Dynamics of the flip long jump

The long jump is a track and field event in which athletes attempt to jump as far as possible from a take-off point into a sandpit. In the 1970s, an athlete called Tuariki Delamere tried to introduce a ‘front flip’ technique where one goes into a front tuck instead of using a regular technique like the hitch kick and argued that it would allow for a longer horizontal distance. Several reasons have been proposed for the superiority of this technique, including the reduction of drag force during flight and increased angular momentum at take-off. We show that the air resistance makes a negligible contribution to the horizontal distance covered whereas a larger angular momentum can increase the distance of a jump. We explain why the technique was banned and discuss whether our argument implies that the technique can be deemed to be superior in practice.


Introduction
The long jump is a time-honoured event which uses standard, well-established techniques which are followed by almost all competitors.Two standard long jump techniques are the 'hitch-kick' and 'hang' techniques (a schematic of the former is shown in figure 1) [1].On the other hand, athletes frequently experiment with new or exotic techniques which allow for improved performance whilst staying within the rules of the relevant governing bodies.The idea of performing Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. a front tuck during a long jump was first introduced in a track and field championship in 1974 by Tuariki Delamere with the goal of achieving an increased horizontal distance (a schematic of the flip long jump is shown in figure 2) [2].Delamere was likely hoping to imitate the success of Richard Fosbury, who had introduced a new technique for the high jump in the 1968 Summer Olympics which was then adopted universally in the sport (known as the Fosbury Flop) [3].The new technique was almost immediately banned, however, mainly for safety reasons.Since the take-off appears to be the same, why would performing a rotation potentially enable the athlete to travel further?
In order to answer this question, we first begin by explaining the basic principles which an athlete uses to project themselves over as long a Figure 1.Schematic of the hitch kick technique.Note that the rotation of the legs and arms causes a rotation of the head and neck in the opposite direction [1].Reprinted from [1], Copyright (1993), with permission from Elsevier.horizontal distance as possible.There are four main stages to a long jump: approach, take-off from a wooden board, flight through the air and landing.The most obvious way to increase distance is to increase the horizontal approach speed before reaching the take-off point.If the athlete approaches with more speed, they will take off with more speed and achieve a greater distance.For this reason, long jumpers sprint along a track towards the take-off board to gain as much speed as possible before jumping.At the same time, the athlete must 'gather' themselves into the correct position for take-off and make sure that they place their foot on the take-off board.The traditional folklore states that the jumper reaches their maximum horizontal speed four or five steps before the board and then use the last set of steps only to get into a favourable position for take-off.Recent research has shown that this usually does not occur and that in most long jumps the highest horizontal velocity is achieved at the penultimate step before the take-off board [1].
The horizontal take-off velocity is obviously not the full story, since one must also have a vertical velocity component to enable the athlete to maximise horizontal distance by 'hanging' in the air for as long as possible.This vertical lift is obtained using good take-off technique, as the athlete must achieve a strong push (or impulse) away from the board to get a significantly large vertical velocity component.During take-off, the athlete also tries to jump in a position which is as upright as possible to ensure that the initial centre of mass is high, enabling the jumper to maximise the time that the centre of mass remains in the air [4].Anything which helps in efficiently displacing the vertical position of the athlete's centre of mass will also help to increase the horizontal distance covered.This includes increasing the height of the athlete, since a longer body means that the centre of mass can be displaced higher (for this reason, long jumpers are usually quite tall).A taller jumper is also more easily able to allow their centre of mass to descend as low as possible before contact is made with the sandpit.
We will argue shortly that it is the take-off stage of the jump which makes the difference between flip and regular techniques, since it is here that the angular momentum during the flight phase is imparted.Several sources suggest that there could be two factors which could contribute to a potential increase in distance using a flip jump: the influence from air resistance and the more efficient landing which is possible due to rotation [2,5].In section 2, we will use a slightly formal mathematical argument to confirm the intuition that air resistance has a negligible effect on distance covered during a flip jump.This section may be skipped by students and teachers who are willing to accept that a mathematical argument can be used to confirm the intuition that one might expect.In section 3, we will resume discussion of the forces which are generated during take-off and explain why an analysis of the resultant angular momentum imparted in both regular and flip long jumps suggests that the flip technique should be superior.In section 4, we will discuss reasons for banning the technique despite its apparent superiority before finishing with conclusions in section 5.

