Going for Gold: the Mathematics of Sporting Glory - Amandine Aftalion

What this episode is about

The discussion highlights the intersection of mathematics and sports, focusing on how analytical modeling can enhance performance in various athletic disciplines, particularly running and downhill skiing. Key points include insights on optimal running techniques, comparisons of speed across sports, and the surprising dominance of downhill skiers in terms of velocity, underscoring the role of physics in athletic achievement.

Key takeaways

• Curiosity in science can turn everyday wonders into groundbreaking insights, especially in sports and physics.
• Fastest sports? Badminton flies under the radar while downhill skiers redefine speed—250 km/h isn’t just for cars!
• Unexpectedly, wheelchair athletes outpace marathoners due to fewer bends, challenging assumptions about running speed.
• Centrifugal force isn’t just physics jargon—it's a sprinter's secret weapon against gravity on tight tracks.
• Optimizing running performance is just like economic decision-making—choices and constraints dictate champions!

Who speaks in this episode?

• Amandine Aftalion (Host) · 42 min speaking

What is discussed, and when

1. 0:00 – 5:00: Julian Bend
2. 5:00 – 10:00: Paris Olympics
3. 10:00 – 15:00: Paris Olympics, Hudson Smith, Nicole Polytechnique
4. 15:00 – 20:00: Hudson Smith, Tokyo Olympic
5. 20:00 – 25:00: Inga Brixton, Paris Olympics
6. 35:00 – 40:00: London 2012
7. 40:00 – 44:31: Paris Olympics, Leo Maron

Notable quotes from "Going for Gold: the Mathematics of Sporting Glory - Amandine Aftalion"

• Well, the answer is related to the French title of my books which is why Julian bend because marathon or in cities so there are not so many bends and wheelchairs are.

  about Julian Bend · at 4:08

• This we are able mathematically to compute a kind of digital twin of the athlete and understand how champions for the moment or students in Nicole Poly techchnique run. And the point of this digital twin of course is once we have data.

  about Nicole Polytechnique · at 12:37

• World championship. The blue is femal just to show you that they accelerate up to 60 70 m and then they go down. Hudson Smith he goes up to nearly a little bit than 11 m/s and…

  about Hudson Smith · at 16:09

• Bigger whatever. How does it change performance? And what we understood is that Inga Brixton is someone who had a very good V2 kinetics. That is his the rate of oxygen that he transforms into energy starts very quickly. because of that.

  about Inga Brixton · at 21:23

• Is earning 0.1 or 0.2 second because of the track then you understand things better. So this is what happened is that Kohl Hawker was in the inside but in the inside the…

  about Paris Olympics · at 26:06

• Someone then it takes the place of the vortices and it produces the drag. So what happened for instance in London in 2012 at the Olympics is that they heated the vel drrome in order to reduce the drag because at 28°.

  about London 2012 · at 35:41

• In order to win time on this part of the case which is the best one and this is where French swimmer Leo Maron is very good. So before I conclude I should say something else…

  about Leo Maron · at 41:26

Who and what is mentioned in this episode?

People

• Hudson Smith — Person · Referenced athlete (case study in sprint modeling) · 3 mentions · first at 13:33
  Used as an example 400m sprinter to illustrate acceleration/deceleration profiles and drafting/psychology effects in racing.
• Inga Brixton — Person · Referenced athlete (case study in middle-distance digital-twin analysis) · 2 mentions · first at 21:06
  Central example for the speaker’s 1500m ‘digital twin’ modeling, especially oxygen-uptake kinetics and race tactics (staying behind vs leading).
• Leo Maron — Person · Referenced athlete (swimming example) · 1 mention · first at 41:32
  Cited as a French swimmer who excels at underwater phases, supporting the physics explanation of why slightly underwater swimming can be faster.

Books & papers

• Julian Bend — Book · Referenced work · 1 mention · first at 4:11
  Mentioned in the context of understanding sports performance and the physics behind it.

