Let’s Twist Again: Throwing, Jumping, and Spinning
- Extra Reading
Throwing things, and jumping up and down or along, lies at the root of many Olympic events. In the gymnasium, the velodrome, and the diving pool we also see the key role of rotation in dramatic displays of strength and speed. What light does simple maths shed on these movements and the stress they place on equipments and the human body? Why do high jumpers use the Fosbury flop and long jumpers cycle in the air? How high can rugby players jump in the line out? These are a few of the questions that maths can help us answer.
This is part of Professor Barrow's Maths in Sport series. The other lectures are:
How fast can Usain Bolt run?
David and Goliath: Strength and Power in Sport
Citius, Altius, Fortius: Records, Medals and Drug Taking
21 February 2012
Let's Twist Again: Throwing, Jumping and Spinning
Professor John Barrow
The top of the diving board in Barcelona that was used for the Olympic Games reaches ten metres in height. The diving pool is almost on a mountain, on the outskirts of the city, so when you go on top of that diving board, you look down over the whole of the city, and you can see the cathedrals and the very high buildings there. So, if you suffer from any form of agrophobia, you do not want to be in that competition.
The simplest and most obvious example of spin that you encounter in all sorts of different sports events is what happens before the event starts. So, whether you are playing football or hockey or cricket, there will be usually some attempt to simulate a random process in order to decide who is going to start from which end, who is going to kick-off, for example, and tossing a coin is the traditional way to do that.
If you think about it, this is not a random process. When you throw a coin in the air, there is an initial condition, there are Newton's laws of motion which govern precisely how the centre of gravity of that coin will move, and how it will rotate and turn over as it flies, and the configuration it is in when it comes back down on your hand is uniquely and completely determined by a solution of those equations. So this is a deterministic process.
And you can make a few observations about it. School students may know that, if something is moving with constant acceleration and we throw it upwards, it feels the force of gravity, acceleration due to gravity against it, then the height that it will reach after time t is the initial height, plus its launch speed times the time, minus a half g times the time squared, and if you then think it will come back down again to the hand, so when it comes back down again to height H zero, we can solve this equation, and Vt will be a half gt squared, and that just gives you the time 2V over g. So this is the time that the coin is in the air before it comes back down to your hand, and this is going to be about 0.4 of a second, typically. If you throw the coin up with a speed of about 2 metres per second, you have got about 0.4 of a second before it comes down.
But if we spin it when we throw it up, so it turns over some number of times – let us call it R times – so it has got revolution rate of R and it turns over N times, then the number of times it turns over is just going to be the time it is in the air times this number of revolutions. And what you need here to make this decent random-like process is a lot of turnovers and if you want it to turn over more than twenty times, you have got to launch it with an angular speed that is bigger than about 50 rotations per second. If you do that, it will go up very high and you will get a lot of turnovers. You can see, intuitively, the more turnovers you get, the more random in some sense it appears, and the reason is because you would have to choose the initial speed and the initial turnover rate very precisely to be able to predict whether one side is coming up as opposed to the other. Obviously, if you just throw it and it turns over once, you know exactly what side is going to land on your hand, and you might even be able to watch if it turns over just twice. But, if it turns over 30 times, you would really have to adjust the initial launch very carefully.
Just by doing a bit of counting, you can see that, if you make it turn over once, then it is going to come down with the same face up. If you have got two to three, four to five, six to seven, then you will catch it with the same face up that you threw it; if you pick the in between ones, three to four, five to six, seven to eight and so on, you are going to land with the opposite face upwards. But once you start to get very large, you cannot sense and tune the initial launch speed and the spin accurately enough in order to be able to predict which of the sides will come down, because you would have to distinguish between the number 2n and 2n+1, when little n gets very large, and those two numbers are of course then relatively very close together.
Another interesting point about tossing coins is that, there is a sort of intuition that somehow you could bias the coin and it would be more likely to come down on one face than another if you biased it. This is rather misleading because, if you change the weight distribution in the coin in some way, you make it asymmetrical, all that happens is that it rotates about a different symmetry axis that it would have done if it were perfectly symmetrical. So, in this sense, you cannot have a biased coin. When you flip it, changing the mass distribution just alters the axis about which it rotates and then falls with equal probability.
If you spin the coin on the table, well then that is a different story.