Influence of air resistance
As mentioned in the previous section, most of the existing literature on the flip long jump assumes that reduction of drag is negligible as a possible benefit for the flip technique [3].There are some sources, however, which suggest that reduction of wind resistance is a contributing factor to the effectiveness of the technique [2].It can be shown formally that the influence of air resistance on the horizontal distance covered is negligible by following classical analysis along the lines of that of Lamb and Ward-Smith [4,5].The motion of the long jumper is considered in the xyplane, where x and y represent the horizontal and vertical coordinates, respectively, with positive y measured upwards.The motion of the centre of mass is governed by the equation for balance of forces where V is the velocity vector for the long jumper, m is the mass, g is the acceleration due to gravity, and D denotes the aerodynamic drag.The first term represents the rate of change of momentum, the second term represents the forces due to gravity, and the third term represents forces due to drag.
Re-writing the balance of forces in terms of horizontal and vertical components, we have where θ is the angle elevation of the long jumper's trajectory and ρ is the density of the ambient air, S is a reference area, and C D is the drag coefficient.Notably, C D can be changed by altering the configuration of the body during the jump, but it will be assumed here that C D is a constant for both a regular and a flip long jump.For simplicity when analysing drag forces, it is assumed that vertical displacement of the centre of mass between the start and the finish of the jump can be neglected.
Without repeating the analysis of Lamb, it can be shown that solving equations ( 2) and (3) for a constant value of K produces a trajectory which may be described by an expression of the form where u 0 denotes the horizontal component of the velocity at take-off and v 0 denotes the vertical component [5].After taking a Taylor expansion and truncating up to terms of third order, we arrive at where θ 0 is the initial angle of elevation at takeoff.The first two terms give the usual parabolic trajectory of a projectile being fired forwards through a vacuum and the third term gives the first order correction due to the effects of air resistance.For input parameters, we will mostly use values provided by Ward-Smith using the winning jump achieved by Bob Beamon at the 1968 Mexico Olympics [4].The density of the surrounding air in our case is atmospheric density, which corresponds to ρ = 1.225 kg m 3 .Using the conditions of Beamon's jump, we also have u 0 = 9.45 ms −1 , m = 75 kg and g = 9.8 ms −2 .Values for angle of take-off amongst high-level professional long jumpers typically vary in a range between 15 • and 27 • , so we will take the mean value θ 0 = 21 • [6].For the product SC D , Brearley and Ward-Smith assume a value of 0.36 which was originally derived for racing cyclists in the touring position [7].Consideration of videos of a flip long jump suggests that during the jump the athlete reduces their cross sectional area by around a half, so we will assume for this jump that SC D = 0.18.In figure 3, the trajectory of a long jump given by equation ( 6) is plotted using these input parameters.The blue curve corresponds to SC D = 0.36 and the red curve corresponds to SC D = 0.18.It can be seen that the two curves are almost indistinguishable.This is not necessarily enough to argue that we can neglect the effects of air resistance, since Ward-Smith has shown that a reduction of the gas density from 1.225 kg m −3 to 0.98 kg m −3 in the same analysis could lead to an increase of 2 cm in the distance jumped.However, the situation is quite different in this case, since the decrease in gas density acts over the entirety of the trajectory.In our case, we unrealistically assumed that SC D is a constant and is reduced across the whole trajectory, but inspection of a video of the flip technique (or figure 2) shows that the reduction in the drag coefficient only occurs over a fairly small part of the trajectory where the body is folded and changed in configuration [2].In combination with the previous analysis, this is enough to show that any aerodynamic benefit obtained by using the flip technique is negligible, or at least negligible enough to mean that specific training dedicated to learning and perfecting the technique would likely not be justifiable.

