Arsène Chemin

Pauline Besserve

Aude Caussarieu

Nicolas Taberlet

come

Magnetic Cannon: the physics of the Gauss rifle Arsène Chemin, [START_REF] Rabchuk | The Gauss Rifle and Magnetic Energy[END_REF] Pauline Besserve, [START_REF] Rabchuk | The Gauss Rifle and Magnetic Energy[END_REF] Aude Caussarieu, [START_REF] Rabchuk | The Gauss Rifle and Magnetic Energy[END_REF] Nicolas Taberlet, [START_REF] Rabchuk | The Gauss Rifle and Magnetic Energy[END_REF] and Nicolas Plihon 1, a) Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, Département de Physique, F-69342 Lyon, France (Dated: 13 December 2019)

The magnetic cannon is a simple device that converts magnetic energy into kinetic energy: when a steel ball with low initial velocity impacts a chain made of a magnet followed by a few other steel balls, the last ball of the chain is ejected at a much larger velocity. The analysis of this spectacular device involves understanding of advanced magnetostatics, energy conversion and collision of solids. In this article, the phenomena at each step of the process are modeled to predict the final kinetic energy of the ejected ball as a function of a few parameters which can all be experimentally measured.

I. INTRODUCTION

A. What is a magnetic cannon?

The magnetic cannon, sometimes referred to as the Gauss rifle is a simple device that accelerates a steel ball through conversion of magnetic energy into kinetic energy [START_REF] Rabchuk | The Gauss Rifle and Magnetic Energy[END_REF][START_REF] Kagan | Energy and Momentum in the Gauss Accelerator[END_REF][START_REF] Chittasirinuwat | More fun and curiosity with magnetic guns in the classroom[END_REF][START_REF] Ucke | Die Magnetkanone[END_REF] . The energy conversion at work is reminiscent of other electromagnetism-based accelerating device, such as rail-guns [START_REF] Starr | A low voltage "railgun[END_REF] . Figure 1 shows a time sequence (from top to bottom) of a typical setup where a line of four balls (the first one being a permanent magnet) is resting on a rail. When an additional ball approaches from the left with a low initial velocity, it experiences an attractive magnetic force from the magnet, collides with the magnet, and the final ball on the right is ejected at high velocity. Note that, to highlight the various sequences in Figure 1, frame-times are not equi-spaced. The video from which these frames have been extracted is provided as a supplementary material. To understand the physics of the Gauss rifle, the process may be divided into three phases: (i) acceleration of the ferromagnetic steel ball in the magnetic field created by the magnet (frames I to III in Figure 1), (ii) momentum propagation into the chain of steel balls which is similar to the propagation in the Newton's cradle (frame IV), (iii) ejection of the final ball escaping the residual magnetic attraction (frames V and VI). While Figure 1 highlights a specific example, we will focus on the more general case described in Figure 2, where the initial chain is formed of n steel balls in front of the magnet and m balls behind (each ball of mass M having a radius R). The magnet is a strong Nd-FeB permanent magnet which is maintained on the rail, strongly enough to prevent the chain from moving in the leftward direction in the acceleration phase, but loosely enough to allow momentum propagation in the chain (a patch of putty can be observed in Figure 1). Note that the dipolar axis of the magnet is spontaneously aligned with the axis of the steel ball chain to minimize potential energy.

a) Electronic mail: nicolas.plihon@ens-lyon.fr The acceleration phase is governed by the magnetic field created by the magnet [START_REF] Castañer | The magnetic dipole interaction as measured by spring dynamometers[END_REF][START_REF] Derby | Cylindrical magnets and ideal solenoids[END_REF] and the magnetization of the impacting steel ball. The determination of the magnetization of the incoming ferromagnetic steel ball differs from classical problems in which material (dia-or paramagnetic balls 8 ) or geometry (large sample of ferromagnetic materials [START_REF] Prasad | An Experiment to Determine the Relative Permeability of Ferrites[END_REF][START_REF] Borin | Measuring magnetic properties of ferromagnetic materials[END_REF][START_REF] Saslow | How a superconductor supports a magnet, how magnetically 'soft' iron attracts a magnet, and eddy currents for the uninitiated[END_REF] ) are different from ours. The gain of magnetic energy during the acceleration phase is called U n to indicate the dependence on the number of balls screening the magnetic field of the magnet. Part of this energy is converted into kinetic energy, which will add up to the initial kinetic energy of the incoming ball K init , resulting in an impacting kinetic energy K impact . The energy transfer in the chain is reminiscent of the Newton's cradle [START_REF] Herrmann | How does the ball-chain work?[END_REF][START_REF] Hutzler | Rocking Newton's cradle[END_REF] , governed by Hertzian contact forces involv-

Incoming ball

Ejected ball

Magnet n balls m balls soliton propagation

x x=0 K init U n K impact K eject -U m-1 K final impact FIG.
2. Schematic of the sequences described in Figure 1 and associated kinetic K and (potential) magnetic U energies. Red arrows represent conversion between magnetic energy and kinetic energy.

ing dissipation 14 in an inhomogeneous chain containing a magnet. Part of the impacting energy is transmitted through the chain, resulting in a kinetic energy K eject . Finally, in the expulsion phase, the final kinetic energy of the ejected ball K final is given by the energy transmitted through the chain decreased by the energy required to escape the residual magnetic attraction on the righthand side of the chain, U m-1 . The goal of the present article is to relate K final to K init and the parameters of the system. Note that the energy conversion process at lead in this problem is not in contradiction with the fact that magnetic field do not work on charged particles (the Lorentz force being perpendicular to the charged particles velocity). The analysis of a situation similar to the one investigated here is discussed in details in Griffiths's textbook [START_REF] Griffiths | Introduction to Electrodynamics[END_REF] ; interested readers could also refer to the discussion about magnetic energy provided in Jackson's textbook [START_REF] Jackson | Classical Electrodynamics[END_REF] (p. 224 and following) All effects related to the conversion of magnetic energy into kinetic energy are studied in section II. The transmission of kinetic energy through the chain is then detailed in section III. Finally, parameters influencing the global energy conversion of the system and the understanding of a succession of Gauss rifles are discussed in section IV.

B. International Physicists Tournament

The work presented here was done in preparation for the International Physicists Tournament (http:// iptnet.info/), a world-wide competition for undergraduate students. Each national team is composed of six students who work throughout the academic year on a list of seventeen open questions and present their findings during the tournament.