Places & events

• Paris Olympics — Event · Key event discussed · 4 mentions · first at 6:54
  Discussed in the context of sports performance and track design affecting athlete performance.
• Tokyo Olympic — Event · Referenced event · 1 mention · first at 16:40
  Mentioned in comparison to the Paris Olympics regarding athlete performance.
  And a friend of mine sent me an information about the Tokyo Olympic, the Tokyo World Championships that ended up last week where the 400 meter champion did didn't manage to.
• London 2012 — Event · Referenced event · 1 mention · first at 35:45
  Mentioned in the context of cycling performance and environmental conditions affecting it.
• Nicole Polytechnique — Place · Referenced location · 1 mention · first at 12:47
  Mentioned as a place where students and athletes are studied in the context of sports performance.

Full transcript with timestamps

0:17 I'm very happy to be here. Actually, I came here in Oxford in 1996 as a very young PhD student. And I think I discovered here how important it is to be curious about everything around you. And how can science bring you information and um explanations about things that you watch that you wonder about and this this book was um after working for about 10 years on sports and modeling of sports. So the first chapter is about um understanding running and how you should go running, how you can run with math and understand what is the best way to run that I will describe. And this book is full of questions that you can wonder about. Some of them you may know the answers, some of them you may not. And the idea of this talk is to present a few questions and a few answers.

1:20 So, uh, the first question is for you. Maybe I won't take the answers because you there are a lot of people here. What is the fastest part? I don't know if you have any ideas. Um, so the answer is is badminton. So, I'm sorry for the distances. I'm still uh thinking in kilometers per hour. But what you have to remember is that golf balls and gi are very quick as well. Sometimes people think that table tennis is very quick. It's not that quick. As you see, it's down down below. Um though the ball is very light, we'll see that it's smooth and it's very different from uh in terms of air resistance. So the other issue is who are the fastest athletes.

2:12 So they are not people with cars. They are not people coming from the sky. They are people you don't find so many in England because uh you don't have these steep slopes but they are downhill skiers. They can go up to 250 kilometers per hour. If you want to jump from sky to reach this velocity you would need to jump from far away in space. Uh because when you jump um you have this spread eagle position and because of the spread eagle position the resistance of the air is very important very big and therefore compared to the downhill skiers who has a top position they go more slowly.

3:00 Another question which is related to things that I will say later is which sport requires the fastest reaction speed. Very good. This is the first time someone gives the an correct answer. So, it's either tennis or baseball because balls are very quick and and you have about 30 to 40 milliseconds to decide whether you go on the left or on the right. Um now the fastest wheelchair athletes outstrip the ablebodied but there's something very strange is that for the 800500 5,000 and marathon they are run more or less at the same average speed but the marathon itself they go more quickly in average speed than the 1500.

3:59 Do you have an idea why they can be quicker on a marathon which is called for able-bodied is not true as at all? Well, the answer is related to the French title of my books which is why Julian bend because um marathon or in cities so there are not so many bends and wheelchairs are very slow when you have to turn. So what about bends? Actually, if you go back to ancient Greece, in ancient Greece, there were no bands. They were running back and forth at that time. So, very bad for performance. Actually, the name stadium, which gives the French stud uh is both the name of the track and the name of the shortest race, which was around 200 m.

4:52 uh 200 m was the size of uh of a foot of a god's foot at that time and so the shortest race you were just doing one way and the others were going back and forth so very bad for performance but at that time that was not the problem because in order to honor gods you had to be the first and so there were no way to measure time no way to measure distance and they didn't really care about uh stopping and having to go backwards. So, it's with the Romans that appeared the circus with the horses and you know from movies that bends were really a problem because they were tight and and this was a difficulty and the issue is when you have bends and you go very quickly then the quicker the more inclined you have to be. And the issue is what about runners? Of course, runners don't go as quickly as speed skaters, but nevertheless, they bend about 10 to 15 degrees uh for sprinters and they lean to the left to go against

