drag coefficient (a number evaluating how well the air flows around his body and clothing, for which I shall pick 1 at complete random). Plugging these numbers into the equation gives a terminal velocity of 36.30m/s, which is an annoying number; because it’s larger than our previous velocity value, calculated without air resistance, of 31.04m/s, this means that Ezio definitely won’t have reached terminal velocity by the time he reaches the bottom of the Campanile, so we’re going to have to look elsewhere for our numbers. Interestingly, the terminal velocity for a falling skydiver, without parachute, is apparently around 54m/s, suggesting that I’ve got numbers that are in roughly the correct ballpark but that could do with some improvement (this is probably thanks to my chosen Cd value; 1 is a very high value, selected to give Ezio the best possible chance of survival, but ho hum)

Here, I could attempt to derive an equation for how velocity varies with distance travelled, but such things are complicated, time consuming and do not translate well into being typed out. Instead, I am going to take on blind faith a statement attached to my ‘falling skydiver’ number quoted above; that it takes about 3 seconds to achieve half the skydiver’s terminal velocity. We said that Ezio’s fall from the Campanile would take him at least three seconds (just trust me on that one), and in fact it would probably be closer to four, but no matter; let’s just presume he has jumped off some unidentified building such that it takes him precisely three seconds to hit the ground, at which point his velocity will be taken as 27m/s.

Except he won’t hit the ground; assuming he hits his target anyway. The Assassin’s Creed universe is literally littered with indiscriminate piles/carts of hay and flower petals that have been conveniently left around for no obvious reason, and when performing a leap of faith our protagonist’s always aim for them (the AC wiki tells me that these were in fact programmed into the memories that the games consist of in order to aid navigation, but this doesn’t matter). Let us presume that the hay is 1m deep where Ezio lands, and that the whole hay-and-cart structure is entirely successful in its task, in that it manages to reduce Ezio’s velocity from 27m/s to nought across this 1m distance, without any energy being lost through the hard floor (highly unlikely, but let’s be generous). At 27m/s, the 70kg Ezio has a momentum of 1890kgm/s, all of which must be dissipated through the hay across this 1m distance. This means an impulse of 1890Ns, and thus a force, will act upon him; Impulse=Force x ΔTime. This force will cause him to decelerate. If this deceleration is uniform (it wouldn’t be in real life, but modelling this is tricky business and it will do as an approximation), then his average velocity during his ‘slowing’ period will come to be 13.5m/s, and that this deceleration will take 0.074s. Given that we now know the impulse acting on Ezio and the time for which it acts, we can now work out the force upon him; 1890 / 0.074 = 1890 x 13.5 = 26460N. This corresponds to 364.5m/s² deceleration, or around 37g’s to put it in G-force terms. Given that 5g’s has been known to break bones in stunt aircraft, I think it’s safe to say that quite a lot more hay, Ezio’s not getting up any time soon. So remember; next time you’re thinking of jumping off a tall building, I would recommend a parachute over a haystack.

N.B.: The resulting deceleration calculated in the last bit seems a bit massive, suggesting I may have gone wrong somewhere, so if anyone has any better ideas of numbers/equations then feel free to leave them below. I feel here is also an appropriate place to mention a story I once heard concerning an air hostess whose plane blew up. She was thrown free, landed in a tree on the way down… and survived.

EDIT: Since writing this post, this has come into existence, more accurately calculating the drag and final velocity acting on the falling Assassin. They’re more advanced than me, but their conclusion is the same; I like being proved right :).

“If I die before I wake…”

…which I might well do when this post hits the internet, then I hope somebody will at least look down upon my soul & life’s work favourably. Today, I am going to be dealing with the internet’s least favourite topic, an idea whose adherence will get you first derided and later inundated with offers to go and be slaughtered in one’s bed, a subject that should be taboo for any blogger looking to not infuriate everybody; that of religion.