Influence of angular momentum
Having argued that the influence of air resistance may be neglected, we will now consider the benefit which may be obtained by considering the influence of rotation on the landing phase.In figure 4, we show that there are two reasons for reduction of distance during the landing phase: contact of the pelvis behind the theoretical point where the centre of mass was supposed to land and falling backwards during landing.It seems intuitively clear that the second type of landing efficiency is very important for the flip jump (when performed correctly), since the additional angular momentum of the athlete should help significantly in ensuring that they do not fall backwards on landing and make a mark before their shoes.
Although the angular momentum cannot be measured directly, it can be shown for completeness that the angular momentum imparted using the flip technique is larger.Readers who are willing to accept that more angular momentum is imparted at take-off when performing the flip technique may gloss over this argument.It is known from classical mechanics that the rate of change for the angular momentum Ḣ of a rigid body is equal to the resultant moment M: where it is understood that the moment and angular momentum vectors are both about a point in the centre of mass frame.Integrating both sides, we obtain where t j is the time at which the foot leaves the take-off surface after making initial contact at t = 0.The resultant angular momentum during flight can then be found by integrating over the times when the athlete is in contact with the takeoff surface.
The resultant moment can be found by considering the horizontal and vertical reaction forces at contact.At the point of take-off, the athlete uses their foot to push forwards and upwards away from the ground and so is subject to both horizontal and vertical reaction forces.In figure 5(left), we show the directions of the horizontal and vertical forces represented as timedependent functions H(t) and V(t).The horizontal and vertical ground reaction forces which can be achieved limit the take-off angle, since the magnitude of the velocity vector determines this angle [9].This means in practice that the take-off angle will be lower than the 'ideal' one which students might expect from the usual trajectory of a projectile.Figure 5(left) also shows the horizontal and vertical displacements of the centre of mass relative to the take-off foot represented as time-dependent functions x(t) and y(t).
The moment M can be written as which can then be substituted into equation (8).
Images of the initial take-off show that x(t) and y(t) look to be almost the same in both cases, so the difference in angular momentum imparted at take-off for regular and flip technique can only come from V(t) or H(t).The athlete is normally not allowed to have too large a value of H(t) as it would cause them to land face first in the sandpit.In the flip case, H(t) is allowed to be larger.This implies a larger resultant angular momentum from take-off as long as V(t) is not changed, since a negative value for the resultant moment indicates forward or clockwise rotation.This increase in resultant angular momentum can play a decisive role in landing efficiency.Even during a regular jump technique, the athlete rotates forwards with a certain amount of angular momentum, but they receive a torque when they impact with the sandpit which rotates them backwards in the opposite direction.Clearly, if an athlete does not have an optimal amount of angular momentum from take-off, they will have a tendency to fall backwards and lose distance.This can be avoided by increasing the total angular momentum so that it is larger than the bodyweight torque during the landing phase.This is shown schematically in figure 5(right).The term 'bodyweight torque' is somewhat non-standard, so we will explain it briefly.A torque in general may be thought of as the rotational response which (Left) Athlete during take-off with the coordinate system which is used to calculate angular momentum [10] and (right) athlete during landing [8].Reproduced from , with [10] permission from Springer Nature.
a rigid body has due to an applied force and is defined as where r is the position vector at which the torque is measured and F is the force vector.As the athlete lands in the sandpit, there is a force at the point of impact at the heels which rotates the athlete counter-clockwise on landing when viewed from the right, since τ is perpendicular both to r and to F. If this torque is greater than the angular momentum during flight, the athlete will fall backwards on landing and there will be a reduction in the final distance.An important suggestion which emerges in the literature on landing efficiency during a long jump is that athletes who use a hangstyle technique should try to increase the angular and linear velocity of the segments of their body during take-off to try to increase their angular momentum, but this is exactly what the flip technique achieves naturally [8].It is important to emphasise that the body does not travel further simply because it is rotating forwards during the entirety of the trajectory.Any effect of rotation on the flight phase of the jump would be due only to air flow, but we have already shown in section 2 that air resistance has a negligible influence during a flip long jump (this could be an opportunity to introduce a classroom discussion of the Magnus effect).