Unlike the typical physics exam, the problems must not only be presented, but also challenged and reviewed by the other participants allowing students to respectively assume the roles of researchers, referees and editors. In addition to the challenge that the tournament represents, it provides students with an exciting and eye-opening experience in which they learn how to design experiments with the aim of solving physics problems, and to constructively criticize scientific solutions.

The authors would highly recommend participation in the IPT as a rare learning opportunity for undergraduate students.

C. What students can learn from this problem

At introductory physics level, this experiment could be used to foster student's motivation while working on an open problem involving energy conservation. In a more classical laboratory work, students could reinvest their knowledge on magnetism to find out whether the incoming steel ball is to be considered as a permanent or an induced magnet through magnetic force and magnetic field measurements. At graduate level, the question of the dependence of the final velocity on some of the parameters of the system might lead to a few days experimental project work in which knowledge on mechanics, magnetism and non linear physics can be reused.

II. FROM MAGNETIC ENERGY TO KINETIC ENERGY

In this section we investigate the conversion of magnetic energy U n into kinetic energy in the acceleration phase, as well as the symmetric problem of the decrease of kinetic energy by U m-1 in the ejection phase. Direct measurement of the spatial dependence of the magnetic force exerted by the magnet on the incoming (ejected) ball and of the magnetic field are in agreement with a permanent dipole/induced dipole modeling.

A. Force exerted by the chain on a steel ball: a permanent/induced dipole interaction Let us first focus on the direct measurement of the magnetic force exerted on the steel ball. Its spatial evolution is measured using a weighing scale mounted as a Newton-meter, as sketched in Figure 3. A steel ball is attached to a heavy plastic block resting on the scale below the magnetic cannon chain. The steel ball attached to the scale is subject to its weight in the downward direction and to a magnetic force in the upward direction. The evolution of the force exerted by the magnet as a function of the distance d between the center of the ball attached to the scale and the center of the magnet, derived from the apparent mass, is displayed in Figure 3. We observe that in the presence of one or two steel balls in front of the magnet, the force is screened but is nonetheless larger than it would have been with a chain of non-magnetic balls, since the ferromagnetic balls channel the magnetic field. A second observation, not illustrated here, is that the number of balls behind the magnet has no noticeable influence on the force exerted on the opposite side. We will now establish the relation between this magnetic force and the magnetic field created by the magnet, in a way similar to that developed by Jackson [START_REF] Jackson | Dancing paperclips and the geometric influence on magnetization: A surprising result[END_REF] , and show that the permanent/induced dipole assumption is accurate. The evolution of the intensity of the field along the magnet's axis as a function of the distance d from its center is shown in Figure 4. The magnetic field intensity was measured using a Bell 7030 Gaussmeter but this could also be achieved using cheap and easy to implement integrated electronic devices ? . This evolution scales as d -3 as expected for a dipolar magnetic field. The magnetic field outside a uniformly magnetized sphere being exactly that of a point dipoleat the center of the sphere [START_REF] Jackson | Classical Electrodynamics[END_REF] , this scaling remains valid arbitrarily close to the magnet surface and the dipole strength of the magnet, M 0 , can be determined according to

B(d) = µ 0 M 0 2πd 3 (1)
where µ 0 is the vacuum magnetic permeability. The best fits according to Equation 1 up to d ∼ 0.2m are shown as full black lines in Figure 4 and lead to M 0 = 3.64 ± 0.1 Am 2 . The origin of the magnetic force experienced by the ferromagnetic steel ball lies in the interaction of the magnetic field created by the magnet and the magnetization of the steel ball. Assuming that the steel ball has an induced magnetic moment m ball (d), the force reads

F M (d) = -∇ (m ball (d) • B(d)) (2)
Equation 2 clearly shows a dependence on the magnetization properties of the steel ball. If m ball (d) is constant and independent of d (i.e. the steel is at saturation), F M (d) is expected to scale as d -4 as for an interaction between permanent dipoles 6 . On the other hand, if m ball (d) ∝ B(d), F M (d) is expected to scale as d -7 as for an interaction between a permanent dipole and an induced dipole. The experimental data presented in Figure 3 are displayed in logarithmic scales in Figure 5 and are consistent with a d -7 scaling when n = 0. This justifies the hypothesis of an induced magnetization proportional to the magnetic field, or equivalently a permanent/induced dipole interaction. A precise computation of the ball magnetization is a rather difficult task, since the field is highly inhomogeneous over the ball volume and the (unknown) magnetic permeability of the steel is expected to play a leading role. However, we will show in the following that a simple model correctly describes our experimental data. Let us first recall a classical result of magnetostatics (refer to p. 199 in textbook [START_REF] Jackson | Classical Electrodynamics[END_REF] ) which gives the magnetic moment of a sphere of relative magnetic permeability µ r immersed in a constant and homogeneous magnetic field B 0 as

m ball = 4πR 3 3µ 0 3(µ r -1) µ r + 2 B 0 . (3)
leading to a magnetic field intensity inside the ball 3µ r B 0 /(µ r + 2). In other words the magnetic field is amplified inside the sphere, by a factor 3µ r /(µ r + 2). In the case of soft steel, one expects values of µ r in the range [50 -10 4 ], which leads to a maximum three-fold increase. Let us make a crude approximation and now assume that Equation 3 remains valid in our configuration where the magnetic field created by the magnet is strongly inhomogeneous. Using the value of the magnetic field at the center of the steel ball, this leads to the following approximation of the magnetic force experienced by the steel ball:

F M (d) = - 4πR 3 µ 0 µ r -1 µ r + 2 ∂B 2 ∂d ∼ µr 1 - 4πR 3 µ 0 ∇ ∂B 2 ∂d . (4)
Using the dipolar model for the magnetic field (Equa-tion 1), the force can be conveniently expressed as:

F M (d) = - 6µ 0 R 3 M 2 0 πd 7 µ r -1 µ r + 2 ∼ µr 1 - 6µ 0 R 3 M 2 0 πd 7 . (5)
Figure 5 shows the spatial evolution of the force according to Equation 4 (black squares) and Equation 5(solid black line) assuming µ r 1. Our simple model is in very good agreement with the direct measurement of the force. When one or two balls are in front of the magnet (i.e. n = 1, 2 in Figure 5), establishing a theoretical expression of the magnetic field created by the magnet and channeled through the balls is beyond the scope of the present article. However, Figure 5 shows that the estimate of the magnetic force given by equation 4 is in close agreement with the direct measurement.