I am not a religious person; despite a nominally Anglican upbringing my formative years found most of my Sundays occupied on the rugby pitch, whilst a deep interest in science tended to form the foundations of my world beliefs- I think (sometimes) to some personal detriment. This is a pattern I see regularly among those people I find as company (which may or may not say something about my choice of friends)- predominantly atheists with little or no religious upbringing who tend to steer clear of religion and its various associated features wherever possible. However, where I find I differ from them tends to be when the subject is broached when in the present of a devoutly Christian friend of mine; whilst I tend to leave his beliefs to himself and try not to spark an argument, many others I know see a demonstration of his beliefs as a cue to start on a campaign of ‘ha ha isn’t your world philosophy stupid’, and so on.  I tend to find these attacks more baffling and a little saddening than anything else, so I thought that I might take this opportunity to take my usual approach and try to analyse the issue

First up is a fact that most people are aware of even if it hasn’t quite made the jump into an articulate thought yet; that every religion is in fact two separate parts. The first of these can be dubbed the ‘faith’ aspect; the stories, the gods, the code of morals & general life guidelines and such, all of the bits that form the core of a system of beliefs and are, to a theist, the ‘godly’ part of their religion. The second can be labelled the ‘church’ aspect; this is the more man-made, even artificial, aspect of the religious system, and covers the system of priesthood (or equivalent) for each religion, their holy buildings, the religious leaders and even people’s personal interpretation of the ‘faith’ aspect. Holy books, such as the Bible or Torah, fall somewhere in between (Muslims believe, for example, that the Qur’an is literally the word of Allah, translated through the prophet Muhammed) as do the various prayers and religious music. In Buddhism, these two aspects are known as the Dharma (teachings) and Sangha (community), and together with Buddha form the ‘three jewels’ of their religion. In some religions, such as Scientology (if that can technically be called a religion) the two aspects are so closely entwined so as to be hard to separate, but they are still distinct aspects that should be treated separately. The ‘faith’ aspect of religion is, in most respects, the really important one, for it is this that actually formulates the basis of a religion; without a belief system, a church is nothing more than a place where people go to shout their views at those who inexplicably turn up. A religion’s ‘church’ aspect is its organised divisions, and exists for no greater or lesser purpose than to spread, cherish, protect and correctly translate the word of God, or other parts of the ‘faith’ aspect generally. This distinction is vital when we consider how great a difference there can be between what somebody believes and what another does in the same name.

For example, consider the ultra-fundamentalist Taliban currently fighting their Jihad (the word does not, on an unrelated note, technically translate as ‘holy war’ and the two should not be thought of a synonymous) in Afghanistan against the USA and other western powers. Their personal interpretation of the Qur’an and the teachings of Islam (their ‘church’ aspect) has lead them to believe that women do not deserve equal rights to men, that the western powers are ‘infidels’ who should be purged from the world, and that they must use force and military intervention against them to defend Islam from said infidels- hence why they are currently fighting a massive war that is getting huge amounts of innocent civilians killed and destroying their faith’s credibility. By contrast, there are nearly 2 million Muslims currently living in the UK, the vast majority of whom do not interpret their religion in the same way and are not currently blowing up many buildings- and yet they still identify as Islamic and believe in, broadly speaking, the same faith. To pick a perhaps more ‘real world’ example, I’m sure that the majority of Britain’s Catholic population steadfastly disagree with the paedophilia practiced by some of their Church’s priests, and that a certain proportion also disagree with the Pope’s views on the rights of homosexuals; and yet, they are still just as Christian as their priests, are devout believers in the teachings of God & Jesus and try to follow them as best as they can.

This I feel, is the nub of the matter; that one can be simultaneously a practising Christian, Muslim, Jew or whatever else and still be a normal human being. Just because your vicar holds one view, doesn’t mean you hold the same, and just because some people choose to base their entire life around their faith does not mean that a person must be defined by their belief system. And, returning to the subject of the ridicule many practising theists suffer, just because the ‘church’ aspect of a religion does something silly, doesn’t mean all practitioners of it deserve to be tarred with the same brush- or that their view on the world should even matter to you as you enjoy life in your own way (unless of course their belief actively impedes you in some way).

I feel like I haven’t really got my point across properly, so I’ll leave you with a few links that I think illustrate quite well what I’m trying to get at. I only hope that it will help others find a little more tolerance towards those who have found a religious path.

And sorry for this post being rather… weird

The Scrum Problem

My apologies from deviating back to a personal favourite- I try too keep rugby out of these posts on the grounds that, in real life, it tends to make things kind of exclusive for people who aren’t into it, but I thought that I might be allowed one small deviation from this guideline. Today, I wish to talk about probably the single most contentious issue in the game today, one that divides, confuses and angers just about everyone involved in it: the scrum.