6:07 what is called centrifugal force. So, I would like to say a few words about what is centrifugal force. If you go in a train that goes very quickly, you will realize that when the train is turning, you have to bend because you want to stay and not fall down. And this is where the centrifugal force comes into play. That is, it will take you uh away because your body has some inertia. Your body would like to stay where it would have stayed if the train hadn't turned. So it's staying on the outside of the band. And the point that I wish was to insist on and I'm giving you the only formula of the talk because you will see it's a a practical information for stud def France and the Paris Olympics is that the centrifugal force is proportional to the velocity to the square and inversely proportional to the radius of the track. So it's really important to remember that if the radius changes then the centrifugal force

7:12 changes. So the smaller the radius the bigger the centrifugal force. And for a sprinter like Houston bolts um if you do a small calculation actually you realize that the centrifugal force is about a third of the weight. The weight is the force that is holding you on the earth. So a third of the weight is something really to take into account when you want to analyze performance. So the issue that I've been uh thinking about in my research is the following is when you go running you fix a distance 100 200 400 800500 10,000. Actually we didn't go up to the marathon we stopped at 10,000 mters.

8:00 And how is it that you can optimize your effort to reach the best time to be the champion to win the race. So of course like everything in life is a kind of economic decision that you have to make. It's a matter of choices and constraints and decide what is best for you. And in order to do that, we've introduced a problem uh made up of equations that rely on three physics laws. The first one is related to Newton's second law of motion. So it's mechanics saying that the variation of your velocity depends on the force that applies on you.

8:45 The second one is energy conservation. That is you breathe you get oxygen you eat and from nutrition you get energy as well what is called anorobic energy the one that doesn't come from oxygen so oxygen is very nice because the tank is infinite the nutrition the tank is finite but both of them they uh contribute to uh your propulsive force and the power uh developed by the propulsive force has two origin that and and hopefully nothing is is lost too much in the in this uh process though. So you have energy and the third equation comes from the fact that neither of you in this audience is a robot. You all have a brain and you run and do sports with your brain. And this is very important that is when you go running when you go doing gymnastics, tennis, baseball, whatever they journalists said go with the mental with the motivation.

9:57 Actually there is something that is going on is that for muscle con for muscle contraction you have to decide your brain has to give an information to the muscle to make an effort. If I tell you to get up to stand up and sit down and stand up and sit down 10 times, uh it may not be that energetic in terms of energy consumption, but you may find this very boring in terms of motivation. And the reason is uh your brain is going to give a lot of information to your muscles, but you don't find this very uh satisfactory.

10:41 So this is where the third equation come into play. What is called motor control. That is something that we developed working with neurossychologists in in Paris working with patients with what is called dopamine deficiency. And for this they understood how to model a propulsive force with respect to effort. And effort is related to the fact that you don't want to vary your force too much. And as soon as you vary force, then this is tiring. And this is something that you can relate to motivation. And motivation can play in three directions in sports. The first one, if you're motivated, you're going to be able to produce an intense effort, very intense. If you're motivated, you're going to be able to react very quickly.

11:42 And if you're motivated, your brain can decide that hurting is not a problem. And that you go on, though it's very hard, very tough, very tiring. And of course this part is very important because when you think of how to do more than what you can do then you can think of medicine that act not only on oxygen consumption on nutrition etc. But they can act also on your brain so that you're not really aware that it hurts too much. So we put all this into the into equations and we don't measure anything on sports people. We have very uh precise data that is position every 100 millisecond. That is position of the athlete 10 times per second. And with this we are able mathematically to compute a kind of digital twin of the athlete and understand how champions for the moment or students in Nicole Poly techchnique uh run. And the the point of this digital twin of course is once we have data uh we're able to change the twin and say

13:03 okay I put him running on another track. I imagine that he improves his training and improves his V2, his running economy, his whatever, how is it going to change the performance? So, um, first question that we are able to answer pretty easily is how you run a sprint. So, for instance, 100 meters. I put you here the data of the Paris 2024, but it's the same for all sprints. is the same for Hudson Smith for a one 400 m. Athletes do not cross the finish line accelerating but decelerating. So I don't like to say slowing down because it gives the impression that they stop their effort. It's really decelerating.

