Enjoying the Tour de France

The Tour de France came to London this year and so it seems appropriate to explore the world of cycling. I’m not a great cyclist: I own an aging Raleigh Traveller which I’ve had since I went up to university and I complain vociferously that my shoulders hurt if I have to ride more than about five miles on it. I’ve also never really been a great watcher of cycling – for most of my life I’ve viewed it as the most boring of sports: surely the trick is to pedal really hard and at the end the fittest person wins? Great for their health, I’m sure, but hardly box office entertainment. This has all changed over the last couple of years, however. I blame the 2012 Olympics: track cycling is so quick that you don’t have time to get bored (and you can see dozens of British gold medals), and I’ve now moved on to watching the Tour de France highlights in the evening and I’m finding the unfolding of a long stage race fascinating. My mind has thus been changed: cycling is amazing (to watch – my bike is still unreasonably heavy and my shoulders too sore to engage with the sport myself.)

The discovery that changed my approach to cycling was that my simplistic reduction of the sport to a fitness competition was inaccurate: in fact the impact of fitness, whilst still important, is often second to teamwork and tactics. On the face of it this seems unlikely: your team-mates can’t cycle for you and no combination of tactics can reduce the height of a hill or shorten a race but what this analysis neglects, and what I had overlooked is Science! (It never pays to overlook science, still less “Science!”).

In order to understand the science that makes cycling interesting one has to understand something of the forces involved in cycling. Simplifying wildly, in order for a cyclist – for example Laura Trott (there really should be a women’s Tour de France) - to make progress up a hill (and cycling always seems to be up hills) she must apply a force (through the pedals, chain, gears and wheels) greater than the forces holding her back. The forces holding her back are gravity pulling downwards and air resistance (there is also rolling resistance from the wheels but this is small enough to ignore). There is also a factor due to any wind but this complicates things without offering any great insight and we shall assume that Laura has chosen a completely calm day for her ride.

 figure 1

Figure 1 is a picture of the cycle with the main forces marked on (including the “reaction” force that pushes up from the ground figure 2and without which the bicycle would just sink into the road). This picture is rather complicated but the important information can be put into a force diagram in which the cyclist is modelled as a point mass (we assume Laura Trott and her bicycle take up no space whatsoever) and in which only the forces are drawn – see Figure 2.

In order to get an estimate for the pedal force required, we need to have an idea of how large the forces acting against the cyclist are. It is reasonably simple to estimate the weight: Wikipedia says that Laura Trott weighs 52kg and that a racing bike weighs 6.8kg. We combine those two and round to a convenient degree of accuracy to estimate that the total weight of cyclist and bike is about 600N (the mass in kilograms multiplied by the acceleration due to gravity which is 9.81ms-2).

Air resistance is created whenever any object moves through a gas (in our case this is air but the physics works the same if you race in an ammonia-based atmosphere1). Air pressure on an object is produced because molecules in the air bounce off the object, pushing it in that direction. If the object is stationary this pressure is equal in all directions but if it is moving it hits more air on the front side than the rear and also hits the particles on the front side harder (because the combined speed is higher on average) than those on the rear side. This means that the air pressure on the front side is larger than that on the rear and there is an overall push backwards. See Figure 3.

figure 3

The faster the cyclist goes the higher the air resistance because they bump into more molecules and because the speed at which they hit them is higher2. In fact, the air resistance is proportional to the square of the speed (so if the speed doubles, the air resistance is multiplied by four). At 30mph (a reasonable approximation to the top speed of a distance cyclist on the flat) the force due to air resistance is about 24N. This means that if P<24 the cyclist will slow down from 30mph but if P>24 the cyclist will speed up. It turns out that the best way of measuring the maximum performance for a cyclist is not the largest force they can generate but the maximum power (which measures how quickly they can turn chemical energy in their bodies into kinetic energy). Power is equal to force multiplied by velocity and so our cyclist requires about 320W to maintain a speed of 30mph on a flat surface3. We will assume that this is the maximum power she can generate. Maximum power output is the key to time-trialling where each cyclist rides a course alone without interacting with anyone else (I stand by my original proposition – this is boring no matter how good Bradley Wiggins is at doing it). Maximum power output depends on the amount of muscle you have (which is proportional to your mass – m) and how good you are at using it (which I shall denote by a fitness coefficient F). Max Power = Fm. For reference F=6 for elite cyclists, 3 for amateur cyclists and probably less than that for me4. In general, then, a large cyclist (such as Chris Hoy) has a larger maximum power than a small one (such as Laura Trott).