Let us move onto more dynamical spins. The classical example of spin in sport is what you find in Ice skating.
The examples that we are going to look at in the first parts of this lecture, they are going to be about how spin is affected and influenced by changing your body shape. So, in many sports, changing the way you distribute the mass of your body – move your arms in and out, do other much more complicated things – changes the way and the rate that you spin.
This very simple example, starts with the conservation of angular momentum, so the moment of inertia proportional to the mass times the extent squared, the radius squared, of the spinning object, times the angular velocity. So this combination remains constant. Once the skater starts spinning, that quantity cannot be changed, but you can change r, you can move your arms in, and that will change omega, the angular velocity. So, what happens in a skating spin, the skater will start spinning by applying a torque with the skate to the ice, and begin with their arms outwards, so their inertia is large, and they will then draw their arms in, so r is reduced, their inertia becomes smaller, and so because this quantity is constant, omega, the angular spin, will go up, and it will go up like one over the distance, the radius, squared.
So, typically, a good skater might be able to reduce that combination, that r squared, by a factor of two, and so produce a very significant spin up as a result of just compactifying their body.
A nice example of spin and inertia is something that we find in cycle wheels. Nowadays, we are used to rather high-tech and fancy bike wheels. Some are disc-like and some are traditional, with rings and very lightweight spokes, just for stability and support.
So, this object here has the inertial properties of a disc, with the mass uniformly spread throughout the whole of the circular disc, and this has got, essentially, all the mass just in a ring around the outside. The inertia of this is twice the inertia of this, if their mass and their size is the same.
Well, suppose that you have a bicycle – we will ignore the rider for the moment – and the bike is made of a frame and it is made of two wheels. It is interesting to know that, if you were trying to make your bike go faster for the amount of energy you are applying cycling it, what should you do: are you better off trying to save weight on the frame or on the wheels? Well, the wheels and the bike are going to move along with a linear speed v, and let us suppose that the radius of the wheels is r, so the angular speed which they go round is omega, which is v over r.
What is the total energy that you need to make the bike go? Well, you have got a kinetic energy, a sort of half mass times velocity squared, but the mass, you have got the two wheels, each of mass m, and you have got the mass of the frame, and you can include the rider with the frame if you want, so this would be the mass of the frame plus the rider. But, as well as moving the wheels along in a straight line, you are also spinning them around, so you need a bit more energy to do that: you need the rotational energy of motion of the two wheels, and instead of being a half times the mass times the speed squared, it is a half times the moment of inertia times the angular speed squared.
The moment of inertia, as we just mentioned, it looks like mass times radius squared, and it has got some number in front. That number is one, for the wheel that is a ring, and it is a half for the wheel that is a disc. So, let us just put b in there – b is a number that is somewhere between a half and one. So, if we pop all this up here, tidy it up a bit, you can see the total energy that you need is proportional to the mass of the frame, plus twice the mass of the wheels, times this one plus b. One plus b could be anywhere between two and one and a half, so this factor here is between three and four. So you see that, if you can save mass on the wheels, you save much more energy than saving mass on the frame, because the wheels do not only move in a straight line, but they move in a rotational motion as well. So the same mass reduction on each wheel gives you three or four times as much benefit as it would just on the frame.
This is a rather spectacular example, not spinning this time, but again, of changing one's body shape in a spectacular way. This is a famous ballet move, if you like, the grand jeté, so if you see this performed, the ballerina will launch herself across the stage, in that direction, in this configuration, with legs completely horizontal. You see it with skaters as well. But they will appear to float in the air, so they will appear to hang in the air before they come back to the ground, and the question is: is this purely an illusion or is it a fiction? Because, what do we know about launching projectiles? Once they have been launched, nothing that they can do can change the trajectory of the centre of gravity of the moving object. It will move in a parabola and come back to the ground and whatever you do when you are in the air, you cannot change the trajectory of your centre of gravity.