Why was the flip technique banned?
If the flip technique theoretically allows for a longer horizontal distance, why was it banned?Grant Birkinshaw, a track and field historian who competed as a long jumper in the same era as Delamere, suggests that it was banned because Delamere was the only person who was using the technique [11].Since the ruling authorities were unsure how to regulate the new technique, the simplest solution was to impose a blanket ban on the somersault jump at all sporting events, which would not affect anyone apart from Delamere.This is to be contrasted with the Fosbury Flop, which was adopted universally by all high jumpers after it was introduced at the 1968 Summer Olympics.The other major justification for is the potential danger of flipping and possibly landing head-first in a sand pit.Delamere makes the argument that if a somersault long jump is dangerous, then the ruling authorities would also have to ban all gymnastics events.However, Delamere himself identifies a key problem that he was never certain on the technique for performing a front tuck because he was not trained as a gymnast.He also admits that he was uncertain about his exact position and orientation with respect to the ground during the jump and states that this is the reason why his flip jump did not break the existing record for the long jump, since he fell backwards on landing due to sub-optimal rate of rotation.This is an interesting point, since such a lack of awareness of the location of the ground forms a key component of the 'twisties', a phenomenon where gymnasts suddenly feel unable to maintain control of their body during aerial maneuvers in competition [12].The 'twisties' are considered to be very dangerous, since they greatly reduce the gymnast's chance of landing a maneuver safely, which could in turn lead to a serious injury.As an example, Simone Biles (considered by most sources to be the greatest gymnast of all time) withdrew from the women's artistic team all-around event at the 2020 Summer Olympics because of a case of the 'twisties' which led her to believe that she would not be able to complete her routine safely.If professional gymnasts do not consider it safe to perform somersaults without being completely certain about the orientation of their body with respect to the ground, it seems reasonable that long jumpers who are generally not trained as gymnasts should not be allowed to perform front somersaults into a sandpit, especially when the somersault is performed from such a high take-off speed.Even if a long jumper trains in gymnastics or practices the technique, there are still specific dangers associated with a flip jump.The result of a bad take-off in competition is mostly innocuous when using a regular technique but could lead to an athlete under-rotating and landing on top of their head when performing a front somersault.As we have stated in the previous section, the balance of horizontal and vertical impulses could be quite delicate in a flip jump such that it would be easy to under-rotate or overrotate, potentially leading to a critical injury.

Conclusions
In this article, we have discussed possible reasons as to why a flip long jump should be superior to a regular long jump, suggesting that the improved landing efficiency justifies the technique.However, the justification in terms of angular momentum is quite simplistic and further investigations may be necessary to clarify the exact relationship between angular momentum and landing efficiency [8].This could be used as a good opportunity to explain to students that simple physical models may provide useful insights but give an incomplete picture of what happens in the real world.In particular, the differences in the biomechanics between the take-off for both regular and flip techniques are complicated such that it is hard to definitively claim that the flip technique is superior, even if an argument based on angular momentum suggests that it is.Detailed simulations found that the flip jump cannot be deemed superior, since one can only produce the rotation necessary for a full flip at the cost of having to decrease the vertical reaction force.This reduction of vertical impulse then reduces the take-off angle, leading to a decreased horizontal distance which cannot be easily compensated for without going to the limits of human performance [13].
This finding is supported by the empirical fact that no professional long jumper (including Delamere himself) has ever used a flip jump to equal the distance which they were able to achieve in competition using a regular long jump technique, let alone exceed it.Supporters of the technique claim that this is only due to lack of practice and that with refinements and focussed training long jumpers could easily surpass the distance possible with one of the regular techniques [10].Added to this is the fact that the take-off stride which is necessary to produce sufficient rotation would have an increased horizontal braking, again potentially decreasing the horizontal distance covered during the jump [14].We hope that this general discussion will help to illustrate to students the 'messiness' of physics and the fact that one needs to be careful drawing conclusions based only on simple physical principles when there are a range of other complex factors involved, especially those involving biomechanics and physiology of individual athletes.

Figure 3 .
Figure 3. Trajectory of a long jump plotted for SC D = 0.36 (blue curve) and SC D = 0.18 (red curve).

Figure 4 .
Figure 4. (Left) The loss of distance dxL due to landing of pelvis behind the theoretical point where the centre of mass was supposed to land and (right) the loss of distance dxL due to athlete falling backward during landing [8].Reproduced from [8].CC BY 4.0.

Figure 5 .
Figure 5.(Left) Athlete during take-off with the coordinate system which is used to calculate angular momentum[10] and (right) athlete during landing[8].Reproduced from , with[10] permission from Springer Nature.