13:50 And if you look at this curve here, you will notice that the maximum of velocity is really long before the finish line because it's around 60 70 m. All sprinters use inbuilt or whoever you want, they keep accelerating up to 70 mters and then they are not able to accelerate anymore. And this is true on the 400. On the 400 they accelerate up to 60 70 m and they do the rest that is 330 or 340 m decelerating. Why? Why is it the best strategy? Well, it's the best strategy because you have a brain and the best way to do an intense effort is like if you want to destroy the door of the going going out if ever there is a fire. Uh the best way to do it is to push very strongly, very intensely, very shortly. You're not going to start pushing more and more and more and more. We realized that the best intense effort is to do something um very strongly and this is the same for running. If you go running very

15:04 strongly, the best strategy and we managed to compute it mathematically is to start with the most intense force that you can you can use that you can produce. And of course this force you you cannot maintain it for 10 seconds. um what you could maintain is a smaller force but if you do a smaller force the fi final time would be much bigger. So the best thing is to produce the biggest force that you can and this biggest force uh because you have a brain and information coming from the brain is going to be intense and then the force is going down and this is why the sprinter uh only can only accelerate for 60 70 m and then uh the velocity is going down and starting slower and accelerating all along would produce a bigger finite time.

15:59 So here I've put simulations for the 400 just to show you. So the red is Hudson Smith that was in the 2022 Munich uh world championship. The blue is femal just to show you that they accelerate up to 60 70 m and then they go down. Hudson Smith he goes up to nearly a little bit than 11 m/s and he ends up at 7 m. So it's not a matter that they slow down really. It's just that they cannot maintain this extreme velocity that they produce at the beginning of a sprint. And a friend of mine sent me an information about the Tokyo Olympic, the Tokyo World Championships that ended up uh last week where um the the 400 meter champion um did uh didn't manage to break the record held by uh the German champion but it was said uh she she's going very fast because she was going much faster for the end of the race. So when you see this, you understand that it's not she's going faster, it's just that she's

17:12 decelerating less. And um people in the old days, they were starting very very strongly. And now the 400 for we men tends to be uh with a smaller deceleration than it used to be before. So another issue is why does running behind someone allows you to improve your performance? So this is something that people have measured for a 400 that is a lap. Uh if you go behind someone you're I mean not me I say you maybe you at least not me. Uh but let's say uh Hudson Smith or whoever uh running properly goes uh behind someone who gives you gives him the right spacing he can earn uh 1 second. So 1 second is a lot on a 400. Uh the the the reason is this is psychological that is um if you see someone be before you you don't have to think about what you're doing you're just following. So it's like you're attracted you're attached toward to the person beh in front of you and this reduces the effort that you have to

18:27 produce. It doesn't reduce the energy because it doesn't change the energy but it just reduces the mental activity that you have and therefore the effort that is all the transmission of information between the brain and the muscle is reduced and it allows you to uh keep uh energy effort for the end of the race and therefore uh get to a bigger time if you want To be convinced of that, you have to imagine if you go to a new city and you go from one point to another, the first time you look everywhere and when you are on the way back and you recognize streets or or monuments, then you will have the impression that the way back is a little bit quicker. It's just because your brain is working less.