Well, although that is true, if you think about throwing something that is asymmetrical – suppose you throw a hammer or a tree trunk or a shuttlecock. Imagine throwing a mallet or a hammer… You know how it spins and flies. The head of the hammer has perhaps a rather complicated trajectory, even though its centre of gravity is moving still in a simple, nice parabola. So, what the head of the ballerina does, does not necessarily have to be as simple as what the centre of gravity of the ballerina does. When the ballerina launches herself in the air, she raises her legs up to the horizontal position, in this spectacular way, and throws her arms above shoulders, and this raises the centre of gravity, the centre of mass, of her body, relative to the position of her head. Then, when she comes back down, she will lower her legs and lower her arms, and the centre of gravity of her body will fall. So, by changing the location of her centre of gravity, she creates the illusion that the head is moving horizontally.
But in fact, it is not just an illusion. You can do a rather detailed study of these jumps and attach sensors to the ballerina. Her centre of gravity is following a parabolic path, but this is the displacement above the ground of her head with time, and you can see it does indeed follow this horizontal trajectory for a significant fraction of a second, 0.3, 0.4 of a second, before descending, and it is just created by changing the location of her centre of gravity.
You see this sort of thing in basketball. Michael Jordan appears to hang in the air. Is this just an illusion? If you talk to many basketball fans, they tend to think that it probably is. It is not. It is exactly the same effect. You are playing the same game: altering the location of your centre of gravity when you jump in the air, so that your head appears and stays horizontal for a significant split second before you descend.
Well, some people, or some living things, are much better at doing these sorts of contortions than us, and the famous one is the falling cat. So, if you drop your cat – I do not recommend this, but if you drop your cat from above your head height, then you have probably seen this sort of thing and know what the cat does. It always seems to be able to land on its feet. No matter how it is dropped, it will change its body shape, it will rotate, and it will always come nicely to rest, not rotating, on all four feet. It has various advantages that we do not have: it has got a very flexible spine and skeleton; and it has an innate sense that seems to enable it to tell the difference between up and down.
What is going on here is a problem of changing the cat's body weight distribution, changing its inertia. So, you remember that rule: the moment of inertia times the angular velocity is a constant. If you can reduce your moment of inertia, make your mass distribution more compact, you will spin faster; if you can make it more extended, you will spin slower. What the cat does is it splits its body into two parts: so, it starts off by rotating its back half and its front half about slightly different axes, in different directions. It starts in by tucking in its front legs, and that means it is reducing the inertia of the front part of its body, and so it rotates a bit quicker, gets it round into the right position quickly that it wants to be in. It pushes out the back legs, so the inertia of the back part of the body becomes larger and it spins more slowly. In this way, it changes the position of the front of its body relative to the rear, and that gets it into this sort of configuration. Then, it does the opposite: it extends the front legs and pulls in the back legs, so the front half of the body's inertia is going up, spins more slowly, the back part, it goes down and it spins more rapidly, and that brings it round into this position, which is the one it wants to be in. So it ends up in a situation where it has got very little rotation at all and it just lands still. In this whole process, angular momentum is conserved. There is nothing that the cat can do to create or offset overall rotation, but it can rotate a little bit in that direction and a little bit in that direction, so when you add the two together, it is zero. It it can keep adding together two different senses of rotation, so the sum is always zero.
Now, in sport, you see a variety of this activity in the diving pool. If you look at pictures of divers, and there are quite a lot in newspapers on sports pages these days because of people like Tom Daley, what you will see is a diver in this type of configuration. What is happening here, after they have taken off and they are falling, they will put one arm up and one arm down, across the body. So what they are doing again, they are creating this sort of clockwise rotation here, and the rest of the body – so the arms are moving in a clockwise rotation, and the rest of the body moves with an equal and opposite rotation, in the anticlockwise direction, so the sum is zero. By changing the body shape in that way, they can get that rotation going, and then, in the next stage, they will stretch the body out, streamline the body, and that will stop the rotation. So they make r very large, the rotation goes quickly to zero, and they end up, just like the cat, landing, entering the water, with no rotation. That is the big goal here: to enter the water vertically, but with no rotation at the end. You start with no rotation, then you create equal and opposite rotations in different directions – sum is still zero, and then you revert to zero total rotation at the end.
Diving is a curious business. If you look at the Olympic programme, diving is part of the aquatics programme, but the whole activity takes place in the air, not in the water, so this event is nothing to do with water, although it is characterised as a water sport, so I have never really understood that.
If you ever watched high-level trampolining, which is a very spectacular sport: world-class trampoline in the men's event, those people will go nine metres in the air. It is like doing high-board diving, but you have time to do exercises on the way up and on the way down, so it much be a much more satisfying activity than the diving.