The scrum has always been a historic feature of the game of rugby- perhaps a historic callback to the old ‘scrums’ of viciously fighting players that formed the origins of the game of football, in the context of rugby it has proved contentious since the very first international ever played. England and Scotland were playing one another and, at the time, both played under different rules, so it was agreed that they would play under English rules for the first half and Scottish ones in the second. The game was around an hour old, tied at 0-0 (yeah it was a bit rubbish in those days), when the Scots won a scrum on the English five metre line. Rather than feed the ball into the scrum, the Scots instead began to push. The unsuspecting English forwards were caught off guard and forced back over their own line, whereupon the Scottish scrum-half grounded the ball. Whilst totally illegal under English rules, and thus generating a barrage of complaints, the Scots had one fair and square, starting off a bitter rivalry against ‘the Auld Enemy’ that continues to this day.

The scrum has developed a lot since those days (everyone now plays under the same rules for one thing), but perhaps the most important development for the modern game came in the 1990’s, specifically within the New Zealand team at the time. The All Blacks were a talented side, but their major disadvantage came up front, for whilst their front row players were skilled, Sean Fitzpatrick and company were not the biggest or heaviest front row around. Whilst not a disadvantage in open play, at scrum time it was feared that they would crumble under their opponent’s superior weight, so they had to find a way round that. In the end, they resorted to a bit of trickery. The structure adopted at scrum time by most sides of the age was to come together gently, get settled, then let the scrum half put the ball in and start to push, twist, and cheat in all the million ways discovered by front rowers over the years. However, what the Kiwis decided to do was hit the engagement hard, smashing their opponents back to get a good body position early. Then, the scrum half would feed the ball in almost immediately, allowing them to start pushing straight away and keep their opponents on the back foot, thus not allowing them time to get themselves settled and start to push back. It worked like a charm, aside from one small drawback. Everyone else started to copy them.

Even with trained wrestlers, there is only so much damage that sixteen men can do to one another when simply trying to push one another back. However, when not much below a tonne of meat slams as hard as it can into another tonne smashing back the other way, the forces involved in the impact is truly huge, and suddenly the human spine doesn’t seem all that strong. Not only that, but the slightest misalignment of the impact, and that amount of force means there is simply no way for it to all settle down nicely. Combine this fact with the immense muscle building and weight gain programs now demanded by the modern, professional game, and the attention to detail of modern coaches to get that extra edge in the impact, and we reach the inescapable and chaotic conclusion that is the modern scrum. In the last world cup in 2011, in matches between top-tier countries 50 scrums out of every 100 collapsed, and there were 31 resets and 41 free-kicks or penalties per 100. The stats were virtually the same during this year’s Six Nations, in which nearly half of all scrums resulted in the ball not coming back and creating one match (Ireland v Scotland) that spent over a quarter of its playing time spent scrummaging, resetting or collapsing.

This is despite the fact that the face of the game has changed very much against the set piece in the modern era. In the early 1970’s, analysis suggests that the average number of set-pieces (scrums and lineouts) in a match was nearly triple its current value (mid-thirties), whilst the number of rucks/mauls has gone up sixfold since then. Even since the game first turned pro in the mid-nineties, the number of set pieces has dropped by a third and the number of successful breakdowns tripled. The amount of time the ball spends in play has also risen hugely, and some are even arguing that the scrum as we know it is under threat. Indeed, in last year’s Six Nations the scrum was only the deciding factor in one game (England v Ireland), and as Paul Wallace astutely pointed out at the time that Ireland getting pushed about for the entire match was their reward for playing by the rules and not sending a front rower off ‘injured’.

Then there are the myriad of various intrigues and techniques that have lead to the scrum becoming the unstable affair it is today. Many argue that modern skintight shirts don’t allow players to grip properly, forcing them to either slip or grab hold of easier and possibly illegal positions that make the scrum decidedly wobbly. Others blame foot positioning, arguing that the modern way of setting up one’s feet, where the hooker demands the majority of space, forces the backs of his props to angle inwards and making the whole business more dangerous and less stable. Some blame poor refereeing for letting scrummagers get away with things that are now becoming dangerous, destabilising habits among front rowers, whilst others may counter this by considering the myriad of confusing signals a referee has to try and keep track off at scrum time- two offside lines, straightness of feed, hooker’s feet up early, incorrect back row binding, illegal front row binding, whether his line judge is signalling him and whether anyone’s just broken their neck. This is clearly a mighty confusing situation, and one I’d love to be able to start suggesting solutions for- but I think I’ll leave that until Saturday…