19:23 And when your brain works less then your activity your effort can be bigger. So it's the same uh for runners is the same for all sports people and especially so when uh you are on uh eight on a track with eight lanes. We have managed to compute what are the best lanes. So of course the problem with the outer lanes is that you're running blind. The problem with the inner lanes is that the centrifugal force is very strong because as I showed you at the beginning the centrifugal force is inversely proportional to the radius. So the smaller the radius the bigger the centrifugal force and in the end the best lanes for standard tracks are lanes four five six seven and three are more or less equivalent. eight and two are more or less equivalent and lane one is terrible for performance. Um so there's no hope to see uh a winner in lane one because of that and also because the draw favors the center lane that is the

20:29 four best are put in the central lanes. So in the end the winner is always there. So um we have been working on the 1500 for quite some time um and analyzed uh especially Ingga Brixton and his the the the point of our um of our studies is that we manage with our digital twins to uh get back information about things that you cannot measure and especially V2 that is the rate of oxygen you transform into energy and we were able to analyze iz this because we had a lot of data for Inger Brixton. So I'll just show you the kind of curve that we're able to produce. This is the velocity for a 1500. And we managed to imagine okay imagine that he has a bigger V2 that he has a bigger force that he has a bigger whatever. How does it change performance? And what we understood is that Inga Brixton is someone who had a very good V2 kinetics. That is his uh the rate of oxygen that he transforms into energy

21:41 uh starts very quickly. because of that it means that he doesn't have so maybe I should explain this is the velocity versus the distance so at the beginning they always start the 1500 accelerating a lot a lot more than the mean velocity or the the the real pacing that they are going to follow why because this is the best way to start the oxygen system quickly so the point is Brixton has a very good kinetic so normally his best way to on is to stay behind. This is what this is how he wins because he stays behind which has two advantages.

22:20 The first one is that he's taking into advantage the fact that he's behind and the second one is that he has his kinetics starting quickly. So he doesn't need to uh spend anorobic energy to to do a a big quick a start a quick start sorry. And the point is then he you know uh on a 1500 they start accelerating on the last two laps. Uh the last two laps are a bit more quickly run a bit more quickly than the second one. And the point is what happens now if as in Paris 2024 he starts first. Well what happens is that Josh Kerr is coming from the outside. Cole Hawker is coming from the inside and normally if the strike if the track had been a standard one Cole Hawker should never have been able to win uh because of the centrifugal force and what happened is Cole Hawker won and he said I don't know how I managed to do it. Sebastian Co uh the former 1500 champion and president of U World Athletic said I don't know

23:33 how he managed that and nobody said it at that time but I give you the information. The information is that the Paris truck was not standard. So you're going to tell me okay the French they always do things differently from the others. Yeah that's not completely wrong. Uh the issue is the following is that on stat of France they often play rugby and they had this rugby pitch and in order to encompass the rugby pitch you cannot have a standard track. So what do I mean by a standard track? All tracks all athletics tracks are 400 meters long no problem. The issue is you can vary the length of the straight and the radius of the bend. So the rate the standard tracks the radius is 36.5 m because twice times 36.5 is 73. That's the width of a football uh track. So if you want to encompass football, you do a standard track 36.5 m for the for the bend and this gives you with the 400 calculation the length of

24:42 the straight. Now if you want to have rugby you have to do something else. In the old days they were building what is called double bend tracks which is very bad for performance which is the case of the Berlin track for instance. But in Paris they did something else which is the same as Monte Carlo is they decided to do a ben which is a little bit bigger. So instead of 36.5 they have 37.5 1 m not much but when it's in terms of centrifugal force where we divide by the radius and they are going at velocity which are of order six or 7 m/s then this is where you can earn a hundth of a second and uh so it means the bend is a little bit bigger and the straight is a little bit shorter so it's very different from performance I'm working on that at the moment is showing exactly uh how it made a difference and especially on 200 400500 it was impressive that um a lot of athletes have beaten their personal

25:48 best. So if you try to do statistics of course on eight people on a final then you're going to say okay maybe five or six have beaten their personal best and so what it's the Olympics etc. But if you go to the digital twin model and you realize that exact that the digital twin is earning 0.1 or 0.2 second because of the track then you understand things better. So this is what happened is that Kohl Hawker was in the inside but in the inside the centrifugal force was little bit smaller than on a standard track and what he could never have reached on uh a standard track and Josh coming from the outside should have won because uh it was slightly different he hawker managed to win.