How much time have you got to do things in the high-board? Well, it is ten metres, of course. If you just drop off, from rest, that ten metres, the time it takes to reach the pool, is a half gt squared, so if we solve this using the acceleration due to gravity, 9.8 metres per square second, you have got about 1.4 seconds before you hit the water, and the speed with which you will reach the water, if you are starting from rest, is just acceleration due to gravity times the time you are in the air, fourteen metres per second – it is about 31 miles per hour.
You are not allowed to apply any sort of torque with your feet when you leave the board – that is not allowed, so you cannot generate spin and rotation using your feet.
If you were doing a number of somersaults in that period, suppose you want to fit in 3.5 somersaults, in your descent, then that means you have got to be spinning at 2.5 revolutions per second in order to have enough time to fit them in. That is 150 revolutions per minute. If you look at your CD player, the outer edge of the spinning disc is going round about 200 revolutions per minute. So, top divers really are spinning extraordinarily fast – there is huge stress.
I remember, a few weeks ago, we saw a picture of Boris Johnson, the Mayor of London, on top of the ten metre diving board at the Aquatics Centre, with Tom Daley and others, and I see everybody really wanted to push him off, but they just could not, in his suit, and he sort of looked over the edge and you kept thinking, "Go on, go on!" but they never did.
Well, this is the high-board, so you notice you have got about 1.4 seconds to do these complicated things.
If you go to the springboard, you have actually got more time because you go up and then you come down. Even though it is only three metres above the pool, you have got more time to do things with the springboard.
When you take off the springboard, typically, your launch is going to be at about five degrees, or a tiny bit more. You do not go vertically upwards. If you do, you do what this fellow here does – he comes straight back down on the board, rather than clearing it.
The springboard is slightly more complicated in one respect, that if you are an engineer, a springboard is what you call a cantilever. So, this is a beam. A typical competition springboard is about 4.9 metres long, and 50 centimetres across, and it is made of aluminium. It is fixed at one end, and there is adjustable roller here. When you go up to dive, you can use your foot to just change the lever to move that position of the roller, because, if you pull it in, that will make the board springier, if you push it out, it will be relatively stiffer, and you do that because lightweight divers and heavyweight divers have different requirements, as we will see in a minute, for what they want the board to be able to do.
So, you then run along the board, bouncing along the board, step by step, and you hit the end, and up you go. Now, anything that has a vibrational possibility, like the end of this board, has the possibility to create a resonance. If you were just to touch this and start it vibrating, when you are not standing on it, it will vibrate at a natural frequency, which is determined by Newton's second law…so the mass times the acceleration, d squared x by dt squared, where x is the deflection of the board, will be proportional to the deflection. And the constance of proportionality, which tells you how quickly it comes back when you deflect it, is a measure of the stiffness of the springboard. If you move the board out a long way, you shift the roller back here and there is a lot of board out, it will be much springier - there will be a larger k. If you move the roller this way, there will be less board out and it will be stiffer, and that will be a smaller k.
The natural frequency is just given by the square root of this k and the mass of the board. So the mass is fixed, in practice, so the square root of this quantity is what we call the natural frequency of the board. And its value for a typical springboard, with a 65kg diver bouncing on it, is about 3.6 per second, so that is the natural frequency.
If you have something with a natural frequency and you start to force it, you are familiar with what happens. Suppose you sit on a child's swing. If you allow the swing just to go backwards and forwards, it has a natural frequency of oscillation. If you now start to pump the swing, to try to go higher and higher, you know that there is a nice way to pump it. If you hit it just right, you go much higher. If you are out of phase with the way it is trying to oscillate, somehow you just deaden the swing. So what is happening when it is just right is that you are trying to pump the swing, force it, at exactly the natural frequency, and the two add together and reinforce. If you are out of phase, they cancel out.
The springboard diver, when he bounces along on the board, wants to try and hit the board with steps and forcing at exactly the natural frequency, and when that happens, that is when you hit the board just right and up you go.
In practice, you can tell when people are hitting the board at the natural frequency just by the sound. So when it sounds just right, you have hit it at the natural frequency, but when you get that awful sort of bluuuuung, that is when you have not hit the natural frequency, and you somehow just do not go up with the speed that you expect.