Air resistance is crucial to what makes cycling interesting because air resistance is enormously reduced for someone who figure 4follows just behind the front cyclist. Because the cyclist in front has pushed through the air, the following cyclist is riding through a pocket of lower air pressure and therefore meets less resistance (see figure 4). This is called “drafting” and is used by geese flying in a V formation as well as cyclists. The effect of drafting on air-resistance depends on a number of factors (primarily the distance between the two cyclists) but it is significant – reducing the resistance by about 30%5.

The effect of drafting on air-resistance depends on a number of factors (primarily the distance between the two cyclists) but it is significant – reducing the resistance by about 30%5.

The impact of this effect on racing can be seen most clearly in an individual sprint race where two cyclists race against each other in a velodrome. Neither cyclist wants to be the one in the lead as the other will stay just behind and use 30% less effort until the last few seconds when they will be able to use the energy they have saved to surge past for the win. The beginning of the race is therefore cycled very slowly (so slowly that cyclists often stop and there is a specific rule in place to stop you cycling backwards) until one of them thinks they can get a good enough start to negate the effect of drafting and sets off at top speed. Judging the right moment to make a break is crucial as unless you can actually get away from the opponent they are likely to win by drafting: tactics and good judgement are at least as important as small differences in fitness.

In a team sprint things are different: two teams of three riders compete against each other but they start on opposite sides of the velodrome and so don’t have any impact on each other. What happens instead is that within each team one rider will lead for the first lap and then drop off whilst the second rider leads for the second lap. The last lap is raced by a single rider on his own. The total ride is much faster than a single rider would be able to manage over three laps because the third rider has been able to take advantage of drafting for most of the race – thus saving energy. The individual and team sprint are Olympic events that take place over short periods of time but the same effect can be observed in the flat stages of the Tour de France. Teams work together to allow their best sprinter to take advantage of drafting until the last few hundred metres. The final sprint comes down to tenths of seconds after several hours of racing and in order to win the sprinter needs to have as many team-mates as possible still with him when the race reaches its last kilometres during which they operate exactly as in a team sprint in the velodrome.

Another interesting aspect of the Tour de France that can be understood as a consequence of drafting is the breakaway. This happens on every stage where a group of riders decides that they want to go faster than the peloton (the main group of cyclists) and gets a lead of several minutes. The breakaway tends not to have any of the “important” riders in it because the race leaders keep an eye on each other and race after their competitors to make sure they don’t get away but they don’t worry about riders they don’t consider to be a threat. A breakaway can open up a gap of several minutes lead on the peloton but they nearly always get caught by the end of the race because the peloton can go faster (considerably faster – commentators compare it to a killer whale inexorably catching up with a smaller fish which is then swallowed up. This metaphor adds a sinister tone to this part of a race. I recommend turning the commentary off and instead playing the theme tune from "Jaws”). There are two reasons for the higher speed of the peloton, both to do with drafting. The first reason is that in the peloton there are many more riders to take turns at the front of the group and so each rider spends less energy on average keeping the speed high; the second reason is that in a peloton the riders are even more sheltered from air resistance than if they are just riding behind a single rider and the benefit is closer to 40% than 30% which saves even more energy. One successful breakaway was in the men’s 2012 Olympic road race when Mark Cavendish was expected to star in a sprint finish. What happened, though, was that there weren’t enough racers in the peloton who were willing to take a turn at the front and it was left to the British team to keep the pace high: other teams either had racers in the breakaway group (and therefore no real interest in chasing it down) or were hoping that they would be able to take advantage of the British riders’ hard work and then sprint past at the end when their legs were tired. Unfortunately, for both these riders and for British hopes, this meant that the peloton didn’t take full advantage of its size and the sprint finish was between the riders in the breakaway. Usually, however, there are enough teams whose interest lies in the peloton that the breakaway gets caught before the end of the race.

If all stages in the Tour were flat then they would all end in sprint finishes, the yellow jersey (awarded to the leading rider text 1overall) would be decided by tenths of a second and riders would be built like Chris Hoy (because the bigger you are the larger your maximum power output, and hence maximum speed, can be – for a fuller analysis see the box on the right). Actually many of the stages contain or finish on a steep hill which tends to spread riders out and so by the end of the three weeks the top riders are separated by a few minutes. These top riders also tend to be small and light and we can again use Science to understand why.

On the flat, you will recall, the maximum speed is reached when P (pedal force) = A (air resistance). A large rider will be slightly less aerodynamic (which will increase A) but will have a much larger power output (and so a larger P). Mass also affects acceleration (large masses get up to speed more slowly) but I’m not going to examine that effect in this essay.