26:37 Um so let's uh stop about running and move to other sports. So I've put a little bit of cycling why on a bike the faster you go the more stable you are and I put this quotation by Albert Einstein life is like a bike you have to move not to lose your balance so the issue is if you have kids or if you are a kid and you started biking it's very difficult at the beginning because you have the impression you cannot find your balance the handlebars are moving left and right And you have the impression that you can maintain the handlebars more or less straight as soon as you reach a high velocity. But of course when you're starting biking or when you're in Paris on a vib which is very heavy um it's very difficult to start with with a big velocity. So the explanation is again related to centrifugal force and related to equilibrium. So if you if you turn your handlebars, you realize that you increase the the area of equilibrium.

27:45 The idea is the following. Um when when you have uh the system which is the the bike and and the human being, it has a center of gravity which is more or less the center of mass. In order to be at equilibrium, the center of gravity has to be above the geometrical figure that you obtain by joining all the points in contact with the the ground. So if you turn the handlebars, this increases the size of this figure and you have more positions of equilibrium. If you have seen clowns at the circus, clowns used to have I say used to because in France, I don't know about Britain, they have forbidden animals in circus. So, there are fewer and fewer circuses. But, uh, at least clowns used to have very long shoes. And with these very long shoes, they could No, Daryl, I'm not doing it.

28:39 I'm not falling down. uh they could bend a lot because the center of gravity could go, you know, if my shoe goes until the the end here, then I could bend a lot because my center of gravity will stay above my points of contact on the ground. So, it's the same for the bike. When you turn the handle, the handlebars, you increase uh your equilibrium in some way. But on the other hand, uh if you fall down, then you're producing uh a centrifugal force. And as we said, it's proportional to the velocity and inversely proportional to the radius. So what you have to think about is that if you turn the handlebars a lot, then you're going to do a very small circle. Um it means you need a small velocity. And if you if you have a big velocity, this allows you to a big radius. And a big radius is close to the straight line. So this is why uh you have to to go to a large velocity in order to not to turn the handlebars too

29:44 much. So another question related to basketball. Why do basketball players seem to hang in the air? Michael Jordan is often said to fly. Um the reason is the following is again the center of gravity uh explanation is when a basketball player is jumping um his center of gravity is following a curve which is in red here which is called a parabola. The point is when I move my arms for instance if I raise my arms then my center of gravity is going to move up on my body. the same if I fold my legs my center of gravity is moving up on the bo my body. So if at the same time as the center of gravity is going up here the player move up his legs and move up his hands sorry and fold his legs then it turns out that the center of gravity is following the parabola but the head is staying at the same level and because what you're watching is the head because on TV there is no arrow or cross to indicate the center of gravity.

30:59 So you're just watching the head and impressive the head stays at the same height. So he's flying just a physics effect. And it's the same for ballet dancer when they do a gro they raise their arms at the top of the parabola which gives the impression of flying of staying with the head at the same level. So you can practice that uh now and give the impression of flying to your neighbors. Uh let's go into throwing. Um again when the weight is dominant that is there are no effects the trajectory of whatever you throw is a parabola and the longest range is reached for the middle angle that is 45°.

31:45 This is not the case for golf for instance where the optimal angle to throw very far is about 26° and the reason is uh they are effects due to the air. So for small or light objects air resistance what is called drag slows down movement. And the reason is that when the ball the the table tennis ball on the top is moving from the right to the left, it encounters the air and the air is going to make vortices or turbulence behind it. The smoother the ball, the bigger the turbulence. And this is why if you have a ball which has defects which have which has irregularities then it will move more quickly. The resistance of the air will be smaller because actually the little vortices that are forms around the ball they are going to produce like a skin of air around and the ball will be able to move more quickly. So this is what happens with a tennis ball that has air hairs and with a golf balls that has dimples.