B. Conversion of magnetic energy into kinetic energy

The available magnetic energy in the presence of n steel balls in front of the magnet is computed from the above spatial evolution of the magnetic force as

2(n-1)R -∞
F M (x)dx; the upper limit of the integral being the minimum approaching distance of the center of the incoming ball from the center of the magnet. This can be done either by integration (i) of the measured force profile, (ii) of the force expressed as a function of the gradient of the square of the magnetic field profile according to equation 4, which reads U n = 4πR 3 B 2 (2(n + 1)R)/µ 0 . Table I summarizes the estimations of the available magnetic energy according to these computations for several values of n. As expected from the previous subsection, a very good agreement is observed between these values, and the available magnetic energy can be conveniently computed from the magnetic field measurement.

Un n = 0 n = 1 n = 2 From force meas. 72 ± 3 mJ 7.2 ± 1 mJ 1.4 ± 0.2 mJ From field meas.

75 ± 25 mJ 6.4 ± 1 mJ 1.6 ± 0.4 mJ TABLE I. Available magnetic energy estimated from direct magnetic force measurement, or from a permanent/induced model involving the direct magnetic field measurement. The larger errors reported on last line lie in the low spatial resolution of the direct magnetic field measurements.

Note that a third computation of U 0 can also be performed by the integration of equation 5 when n = 0 as U 0 = µ 0 M 2 0 /(64πR 3 ) = 95 ± 5 mJ. This larger estimate can be understood from an overestimation of the magnetic field in the vicinity of the magnet using the dipolar approximation (as expected the dipolar approximation is not valid close to the magnet). Since most of the acceleration occurs very close to the magnet, this leads to an overestimate of 25% of the available magnetic energy. However, as shown below, the expression of the force given by equation 5 is useful to predict the time evolution of the speed of the impacting ball.

A partial conclusion can be drawn here for the optimization of the magnetic cannon. U n and U m-1 represent respectively the gain and loss of magnetic (potential) energy. As U k strongly decreases with k, the maximum increase of magnetic energy is achieved for the lowest value value of n, i.e. n = 0, while the minimum losses are obtained for large values of m. The optimization of the number m of balls behind the magnet will be addressed in section IV.

Let us now consider the conversion of the available magnetic energy U n into kinetic energy. The incoming ball is subject to magnetic and friction forces. Friction forces may however be neglected: the magnetic force is of the order of 10 N, while for M = 28 g balls and a maximum velocity of order of 3 m.s -1 , the friction force is estimated around 0.03 N and the viscous drag force around 10 -4 N. Because the magnetic force strongly increases as the distance between the incoming ball and the magnet decreases, most of the acceleration occurs in close vicinity of the magnet. As a consequence, the magnetic energy is mostly converted into translational energy and the rotation of the incoming ball can be neglected.

The work-energy theorem leads to an evolution of the ball velocity ẋ as

ẋ(x) = 2 M x -∞ F M . dx + ẋ2 -∞ (6)
where ẋ-∞ is the initial velocity at large distance from the magnet (the initial velocity being null in the case of a single cannon, but may be non-zero when several successive rifles are investigated, as in subsection IV B). At FIG. 6. Spatial evolution of the incoming ball velocity (n = 0): experimental values from high-speed camera (dots), permanent/induced model (solid red curve) and final velocity estimated from integration of the magnetic force measurement, assuming a complete conversion of magnetic energy into kinetic energy (blue diamond).

the impact, this expression reads K impact = K init + U n , i.e. a unity conversion factor. The assumptions leading to this simplified expression have been experimentally verified with n = 0 and ẋ-∞ = 0. A high-speed camera (8000 frames per second) images the incoming ball during the acceleration phase. The velocity of the ball is computed as the derivative of the position of the center of the ball, extracted using ImageJ, a free software developed by NIH. Figure 6 displays the experimental velocity of the ball (dots) and the theoretical curve predicted by equation 6 using the dipole approximation of equation 5 (solid red line). The impact velocity computed from the integration of the direct force measurement (and assuming a complete conversion magnetic energy into kinetic energy) is also displayed as the blue diamond symbol. The good agreement of the measured final velocity with these estimates shows that friction may indeed be neglected and that the available magnetic energy is fully converted into translational kinetic energy. The red curve of Figure 6 shows that the permanent/induced dipole hypothesis enables to correctly predict the evolution of the impacting ball velocity; however, this model slightly overestimates the velocity, as expected from the overestimate of the available magnetic energy discussed above.

As a partial conclusion here, we showed that the whole available magnetic energy from the attraction of the magnet is converted into kinetic energy. Moreover, we provided a simplified expression of the force exerted by the magnet on the steel ball which leads to a theoretical expression of the magnetic energy U n . Similar arguments can be applied in the ejection phase, where the kinetic energy of the ejected ball is given by K final = K eject -U m-1 .

III. NESTERENKO SOLITON: FROM NEWTON'S CRADLE TO GAUSS CRADLE

Following the impact of the incoming steel ball, the energy propagates in the ball chain similarly to what occurs in Newton's cradles [START_REF] Herrmann | How does the ball-chain work?[END_REF][START_REF] Hutzler | Rocking Newton's cradle[END_REF] . However, in the magnetic cannon the chain is inhomogeneous (presence of a sintered NdFeB magnet). This section develops a classical model based on Hertzian contact and discusses briefly the Nesterenko soliton. Experimental yields accounting for the presence of the magnet are then presented.

A. Nesterenko soliton: propagation of a non-linear wave

Let us consider a chain of N balls of radius R allowed to translate along the x-axis and let the position of the balls be x i (see Figure 7 (a)). The force acting between two balls in contact is given by the Hertz law 18 :

F = E √ 2R 3(1 -ν 2 ) (x i -x i+1 -2R) 3/2 ,
E and ν being respectively the Young and Poisson moduli of the ball material.

In the case of the steel balls used in our experiments, an upper bound of the compression δ = x i -x i+1 -2R of the balls during the propagation of the wave can be estimated assuming equality of the compression energy 2E √ 2Rδ 5/2 /(15(1 -ν 2 )) with the kinetic energy of the impacting ball (of the order of 0.05 J). The corresponding force is of the order of magnitude of 1000 N. This largely exceeds the magnetic force experienced by the steel balls (even when in direct contact with the magnet, typically 25 N). This demonstrates that the properties of the solitary compression wave are unaffected by the magnetic forces acting within the chain (although obviously the velocity of the impact ball strongly depends on the magnetic force).