On a hill, however, the weight of the rider counts against them – instead of pulling the bike straight into the ground it also pulls a little backwards. The amount that pulls backwards depends on the angle of the hill (the larger the angle/ steeper the hill the more backwards force there is) and this backwards force is equal to Wsin(x) and so the maximum speed is reached when P=A+Wsin(x). We can see the impact of this by supplying some numbers. If we go back to Laura Trott with a weight of 600N and a maximum power output of 320W and suppose she goes up a hill with an angle of 15°6 we can work out what her maximum speed is. The first thing to note is that 600sin(15) is 155.3N and so the pedal force has to be at least this in order to get up the hill. With a power output of 320W and a pedal force of at least 155.3N, the speed can’t be any larger than 2.06ms-1 (which is 4.6 miles/hour) and at this speed the air resistance is tiny (0.56N) so we can ignore it and calculate that Laura’s top speed going up this hill would be just over 2ms-1 (this really is quite a steep hill6).

This calculation explains why riders are more likely to make a break on a hill than on the flat: on a hill the chasing rider has verytext 2 little advantage because the hard work is done overcoming gravity rather than air resistance; this means that a rider who is able to generate a little more pedal power for their mass will be able to get away from their competitors and gain a significant amount of time. Heavy riders don’t have any advantage over light ones going uphill because their weight pulls them back just as much as their extra power pulls them up (for an analysis see the box to the right).

One last phenomenon of the Tour de France that can be explained by an understanding of Physics, and particularly of drafting is the formation of “echelons”. This happens when there are strong cross-winds (blowing at 90o to the road) and describes how the riders get strung out from one side of the road to the other. The reason for this is that if there is a cross-wind, in order to get the maximum benefit from drafting you don’t want to be directly behind the rider in front but behind and away from the wind. An echelon is a diagonal row of cyclists running from one side of the road to the other and it has a very interesting effect on the race: on a normal day (with no cross-wind) the peloton can contain as many riders as want to join it but on a windy day the echelon is full once it runs from one side of the road to the other. If twelve riders fit in an echelon then the thirteenth rider can’t take advantage of drafting properly and won’t be able to keep up: instead she will have to form a new echelon with the riders behind. Large gaps can form between echelons and a rider’s ability to win the stage or to gain time in the whole race depends enormously on getting into the right one (see Figure 5).

figure 5

Science, with or without an exclamation mark, has therefore led to a combination of circumstances which make bicycle racing much more than a fitness competition and understanding the science the cyclists are battling against can help to explain what is going on in a race and to increase the entertainment of watching it. Unfortunately no study of Science can explain why Mark Cavendish and Chris Froome chose to come off their bikes this year and I fear that the British dominance of the Tour has come at least to a hiatus, if not an end. Nor can study increase my F value so maybe I should stop blaming my shoulders, jump on my Raleigh Traveller and take advantage of the sunshine.

 figure 6

1The Biology, however, is completely different. In an ammonia-based atmosphere as the cyclists, unable to breathe, would never reach the end of the race. This shows the limitations of Physics.
2This is a gross simplification of a very difficult problem. Air resistance is more complicated than I’ve described and the equations of fluid dynamics verge on the intractable (simply proving that they are always soluble and that the solutions are reasonably well-behaved will earn you $1,000,000 – see Navier-Stokes)
3For comparison, it uses about 30W for a 70kg adult to travel at normal walking pace (5km/h)
4These figures are for the maximum power that can be produced for an extended period of time. More power can be generated in a sprint situation.
5 Interestingly, drafting also decreases the resistance for the cyclist at the front by a small amount because having someone filling the space behind you increases the number of air particles bumping into your back – pushing you forwards. The importance of keeping the distance from the lead rider small is why competitive cyclists ride so close to each other. You might think that this increased the risk of collisions and accidents. It does.
615° can be visualised by stretching your arm out in front of you with your hand stretched open. If you twist your wrist so that your thumb is on the horizon then the tip of your little finger is about 15o above it.
7This analysis does not explain why lighter riders seem to have an advantage going up hills – to understand this one needs to look closer at how maximum power and mass are related because speed up hills is (almost) entirely dependent on this ratio. I’ve assumed that this ratio, F, is independent of the size of the rider but the physiology is more complicated. It’s actually much harder for a heavy cyclist to have a large F than for a light one (the details of this are FBSI and have to do with lung capacity being an area whilst mass depends on volume). Biology wins again.