33:01 And actually some people have investigated how the dimples should be uh constructed. What is the shape? What is the depth? What is the angle? It turns out that there's no math proof about that. that experimentally uh the dimples shouldn't be too deep rather shallow but the angle of transition should be large and no one knows exactly whether it's better to have between 250 or 500 dimples. So no if you take two brands they will have uh honom structure which is slightly different and depth etc. But the point is they have dimples and the drag instead of being proportional to the velocity to the square is only proportional to the velocity and therefore you can shoot a ball a golf ball much larger with a biggest range than ordinary I would say balls. So this is why if you remember at the beginning the golf ball was one of the quickest object that you can throw. Nevertheless, uh if you decide to throw it at a very

34:15 large angle, of course, it will go with a very different trajectory which is uh called tartaglea uh where you have it's going up but at some point you have the impression if you have seen champions engulf that the the ball is is falling almost vertically. That's because it's a situation where the weight and air resistance more or less compensate and this is where it falls down. I like speaking about France. Uh so what about tour to France? It's really well known that you know being behind is best when you go running. I've explained you it's not only a matter of air but of course in when you go cycling it's really much better to be behind. Not only because of psychological reason but also because of air because if you have someone in front of you then you're protected and it's less effort etc. What you may not know is that the first one is better to for him to have someone behind instead of biking alone. So if

35:25 you go biking uh if you decide to be the first check that there is someone behind you because it will be less effort and the reason is that you know if when there's air there are vortices behind you and if you have someone then it takes the place of the vortices and it produces the drag. So what happened for instance in London in 2012 uh at the Olympics is that they heated the vel drrome in order to reduce the drag because at 28° uh the drag is reduced and what happens on tour to France is that when uh someone leaves I don't know if you've noticed uh he's he's never going alone there's always a car behind with six to eight bikes on the top that is following at a very precise distance and the reason is that uh some Belgian mathematicians have done the computations. If you have a car with six to eight bugs behind this reduces the drag. So nothing in to France is left to to something that is not computed.

36:37 Uh football you know it bend it like Beckham. If you rotate a ball when you shoot it, then it will uh change its trajectory. That is if you want I'll show you to do this. Uh the only thing to do and this is where you have to be good at it is at the same time that you send the ball you have to rotate it because if the ball rotates then the on the top the the velocity of the ball will meet the air counter going on the other way whereas the bottom of the ball you see that the velocity is the same as the velocity of the air. So there will be a difference of velocity between the top and the bottom. And there's a law in physics which is called Bernui law which tells you that a difference of velocity creates a difference of pressure. And a difference of pressure means a force and a force perpendicular to the u velocity of of the ball makes a spiral trajectory and it does this. Of course, it never go

37:44 into the spiral at the end because uh there's something to stop the ball. But this is how you have to imagine the physics in in Okay. So my last uh my last question is about swimming uh because we have a very good swimmer in France. So I like uh ending up with this and especially he's very good at swimming underwater. But actually the question why do you swim faster slightly underwater came from my own experience as a moderate swimmer and in my swimming club there are a lot of kids learning to swim and they I I always hear the parents saying but take your head out of the water take your head out of the water and it always makes me laugh because their kids are not going to drown and if they swim slightly underwater with more or less the equivalent of a body above them.

38:39 It's because naturally they find the best way to swim because this is the best depth to swim is slightly, you know, slightly underwater. This is where the water resistance is the smallest. So if you see people swimming slightly underwater just think that they must have a good intuition of what is going on or they must have do some physics in it to understand what is going on. So the idea is the following. When you have your you you're swimming the water is slowing you down for again what is called drag and is made up of three components. The first one is what is called pressure. That is the the water is pushing more in front of you than behind you. And you want to be as horizontal as hydrodnamic as possible.