The springboard is quite a different business to the high-board and requires an individual diver sensing where is the best place to move the roller. They know what their natural frequency is going to be, and how you want it to fit in to how they want to bound down the board, so they adjust all these things so that they can, rather naturally and with all their strength, hit the board at the natural frequency.
Let us move on and have a look at changing body shapes and so forth when jumping and look at long jumping. Here is my attempt to draw a long jump. There are three parts to the distance that a long jumper will eventually achieve. The long jumpers come sprinting in, the last stride that he or she makes, this is the last foot-plant on the ground, there is the centre of gravity of the long jumper, and typically, they will be leaning forward a bit at that last step, and there will be a little distance between where the foot goes down and where their centre of gravity is. That is important because, once they launch and cannot do anything else to alter the trajectory of their centre of gravity, all their energy of running in will go into moving the centre of gravity in a parabolic path, so the further forward they can get the centre of gravity, the greater the distance they will get it to go for the same input of energy.
Here is that second phase, the almost parabolic trajectory. The centre of gravity does not reach the ground of course. They put their feet forward, and that is the final mark, assuming they do not sit on their backside in the sand, the last mark they make in the sand is what is recorded as the landing distance. Part of the art of long jumping is to make sure that your body shape and forward motion when you land is not going to allow you to fall backwards and make a mark in the sand here rather than here. So the distance you will achieve will be the sum of these three parts: the distance between foot and centre of gravity; centre of gravity to centre of gravity when your foot hits the ground; and the distance between your centre of gravity and the footfall. Typically, if you were a pretty good long jumper, jumping about seven ¾ metres, you would make up that distance by having those ingredients, obviously dominated by nearly six and a half metres of actual jump, and these other little tricks about your centre of gravity will give you about 0.4 of a metre there and about 0.9 or so at the end. So, they are small, but they are not insignificant. This is for a launch speed of about 9 metres per second, at an angle of about 20 degrees.
The distance you will go is just given by the formula for the range of a projectile, which is launched at speed v and an angle theta.
I have said nothing about where you put your takeoff foot. It is up to you where you put your foot. You have obviously got to try and get it as close to the takeoff board as possible. If the takeoff board is down here, this will be a no-jump; if the takeoff board is down here, then you are really giving away distance that you could have. The skill is to make sure that last step occurs right on the board, without overstepping the plasticine.
There are three styles you will tend to see on offer if you watch top long jumpers on the television or in practice. A rather old-fashioned type of technique, which, it would be like using the scissors for the high jump when you first try it at school, so that is the only place you see it, except in some British international jumpers, and it is called the sail. It is just like taking an enormous leap. What is happening here is you are starting to rotate when you takeoff, but you do not do anything to counter that. You are not like the diver, moving your arms. You do not do anything complicated or clever with your body shape. If you are not careful, you will have a tendency to push your feet out at the end, and you will not have enough forward impetus and you will fall backwards.
A slightly more sophisticated version here. You can see how this evolves into what is sometimes called the hang technique. So someone like Tomlinson, the best British jumper, interestingly, uses this technique. He is very large, six foot five inches tall. But what is going on here is that the limbs are all being moved, particularly the arms, the knees are being brought up here, and the aim is to change the body position when you land. You are moving the legs at take-off, in a quite different way…you see one limb going forward, with another limb going back. When you land, usually, with this technique, you do not have much forward momentum, and you tend to see the jumpers coming out of the pit to the side, so that is the trick you use, but instead of waiting to fall backwards, you fall to the side.
The much more sophisticated technique that you see most world-class jumpers used is called the Hitch kick. This technique, if you watch what is going on here, you will see people move one leg backwards and one leg forwards, arms forward again. You are not creating any new momentum, what you do forwards, you do backwards, but, all the time, you do that again, it is as though you are cycling in the air. You are rotating your body around that centre of gravity, which is moving in the parabolic position, so that, when you land, you are tending to fall forwards. If you are jumping a long way and you are very dynamic, you may get several of these Hitch kick movements happening before you finally hit the sand at the end. What is happening here is that you have, at any one stage, a straight leg and a bent leg, and you make sure that the rotation you create in one sense, by one leg, is counterbalanced by the opposite sense of the other. It is just like the other cases that we saw, where you make rotation in two different directions, just like the cat, but you change your body position so that, when you land, you do not fall backwards – you maximise your actual distance in the sand.