Knowing the forces acting on each ball, the equations of motion can be numerically solved. Figure 7(b) shows the numerical solution for a chain of 5 steel balls (E = 210 GPa, ν = 0.3) of radius R = 9.5 mm. This graph shows several interesting features: (i) the final velocity of the impacting ball is negative, i.e. it experienced a rebound (note that unlike in the experiments, the first ball of the chain is unconstrained). (ii) The final velocity of the second-to-last ball is non-zero (although small), i.e. several balls can be ejected. Note that these features are visible in actual Newton's cradle. (iii) Since in this simple case the total energy is conserved, the velocity of the last ball is less than that of the impacting ball (roughly 98.7%). The main conclusion is therefore that, even without any dissipation, the transmitted energy of the ejected ball K eject is lower than the kinetic energy of the impacting ball K impact and a yield η = K eject /K impact should be introduced. For the specific case displayed in As a side-note, it is worth mentioning that an analytic solution to the continuous limit of the equation of motion was given by Nesterenko [START_REF] Sen | Discrete Hertzian chains and solitons[END_REF][START_REF] Nesterenko | Propagation of nonlinear compression pulses in granular media[END_REF] :

∂x ∂t = A sin 4 x -ct L .
This analytic solution is in excellent agreement with the numerical solution (see Figure 7). Remarkably the dispersion is counter-balanced by the non-linearities and this solitary wave travels without distortion. Among other interesting features, it should be mentioned that the spatial extension of the soliton is constant (L 10R) while its velocity c increases with increasing amplitude A.

The duration of the propagation for a given chain therefore decreases with increasing velocity of the impacting ball.

B. Experimental measurement of the energy transmission yield η

The above model does not account for any source of energy dissipation. An experimental determination of the yield η = K eject /K impact is required to derive a global energy balance of the magnetic cannon. Figure 8(a) shows the experimental setup used: the energy of the impacting ball is controlled via the launching height of a pendulum hitting the chain (with no magnet) and the energy of the ejected ball is computed from its impact position on the ground after falling from a table. Similarly to the above mentioned model, none of the balls are fixed. Figure 8(b) shows the measurements of the energy transmitted to the ejected ball K eject as a function of K impact . Several interesting features should be emphasized: (i) the energy transmitted to the ejected ball is proportional to the energy of the impacting ball, (ii) the yield η decreases with the length of the chain, (iii) the experimental yield is lower than 0.975 due to the numerous sources of dissipation: for a chain of five balls, the experimental yield is only about 0.83. The experimental determination of the yield is compatible with an evolution as η = η 0 -0.024(n + m + 1), with η 0 = 0.95 in the case of a chain of steel balls. This result shows that dissipation sources (viscous and solid friction, deformations or imperfect contacts between the balls in the chain) cannot be neglected.

Moreover, in the magnetic cannon, the presence of the magnet does not only introduce an inhomogeneity, but also a magnetic field that magnetizes the steel balls, leading to a strong attraction between the balls -which prevents for instance the rebound of the impacting ball. The strength of the magnetic field does not modify the physical principles at lead in the Nesterenko soliton propagation: the maximum intensity of the magnetic force for a 600 mT field, is three orders of magnitude below the mechanical compression forces. Moreover, the sintered NdFeB magnet introduces an inhomogeneity with distinct mechanical properties (Young and Poisson moduli and density) which may cause a higher dissipation due to the sintered structure of the magnet. The presence of an intruder (the magnet) in the chain also triggers a partial reflection of the wave. In order to estimate this effective dissipation, the experimental setup has been modified as represented in Figure 8(c). Figure 8(c) shows the evolution of the transmitted kinetic K trans as a function of the impacting kinetic energy K impact for a 7-balls chain with and without a magnet. Note that the increase/decrease of energy from magnetic acceleration/deceleration has been taken into account in the computation of the energies. Yet, the insertion of the magnet leads to an impressive yield drop from 81% to 44%, leading to η 0 ∼ 0.61.

IV. OPTIMIZATION OF THE MAGNETIC CANNON

A. Optimization of a single magnetic cannon

As previously stated, the optimal energy gain is obtained with no ball on the left of the magnet (or n = 0), and strongly depends on the properties of the magnet. The optimal configuration also requires to lower the loss of magnetic energy U m-1 , which is obtained with a large number of balls on the right of the magnet (or m 1), and to maximize the yield η of the chain, which requires to minimize the total number m + n + 1 of balls in the chain. The details provided in the previous sections enables us to express the kinetic energy of the ejected ball as a function of the initial kinetic energy and the properties of the system as

K final = η(n + m + 1) [K init + U n ] -U m-1 (7)
The maximal value of K final is obtained when the total losses [1-η(n+m+1)]U n +U m-1 are minimized. Figure 9 shows the evolution of the normalized losses as a function of m for n = 0 (in this configuration, no ball is ejected when m = 1). A weak minimum is observed for m = 3; for values of m larger than 3, losses weakly increase with m. This Figure shows that the loss of magnetic energy U m-1 can be neglected when m > n + 2. The optimal configuration, i.e. maximizing the kinetic energy of the ejected ball, is thus obtained for no balls in front of the magnet and three balls behind the magnet (or n = 0, m = 3), which is the configuration displayed in Figure 1.

The kinetic energy of the ejected ball may be conveniently expressed for a spherical magnet of radius R when neglecting U m-1 in equation 7. The dipolar moment of the magnet can be estimated as M 0 ∼ 4πR 3 B r /(3µ 0 ) with B r the residual flux density (of the order of 1.27 T for a grade N40 NdFeB magnet), which leads to

K final ∼ (η 0 -0.024(m + 1)) πR 3 B 2 r 36µ 0

B. Maximal acceleration achievable using N successive rifles

Once the optimization of one single magnetic rifle has been achieved, a natural question arises: to what extent is it possible to increase the ejected kinetic energy by using a succession of several rifles?