39:35 The second one, the second component is friction, skin friction. That's why um professional swimmers put swimming caps. They shave. At some point, at some time, they used to have swimming suits that mimicked uh shark skin because it flows better. And um these two components, they are smaller when you're underwater than when you're at the surface. And the last one is the wave drag, which is 30% smaller when you're underwater than on the surface. That is when you swim, when you move, you produce waves. And these waves they are going to slow you down which is different from the pressure drag is really the waves that you're producing.

40:21 So all this makes that it's better to be underwater than on the surface. But of course the issue is how do you swim underwater. So there's a nice way of swimming which is called dolphin undulation. And the perfect way to do the undulation is uh to create actually a thrust with an unilateral motion. you create little u vortices behind you that you're going to pu that are going to push you because if you uh if you undulate well enough they're going to push you exactly on your body and propel you uh with the undulate mo movement. So what is to be known is uh long time ago some people managed to uh break records doing uh 50 meters underwater but that was forbidden in competition and now it's only 15 m authorized and this is really something important in optimizing your your swimming is doing this unilateral motion uh as well as possible in order to to to win time on this part of the case which is the best one and

41:32 this is where uh French uh swimmer Leo Maron is very good. So uh before I conclude I should say something else about the Paris Olympics is that okay we had a strange track but that was pretty good but we had a strange pool that was pretty bad and the reason is the following is that in 2008 the regulations changed about the depth of the pool. So 2008 in Beijing, I don't know if you remember, uh this is this was the time of the swimsuits that look like shark suits, shark skin. And at that time, everybody was saying, okay, they swim very quickly because they are like sharks. But there was another reason that was less advertised is that 2008 was the date when um the depth of the pool was increased. uh 2008 depth of pool authorized were 3 mters and 3 mters is very good because when you produce waves they reflect at the bottom but the bottom at 3 mters is far away so they don't have time to come

42:37 back and slow you down. Well in Paris what happened is that you know sports and science are things that are very separate. So they thought three meters deep that's going to cost a lot of money because the pool was a temporary pool. Uh it's going to cost a lot of water and we want to be ecological. So we'll do the minimum authorize 2 m.10 2 m.10 what happened? Adam Pachy went for his 100 meter breast stroke and I don't know if you followed that he did one second more than three weeks before and it was the same for the three uh the first three swimmers in the 100 meter breaststroke they all did more or less one second more than three weeks before and this is how they started asking but what happened what is wrong with this pool and what is wrong was that it was not deep enough and uh professional swimmers now they really produce a lot of waves and at 2 mters the waves reflect and slow them down and they were

43:47 aware they were aware of that so my conclusion is uh don't do like the French in sports uh practice science ask questions around you just watch the world around you and try to understand what is going on and improve your curiosity And there are a lot of things that you can understand with some physics and some mathematics. Thank you very much.

Questions about this episode

Who speaks in "Going for Gold: the Mathematics of Sporting Glory - Amandine Aftalion"?

Amandine Aftalion (Host).

What is "Going for Gold: the Mathematics of Sporting Glory - Amandine Aftalion" about?

The discussion highlights the intersection of mathematics and sports, focusing on how analytical modeling can enhance performance in various athletic disciplines, particularly running and downhill skiing. Key points include insights on optimal running techniques, comparisons of speed across sports, and the surprising dominance of downhill skiers in terms of velocity, underscoring the role of…

What is mentioned in this Oxford Mathematics episode?

8 people, works and ideas are mentioned, including Paris Olympics, Hudson Smith, Inga Brixton, Julian Bend, Leo Maron, Tokyo Olympic. Each one on this page links to its own page with every episode that mentions it.

How long is "Going for Gold: the Mathematics of Sporting Glory - Amandine Aftalion"?

The episode runs 45 min; the transcript is about 6,553 words in 43 timestamped sections.

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