This is a projectile. People are always very interested in working out how do you get the maximum range when you launch a projectile. And if you launch a projectile from the ground, then this is the range that you will achieve if the launch angle is theta, and you know that the sine of any angle has a maximum value of 1, and it has that value when the angle that is being sined is 90 degrees. So, the two theta here means that you will get a maximum range when the launch angle is 45 degrees, if you launch from the ground and you finish on the ground.
There is another question that is of interest to some sports, notably rugby. If you kick the ball in rugby, and maybe you kick it up in the air, you are not usually interested in range. Many times, you are interested in time. You want the ball to be in the air for as long as possible so that your players have a chance to charge downfield and do whatever they do. So, sometimes you want to kick for time rather than for distance.
If we look at this formula, we see something very interesting about it. We know that the sine of an angle is exactly the same as the sine of 180 minus the angle, and so, we would get the same range if we put two theta there as if we put 180 degrees minus two theta there, and so the range is the same for theta as it is for 90 minus theta. You have got two trajectories, two launch angles, which give you the same range. They are different launch angles: one has a low trajectory, at a small angle; and the other has a high trajectory, at a large angle. And we can work out how long you stay in the air for these two trajectories. It is the distance that you travel, divided by the horizontal component of the speed of the launch, which is v cos theta. So, the ratio of the time in the air for the high trajectory to that of the low trajectory turns out to be just the cotangent of the launch angle, one over the tangent of the angle, so that is the cosine of theta over the sine of theta. And the two heights that are achieved, again, there is a pretty result for that: it is just the square of that.
Suppose that your high trajectory was launched at 75 degrees, your low one at fifteen degrees, then the ratio of the times in the air is 3.7, so it is a very significant extra time you will achieve by going for the high trajectory to allow all your sort of huge players to charge downfield and be in position for when the ball comes down and avoid being offside and all those sorts of things. And the height that you achieve, again, there is a factor of fourteen difference between the two heights. This is a rather unfamiliar type of projectile problem. People almost always are interested in ranges, but maximising time, for a certain type of sports activity, is sometimes a good thing to worry about.
Let us have a look at javelin and discus throwing.
These events are complicated aerodynamically. The shot, for example, is very simple because the shot is so heavy, and likewise, the hammer is so heavy, that there are no aerodynamic qualities to worry about. It is launched, and gravity determines the trajectory.
But the javelin is very light – I think it is about 800 grams, that is all, for the men's javelin - and it has an interesting aerodynamic shape, and so its motion through the air is rather more complicated than, say, a cricket ball and certainly a shot.
Many years ago, the javelin event became rather dangerous. I think it was 1984, the world record went up to 104 metres and 80 centimetres, by Uwe Hohn, so this was becoming too dangerous an event to hold in any stadium. If you have an Olympic stadium with somebody threatening a 105 metre throw, this is going to go into the lower seats on the far side of the stadium, so something had to be done, and poor Mr Hohn's world record was crystallised forever. What was done was to shift the centre of gravity of the javelin, to move it by four centimetres forwards, and this was enough to make a 10% reduction in the distance that you would throw the javelin. This was a consequence of the aerodynamic features.
Nowadays, the same problem is still very much in the offing. I think the world record is about 98 metres and 48 centimetres, by Zelezny, although no one has thrown that far for a very long time. It is not far off the 100 metre mark, and if another outstanding thrower comes along, there is going to have to be another change in the centre of gravity of the javelin.
What changing the centre of gravity does is make the javelin come down much sooner, at a much sharper trajectory, sticks in the ground, but does not go as far. What is happening with the javelin is that there is a centre of pressure for the javelin's motion through the air and it is not the same as the centre of mass, and what that means is that there is a overturning moment, a torque on the javelin as it moves through the air, and it is subject to two forces other than gravity. There is an air drag on it, which is not very significant because of its rather streamlined nature, at first, but it also experiences an aerodynamic lift because of the air that it is displacing. Just like an aircraft experiences a lift because of Newton's third law, displacing a mass of air, and so it receives an equal and opposite lift force. Those two forces conjure up this rather unusual type of trajectory of the javelin. What you try to do, so if this is the trajectory of the centre of mass, the centre of gravity of the javelin, you try to launch it in such a way that its actual trajectory lags here by about ten degrees or so, so there is an angle of attack, just like with an air wing, of about ten degrees or so, between the path of the centre of gravity and the actual path of the javelin. What that does is it creates this continual overturning moment, so the javelin is always being brought down, in this sense, which helps to make it eventually hit the ground and stick in the ground. This is the type of trajectory that the javelin will end up following, and as it does that, it sails in the air. The lift factor will allow it to stay it in the air for longer, and you can get a greater distance because of that aerodynamic lift factor. It is another case where you are looking for maximising the distance because you want to maximise the time that you are in the air and maximising the lift force that you experience.