Let us now focus on a configuration with N successive identical magnetic rifles: the ball ejected from rifle i will be accelerated by rifle i+1 according to equation 7. When neglecting losses between two successive rifles, the kinetic energy of the last ejected ball reads

K final (N ) = η N U 0 init + U n N i=1 η i -U m-1 N -1 i=0 η i (8)
The kinetic energy of the ejected ball increases with the number of modules, but since the energy gain is con- stant and independent of the initial velocity while the dominant source of energy losses -through the propagation of the soliton within the chain -is proportional to the impacting energy, there is a maximum achievable kinetic energy

K max = ηU n -U m-1 1 -η (9)
This saturation has been experimentally observed using a chain of 10 rifles composed of one magnet followed by m = 3 balls and separated by 10 cm -note that for this specific setup, balls have a radius of 4 mm. Velocities of the ejected ball were estimated from sound recording of the successive shocks between the ball ejected from rifle i and the magnet of rifle i + 1 and is displayed in Figure 10. This low-cost technique allows to provide the time-average velocity between two successive rifles. The experimental evolution is consistent with the prediction given above, neglecting U m-1 , using U 0 as determined by the direct force measurement and a yield η = 0.6.

V. CONCLUSION

In conclusion we were able to successfully model the kinetic energy of the ball ejected from a magnetic cannon. In the acceleration phase, experimental data show a very good agreement with a simple model of uniform magnetization of a sphere plunged in the magnetic field created by the magnet. In the case where no steel balls are present between the incoming ball and the magnet, this magnetic field is accurately modeled as a dipolar magnetic field. We also provided a model based on Hertzian contact and solid collisions accounting for the propagation of momentum in the chain of balls and determined an experimental effective yield accounting for its efficiency. These ingredients enable us to predict the final kinetic energy as a function of the parameters of the system (geometrical sizes, magnetic properties ) for a single magnetic cannon or an assembly of several modules. Some limitations of our work could however deserve further analysis and modeling. In the presence of steel balls between the magnet and the incoming ball (i.e. n ≥ 1), we derived the magnetic force as a function of the spatial evolution of the magnetic field. However a detailed modeling of the "channeling" of the dipolar field created by the magnet within ferromagnetic balls would make it possible to predict more precisely the energy gain in the acceleration phase. The study of the behavior of the magnetic cannon using paramagnetic or superparamagnetic materials instead of ferromagnetic materials could also be envisioned. The modeling of momentum propagation described in the present article is very similar to the Newton's cradle. Including the cohesion forces from the magnetic field as well as an effective behavior of the sintered NdFeB magnet could be an extension of this work.

FIG. 1 .

 1 FIG.1. Time sequence (from top to bottom) showing acceleration in the magnetic cannon. A line of three steel balls initially stuck to the right of a strong permanent magnet, loosely fixed on a rail, is impacted by a steel ball coming from the left hand side and accelerated in the magnetic field of the magnet. Momentum is transferred into the chain and the last ball on the right is ejected at high velocity. Images from a 2048x360 pixel film acquired at 400 frames per second.

FIG. 3 .

 3 FIG.3. Force measurement setup and spatial evolution of the magnetic force experienced by the steel ball attached to the scale as a function of the number n of steel ball in front of the magnet.

3 )FIG. 4 .

 34 FIG.4. Spatial evolution of the magnetic field created by the magnet -the inset shows B -1/3 (d) -and associated dipolar best fits (black lines).

FIG. 5 .

 5 FIG.5. Spatial evolution of the magnetic force for n = 0, 1, 2: direct force measurements (colored bullets), force computed from magnetic field measurements according to Equation4(black squares) and according to equation 5 for n = 0 (solid black line) showing the validity of our permanent/induced dipole model.

FIG. 7 .

 7 FIG. 7. Time evolution of the velocity of each ball normalized by the incoming velocity. The last ball is ejected at a slightly lower speed that the impacting ball and all other balls move. The open circles represent the analytic solution of the solitary wave.

Figure 7 (

 7 Figure 7(b), the numerical simulation provides an estimate of η = 0.975.As a side-note, it is worth mentioning that an analytic solution to the continuous limit of the equation of motion was given by Nesterenko19,20 :

FIG. 8 .

 8 FIG. 8. (a) Diagram representing the setup for the energetic yield of the chain. The impacting energy is controlled thanks to a pendulum and the ejected energy is measured after a fall. (b) Energy of the ejected ball as a function of the energy of the impacting ball for different chain lengths without magnet. (c) Diagram representing the setup for the energetic yield of the chain with a magnet. (d) Energy of the ejected ball as a function of the energy of the impacting ball for a five balls length without magnet and with a magnet.

FIG. 9 .

 9 FIG.9. Evolution of losses normalized to U0 as a function of m: loss of magnetic energy (blue circles), loss within the chain (red squares) and total losses (black diamonds).

6 FIG. 10 .

 610 FIG.10. Time-average velocity of balls ejected by successive identical rifles (estimated from sound recording) and model with η = 0.6.

ACKNOWLEDGMENTS

This work was supported by the École Normale Supérieure de Lyon and Univ Claude Bernard, Lyon, France. The organization committee of the International Physicist Tournament is gratefully acknowledged, as are contributions of the the French Academy of Sciences and all the other tournament's partners. We acknowledge also A. Bourges, C. Gouiller, A. Guittonneau, C. Malciu, G. Panel and J. Sautel with whom discussions and exchanges were prolific.


Sevinç Göksen-Zayim

Derk Pik

Rijkje Dekker
email: r.dekker@uva.nl

Carla Van Boxtel
email: c.a.m.vanboxtel@uva.nl

Mathematical modeling and the role of language proficiency

Keywords: Mathematical modeling, conceptualization, mathematization, language proficiency

Theory

Modeling is defined as "a practical and creative process in which realistic problems are translated into mathematical form" (cTWO, 2007, p. 25). Usually, modeling problems in context-rich assignments are offered to learners through texts. One of the first obstacles students encounter is reading and interpreting text. In secondary school, being able to read a problem is a decisive factor in solving a problem (Korhonen, Linnanmäki, & Aunio, 2012). The language used at school often forms an obstacle to learning mathematics [START_REF] Van Eerde | The integration of mathematics and language learning in multiethnic schools[END_REF]. It is thus to be expected that language is an important factor at the various phases of the modeling process, although the role of language proficiency may be different. To analyse the different difficulties concerning language, we have to split up the modeling process in different phases. According to [START_REF] Spandaw | Modelleren, van werkelijkheid naar wiskunde en weer terug[END_REF], modeling encompasses various activities that are carried out consecutively, as shown in Figure 1. The phase of conceptualizing: the student is able to understand the text in which the problem is posed in order to make the translation into a conceptual model.

The phases of mathematization and solving: the student goes from the conceptual model to the mathematical model. Then the student will try to solve the problem. Mathematical reasoning occurs in both of these phases.

The phase of interpretation: translating the mathematical model into reality appeals to language skills. This skill is similar, but inverse to the language proficiency in the conceptualization phase.