The discus is slightly different. It is launched in a completely different way, and it also feels a drag and a lift force. The key about both those forces, although the constant proportionalities are different, is that they are proportional in a wind to the speed minus the wind speed squared. So, they are proportional to the speed of the discus, relative to that of the wind, squared.
The interesting thing to notice about these formulae is that the effect of a wind, with speed w, is very significant. If you have got a wind behind you, a tailwind, behind the discus, intuitively you would think, "Oh, that is going to make the discus go further, it is like being a sprinter, with the wind behind you, getting a push". In fact, completely the opposite is true.
So, if you have got a wind behind you, this wind speed, w, will be positive, and so you can see that when the wind speed is positive, your lift force will be less. But if you are throwing into a wind, so there is a headwind, the wind will be negative, w will be negative, different sine to the direction of speed of the discus, and the lift will be greater. If this was ten minus minus ten, this is bigger than if it is ten minus ten. The first important thing to notice about the discus is the headwind is very advantageous.
Top discus throwers become meteorological experts and they are always looking for – they enter the stadium and the first thing they are sensing: what is the direction of the wind? What is the speed of the wind? I can remember cases with Jay Sylvester in the '60s and '70s, who, in America, would seek out particular arenas in the US that were famous for very powerful directional winds as places to set a world record.
Here is some scientific evidence of this factor. So, you look at what happens to the distance thrown by a discus when the wind speed w is given here. This is positive wind speed, so this is when you have got a following wind, and this is with a headwind. And, what you notice, if there is no wind here, you are looking at a throw roughly between 65 and 66 metres. If you were to go back to ten metres per second headwind, then you have really got a very significant, nearly four metre, increase in the distance that you will have thrown; whereas, if you have a following wind of ten metres per second, you are looking at a meter or two smaller distance.
This is a rather odd state of affairs. Let us get a feel for what these wind speeds are like. Twenty metres per second, I mean, this is a ridiculously strong wind – that is why it is not mentioned here. Two metres per second, you remember, as a following wind, for a sprinter, will invalidate a performance as a record, so two metres per second following wind is the strongest you are allowed. But, curiously, there is no wind proviso for setting records in the discus, but you can see, in many ways, the wind is much more significant than it is in the 100 metre sprint. So you can set a world record with a very strong headwind, and it will not be marked or treated specially as wind-assisted, but it certainly is.
Let us say something about football. David Beckham is famous for dead-ball kicks and being able to swerve a football, something that every school child can do as well, but not to such an extent or with such power that he can do it. We know the sort of thing that is done here. If you play football, you know that if you kick this ball in this direction and you apply the outside of your right foot to this side of the ball, you will make it spin in this clockwise direction, and it will tend to swerve off in this right-hand direction; whereas if you kick it on this side with your right foot, the inside of the foot, you will make it rotate in an anticlockwise direction and you will make it swerve in this direction.
So, what is going on there? First of all, let us think about the situation where the ball is not spinning at all, and instead of having a moving ball, we will imagine the air is going past the ball, so we are taking a ball's eye view, as it were, of the situation. Here is the ball, and imagine air is rushing past the ball, then, because of the presence of the ball, the streamlines of the air, as they go past, get squashed together, and when you do that, you create a higher speed, here, for the air flow going past, so it is being constricted in some way, so it goes faster, like putting your finger over the end of the tap, and when you do that, pressure goes down because it has to conserve mass and energy. What is happening to this airflow? It starts with a higher pressure and a lower speed. As it goes past the ball, the pressure goes down, the speed goes up, and then, once it is past, it resumes its lower speed and higher pressure.
If we start the ball spinning, then the situation becomes slightly asymmetrical. If we look down from above, on the ball, you have now got the ball spinning, in this way, and you can see that, up here, the surface of the ball, when it impinges upon the air, is going in the opposite sense to the air that is coming past, so it is dragging the air in that direction, so this air going past the ball has got a lower speed than it would have had if the ball was not spinning. But at this end of the ball, it is spinning round with exactly the same direction of motion as the air that is coming past, so this air is going past the ball with a higher speed. What happens here is that, up here, you have got a lower speed, higher pressure and down here, you have got higher speed, lower pressure, and there is a net force on the ball from here to here because the pressure is higher here than here, so the ball moves sideways. It is created by the spin effect and the effect of the spin on the airflow that is going past the ball.
The big picture, as it were, if you are taking a corner, if you kick the ball with your right foot, you hit the ball, you spin it in an anticlockwise direction, then what you are doing is creating a high pressure area on one side of the ball and a low pressure zone on the other side. As it goes through the air, and that creates a force in that direction, the ball will move round and sail miles over the bar up into the Chelsea stand.
The last topic, a slightly odd, but rather interesting one, is Archery, where, again, there is an issue of swerve, in some sense, but it is very different to the swerve in the last case. There is an old expression, something that has probably been known for hundreds and hundreds of years, as old as the history of archery itself, and it is usually called the Archer's Paradox, and it is the fact that you cannot fire an arrow in a straight line.
If you want to hit the bulls-eye in this target, you cannot fire the arrow in a straight line to the bulls-eye because the bow gets in the way. So, you hold the arrow at the back, with your finger. The side of the arrow has to touch the bow frame, and as a result, the arrow is always fired at a small angle off-centre. So, you pull the bow back at the knock and the arrow is touching the bow frame and, if it just went in a dead straight line, it would hit the target off-centre. If you wanted to hit that target with an arrow that was going in a dead straight line, you would have to aim somewhere else, so you would have to move the bow around.
In fact, archers do not quite do that. What happens here is more complicated. The arrow has a certain stiffness – it is similar to our problem with the natural vibration of the springboard. When you give the impulse to the arrow, it starts to oscillate, and the extent to which it will oscillate will depend on the stiffness of the arrow. So, once the arrow is released, it feels this impulse, in effect, by touching the side of the bow, and the arrow will undulate, and the trick is to make sure you understand your arrow and its stiffness and you have the skill to release it in just the right way, that it has just the right number of oscillations to bring it back on a straight course for the bulls-eye after you let go of it.
You draw the bow back, you then let go of the arrow, and it will oscillate through a sequence of shapes. It will start rather bent, where it receives the force from the bow, it will then reach a straight configuration, it will then bend the other way, and so on. By the time it parts contact with the plane of the bow, you have got to organise things so that it is now moving, in a sense, in a completely straight trajectory towards the centre of the target.
The Archer's Paradox is resolved by the fact that these oscillations are understood by the archer. They are controlled by the stiffness of the arrow. After one or two of them have occurred, they damp out very quickly, aided by the fletchings on the back of the arrow.
The stiffer the arrow, the more it is going to oscillate. The stiffness is given the name of "spine", although the spine is really the flexiness. If you have lots of spine, you are rather flexible, and if you have not much spine, then you are stiff. So, what would happen for an archer whose arrow had too little spine, it would tend to veer off down here –it would be too stiff, and you would not get enough oscillations to bring it back on course. If it had too much spine, so it was just a bit too flexible, too many oscillations, then you would expect to veer off in the other direction. So there is a Goldilocks' level of spine that you get to understand, by using your arrows and the weather conditions and so forth when you are shooting, which will produce just the right amount of oscillation to bring your arrow back on central target by the time it hits the target.
You can see that, to do this over and over and over again, under competition conditions, maybe varying wind and temperature conditions, requires extraordinary skill and sort of stability.
Of course, if you were an archer long, long ago, you can see one reason why a crossbow is a much more appealing idea than a bow. With a crossbow, you really can fire the arrow, the bolt, in a dead straight line at the target, and crossbow bolts are very, very small. You do not have these sorts of factors for the spine and the oscillations. So they are rather short, and you do not want them oscillating and they do not need to.
Robin Hood really had to master this stuff; William Tell, not really quite so much…
© Professor John Barrow 2012
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