Mathematical modeling is absent in the greater part of lower secondary education in the Netherlands. For that reason, this study will emphasize modeling in lower secondary education. This study will answer the following research question: Which language problems occur in the three phases of mathematical modeling in context-rich assignments?

Participants of the study are students from grade six, eight and ten. These students are 11 up to 15 years and participated from four primary schools and four secondary schools located in an urban environment. 260 students performed two mathematical modeling tasks. The first modeling task is directed towards algebra and the other towards geometry. Each individual task consists of two parts. The first part is a modeling assignment in a rich context and the second assignment is a mathematical core assignment, focusing on the mathematical content without context. In consultation with the teacher of each participating class we selected two students (n=26), one with strong language proficiency and the other with weak language proficiency, with whom we performed a task based interview. Six teachers were interviewed about the levels of the designed modeling tasks. 12 students (two from each class, from 11 up to 15 years) were asked to do the assignment while thinking aloud.

Results

The analyses of students' answers, the task based interviews and the think-aloud protocols showed that language is an important obstacle, especially in the conceptualization phase. In all grades most of the students repeatedly reread the text of the modeling task. In many cases learners fail to construct a meaningful representation of the described situation. The transition from reality presented by the text towards a conceptual model often halts halfway. The outcome of the students' answers for the geometry modeling task did not show big differences between the rich context part and the mathematical core part, for all three grades. From the outcome of the task-based interviews with the learners, feasible components and levels for the different modeling phases are formulated. The think-aloud sessions gave more insight in the language problems students encounter while solving the modeling task and showed the strategy that students use.

Figure 1 :

 1 Figure 1: The modeling cycle (Spandaw & Zwaneveld, 2012, p. 240) Language proficiency may play a different role in every phase of the modeling cycle. The phases are indicated by:

Acknowledgment

This research is funded by The Netherlands Organisation for Scientific Research (NWO), number 023.009.047


Carolina Guerrero-Ortiz
email: c_cguerrero@yahoo.com.mx

Pre-service mathematics teachers' learning through designing modelling tasks

Keywords: Modelling task, future teachers, teacher training

The results of analysing the projects for the teaching of mathematics and modelling developed by future mathematics teachers are discussed in this paper. The participants were part of a seminar in which the objective was to develop their mathematical and pedagogical content knowledge related to the teaching of mathematics through the design of modelling tasks. With a qualitative methodology the written reports of and interviews with the pre-service teachers were analysed. The results bring to light some elements associated with their knowledge and how this knowledge is constructed in their experience with modelling activities and then reflected in the creation of teaching tasks.

Introduction

The design of novel tasks and teaching activities represents a challenge for teachers and future teachers. They must be able to involve students in meaningful activities and create environments in which students construct and reflect on their own learning. In designing modelling tasks, the future teachers' knowledge about the context and the mathematics related to it, joined with the knowledge about the teaching and learning of the students becomes structured and shapes the tasks. The design and implementation of modelling tasks has been subject of innumerable studies with attention to the characteristics of the final product; but few investigations consider the processes that are favoured in this activity [START_REF] Czocher | Mathematical Modeling Cycles as a Task Design Heuristic[END_REF], for example the construction of the teacher's pedagogical content knowledge.

When modelling activities are introduced to the classroom, teachers must adopt an adaptive view of new teaching methods. This involves a challenge that requires the mastery of mathematical knowledge associated with the context of the task and adapting that knowledge to the environment in which students will work with the task according to the learning objectives [START_REF] Doerr | Middle grade teachers' learning through students' engagement with modelling tasks[END_REF]. In the setting of a teachers' training seminar, we analyse: 1) how future teachers, when designing teaching tasks, construct and transform their mathematical knowledge associated with the modelling of population growth and 2) how this knowledge is adapted for teaching in a middle school mathematics course. The research question is: What elements associated with pedagogical content knowledge emerge in the design of modelling tasks for teaching mathematics? An important aspect in this study is the design process where participants learn to model to teach modelling. This fact is evidenced in the exploration of several situations and in how future teachers simplify and idealise the initial situations to adapt the task to the knowledge of the students.

Conceptual framework

When studying task design in mathematics education two perspectives can be identified: the design as implementation or the design as intention. The task design analysed here is characterized as design as intention since it addresses the initial formulation of the design [START_REF] Czocher | Mathematical Modeling Cycles as a Task Design Heuristic[END_REF] and the analysis is supported by constructs for studying teacher's pedagogical content knowledge [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF][START_REF] Doerr | Middle grade teachers' learning through students' engagement with modelling tasks[END_REF][START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF]. By teaching modelling it is intended that teachers become competent at modelling activities, they also should be able to link knowledge about modelling cycles, about goals and about types of tasks. Solving, analysing and creating tasks as well as planning and performing modelling lessons are important aspects here [START_REF] Borromeo Ferri | Mathematical modelling in teacher education -Experiences from a modelling seminar[END_REF][START_REF] Huincahue Arcos | El conocimiento de la modelación matemática desde la reflexión en la formación inicial de profesores de matemática[END_REF]. Designing and modifying tasks to be easier or harder for presenting mathematical ideas constitutes a part of the teacher's knowledge [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. When designing tasks for learning mathematics and modelling, the teacher's pedagogical content knowledge comes into play in teaching organization, in the understanding of how to present the concepts to favour learning and when understanding the mistakes associated with the domain [START_REF] Doerr | Middle grade teachers' learning through students' engagement with modelling tasks[END_REF]. In addition, the teacher must know the specifications of the curriculum and maintain a critical position to organize the learning and management the class [START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF]. The practice of modelling leads to the development of modelling skills [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF], so that when the teachers design a task, they structure their specialized content knowledge related to models with the intended knowledge for the students. On the other hand, when modelling students build knowledge depending on the characteristics and cognitive demands of the tasks [START_REF] Henningsen | Mathematical tasks and student cognition: Classroombased factors that support and inhibit high-level mathematical thinking and reasoning[END_REF]. Therefore, it is expected that the tasks created by future teachers support progress in the modelling abilities of the students. According to the model of three dimensions [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF], the modelling competences can be distinguished as: (1) Technical level, related to the kind of mathematics the students use and to how flexible they are in their use of mathematics.

(2) Radius of action, related to the domain of situations in which students can perform modelling activities and (3) Degree of coverage, according to the phase of the modelling process the students are working in, and the level of their reflections.

Context and method

In most Chilean teacher training programs, modelling has not been explicitly part of the education for future teachers. The future teachers take part in several courses of mathematics, didactics and pedagogy. They learn mathematical concepts and theories of education, but they have few experiences with modelling situations; as [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF] pointed out modelling is different than the nature of mathematics. In this context, it is described and analysed the process developed by a group of pedagogy in mathematics students, when they designed an activity for teaching. The experimentation was carried out with a group of six students, who participated in a research seminar for future teachers. Here, the case of a group with two participants is presented. They were selected because of their willingness to participate in the research. In the seminar some topics of differential (and integral) calculus and differential equations were discussed and among these functions, derivatives, logistic equations and systems of differential equations. Some perspectives of modelling [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF][START_REF] Borromeo Ferri | Theoretical and empirical differentiations of phases in the modelling process[END_REF] were also reviewed. During the seminar several discussions about the use of technology and the content knowledge for teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] were carried out. The students explored modelling different phenomena and proposed a teaching activity for middle school level (15 years old). The process of designing a task (for approximately three months) was documented, where participants studied how some growth situations are described in reality. Then, after simplifications and idealizations their initial study took shape of a teaching activity. The design phase considered aspects such as, idealization and exploration of the initial conditions, parameter values and the domain of the solutions. Meetings with students once a week during one semester were developed in where the design process was followed. At the end of the course, they presented the teaching task and a description of the process of designing it in a written report.

Data analysis was carried out following a qualitative perspective [START_REF] Cohen | Research methods in education[END_REF]. First the episodes related to the modelling phases along the process of designing the task were identified. Then, units of meaning associated with mathematics, modelling and pedagogical content knowledge were identified. At the end of the seminar participants were interviewed by two researchers to verify the first interpretations of the author. Finally, the units of analysis identified in the written report and those from the interview were contrasted and discussed in the modelling seminar, where the author took part. Thereby, the validity of the results was verified.

Discussion and analysis of results

In this section, the study carried out by the participants related to the phenomenon in question is initially presented. It is not the intention in this paper to inform about a common process of population growth modelling, however, in order to analyse the knowledge that emerges when future teachers design a modelling task, we present a summary of the facts that they recognise as relevant. It should be noted that most researches on teacher's knowledge do not explicitly address the role of modelling [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF][START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF] but consider it as an element of mathematical practice. Here modelling is posed as a precursor element of the development of future teachers' knowledge. Furthermore, the experiences with modelling activities can be related to the improvement of the mathematical knowledge of the future teachers. Subsequently the teaching task proposed by the participants was analysed where some elements from the pre-service mathematics teachers' pedagogical content knowledge were identified. Three episodes were recognized: first when participants explored the growth of penguin populations reported by the research literature, where the case was analysed by statistical methods. Second, when the relevant information was identified and linked with an idealised mathematical model. Third, when information gathered in the first and second episodes was adapted to create a task for teaching.

Study of real situation

After consulting literature related to different growth phenomena, a growth situation of Magellanic penguins was selected [START_REF] Pozzi | Population regulation in magellanic penguins: What determines changes in colony size?[END_REF][START_REF] Valdebenito | de Magallanes (Spheniscus magellanicus). Colonia Seno Otway [Census of the population of Magellanic Penguin (Spheniscus magellanicus). Colonia Seno Otway[END_REF]. The habitat places of this type of penguin were located on a map. In addition, the total population of penguins in the world was identified (1.5 million pairs of reproductive age). This led the future teachers to focus their attention on the population of penguins in Chile (700 pairs), particularly a population that is located on an ecological reserve that hosts around 400 specimens. Here participants observed the existence of infrastructure designed to allow tourism without affecting the reproduction of the penguins. This fact allowed them to transit from the study of a more general situation to the case of a growing population limited by a border. When doing the first exploration participants observed that studying a population in freedom, implies considering predators and death of penguins due to the fishing process. By exploring the reproduction of the population, they identified the nesting seasons, the number of eggs per pair (2) and the places where the nests were located. They also noticed the existence of other predators like seagulls that attack eggs. Considering this information, participants associated the successful breeding with the number of chicks that survive per nest on each breeding season. These considerations were initially associated with logistic equations and systems of ordinary differential equations (predator-prey model).

Then, an idealisation of the situation was made, when participants saw the case as population growth in a limited place where the necessities of feeding (anchovies, sardines and some crustaceans) also influence the size of the population. Thereby, pre-service teachers made relationships between the competition for limited resources in the environment and the population size; this led them to model the situation using the logistic equation of population growth. The elements used to determine the parameter values in the equations were: the reproduction age (the females reproduce between 4 to 5 years old, while the males reproduce when they are 6-7 years old), and the penguins are monogamous. The average life expectancy is about 20 years. With this information students calculated an approximation of the growth rate.

This phase was a support for pre-service teachers to: a) recognise the reasons why some simplifications are necessary, b) recognise and explain the frequent difficulties students find in mathematizing a situation and c) identify and explain the behaviour of some situations. In addition to the ideas of [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] it is assumed that those elements must be part of the teacher's mathematical knowledge for teaching modelling.

Simplification of the real situation and modelling

When simplifying the real situation, the participants made decisions that were reflected in the design of the task as will be discussed below. In summary, the literature gives information about the penguins' birth and mortality rates per year. Then by calculating an average students approximated this values by a constant (intrinsic rate of increase) to be used in the expressions to describe the population growth. From here, by comparing continuous and discrete exponential growth future teachers were aware of the complexity of modelling realistic situations. They also realised the limitations of modelling population growth with simple models. Hence, when pre-service teachers explored open modelling situations, they built their mathematical knowledge associated with the situation [START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF], and when they used that knowledge to design the task, they strengthened their pedagogical content knowledge related to modelling practices [START_REF] Doerr | Middle grade teachers' learning through students' engagement with modelling tasks[END_REF]. Thus, the experience they got by exploring the characteristics of the phenomena served as the basis for the design of the teaching situation.

Below some of the arguments from the participants regarding to the relevant information and the simplification they made are shown: Initial populationthe literature mentions that in 1993 a population of 400 penguins was estimated in the reserve.

Birth-[...] paired penguins lay 1 to 2 eggs, but not all penguins are in pairs and not all couples get two eggs, for this reason the birth rate can be considered to be 50%. However not all chicks are born, which leads us to consider a birth rate of 40% of the total population. This rate is consistent with the data in the literature […].

Mortality-[...] the longevity due to natural death of penguins is almost 20 years, in addition there are other factors that cause the death of this species [...], the spilled oil and the fishing nets, that add up to 16% mortality [literature information]. If [mortality] is adapted only to natural causes of death (Diseases, poisonings, etc.) an estimate of 14% can be made since the reserve has the necessary care to maintain the health of the penguins.

Predators-The presence of predators was not taken into account for the design of this task, since the penguins are in a reserve and are protected from these threats.

With this information the participants analysed two cases described by the equations y , getting the solutions y . When studying the graphs of these expressions they identified the behaviour of a population modelled by the logistic equation and described it as below:

[...] if the population is less than the carrying capacity, the graph behaves increasingly, but when it approaches (K) the graph begins to stabilise, instead if the population was greater than the capacity of load the graph behaves decreasingly and begins to stabilise when it is closer to K.

In this episode, stand out identification and simplification processes of relevant information and the participants' mathematical work using analytical and graphical representations. The experience gained studying the real situation helped future teachers to identify and anticipate appropriate mathematical representations to be used in the teaching activity. Those elements are related with the Specialized Content Knowledge [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF].

The teaching task

In this section the task designed by two participants is analysed. The learning objectives were defined as: Analysing the behaviour of a population based on birth and mortality rates in relation to the time. The task is the following:

In 1991 a massive death of penguins of the Magellanic species was evidenced. To revert the situation a group of people in 1993 decided to create a Natural Reserve of Magellanic Penguins called "Seno Otway". The reserve has capacity for hosting 6500 penguins. At the beginning the reserve started with an initial population of 300 Magellanic penguins. From a census they calculated a mortality rate of 70 per 500 penguins and a birth rate as 200 per 500 penguins.

Some of the questions that the future teachers posed are:

1) What do you think would happen if the population exceeds the capacity of the reserve?

2) Use the data for graphing the situation.

dN dt = kN dN dt = r 1- N k ae è ç ö ø ÷ N N t ( ) = Ze rt K Ze rt +1
3) Analyse what should happen when reaching the load limit.

Those questions show that teachers focused on the carrying capacity as the relevant aspect that affects the growth of the population. The first question demands an intuitive answer from the students, with the third question they are challenged to prove or reject their initial conjecture by studying the data. To answer the second question, students were encouraged to organizing the data about the time and the population in a table, using this information they could build a graphical representation. Here, the future teachers' knowledge related to the different registers in which a mathematical content can be represented [START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF] and their strength to interpret situations becomes evident. This is also related to the knowledge of some theories of learning as it is shown in the next fragment related to the characteristics of the task:

The activity is designed for the students to articulate the same information in different semiotic representations, and to look for a generality that represents the behaviour of the population [...]

The teachers' knowledge associated with the characteristics of modelling was also evident when they described the processes through which the student should transit when answering the questions. They anticipated that students would go through a spiral-shaped cycle increasing as they require more knowledge to answer the questions. This is shown in an extract of the written inform:

In order to solve successfully the task, it is necessary to carry out several modelling processes that respond to different situations, we could group the questions into three categories, easy (1, 2), medium (3, 4) and hard (5, 6); each of them corresponds to a separate modelling cycle that adds more difficulty as the students respond.

In the previous transcript the knowledge of pedagogical content was evidenced in the design of the task when teachers anticipated the students' reaction in relation to the difficulty in answering the items [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF].

While designing the task, future teachers recognised the importance of students being aware of the idealization processes involved in the analysis of a phenomenon. In the following fragment, when justifying the pertinence of the question number three, they referred to the idealization that implies modelling the situation through the differential equation and compared it with the case modelled by the equation :

[...] the students should reflect on the exponential growth of the population, because so far we have only analysed an ideal case, where no carrying capacity or predators in the area have been taken into account.

In relation to the graphical representation of the behaviour of the population, future teachers pointed out that working with graphics could generate some errors of interpretation when students superficially see the generated curves; this is associated with the knowledge of the structure of mathematics [START_REF] Carrillo-Yañez | The mathematics teacher's specialised knowledge (MTSK) model[END_REF]. They point it out as follows:

They [students] may build a more flattened exponential, arguing then that the growth population should be slower.

dN dt = kN dN dt = r 1- N k ae è ç ö ø ÷ N
In the previous analysis a strong relationship between the (mathematical and extra-mathematical) knowledge constructed by future teachers when experiencing modelling to design a teaching task was observed. And the pedagogical content knowledge was evidenced when they were able to anticipate the students' actions with the modelling task. The task was designed to cross through the modelling cycle in different ways. However, the activity was subordinated to only working with the data in the tasks [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF].

Comments and conclusions

It was evidenced that in the challenge of designing teaching tasks the future teachers experiment with mathematical and extra-mathematical knowledge and bring to light what they learn for planning the teaching. The future teachers' learning endured a transformation from studying pure mathematics to study the mathematical and extra-mathematical content associated with several phenomena. When exploring the situation, the participants' work highlighted activities as identifying relevant information, simplifying the situation, formulating a task, systematising, mathematising and analysing mathematically. Those activities reinforce their knowledge about the mathematical characteristics of the models. They also had the opportunity to link concepts, understand procedures, rules and meanings. The knowledge constructed in this way was a support for designing the task, so that it strengthens their pedagogical content knowledge. In relation to the progress of mathematical competence [START_REF] Blomhøj | Teaching mathematical modelling through project work. Experiences from an in-service course for upper secondary teachers[END_REF], it was observed that the future teachers' performance with the modelling activity and the task designed by them are situated in the technical level, related to the use of mathematics and its representations. However, it is desirable that future teachers widen their radius of action.

The experience in this research showed thatin addition to the knowledge pointed out by [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] for implementing modelling as a vehicle for teaching mathematics, the teacher should be able to: recognize the reasons why some simplifications are necessary, recognize and explain the frequent difficulties students find in mathematizing a situation, identify and explain the reasons for the behaviour of some situations, identify and anticipate appropriate mathematical representations, identify and enhance the transit through the different phases of a modelling process. On the other hand, some tension between developing modelling competences and teaching the content was identified. That is, although the task could be more open allowing students to build their own strategies, the final characteristics of the task limit the actions of the students to whom it will be directed. This situation has a strong relationship with the educational system because most institutions expect that teachers taught the content as it is described by the curriculum where modelling is considered closer to word problems. In the same way, Chilean teachers training programs are more directed toward the teaching of content rather than developing skills. Therefore, our challenge is to encourage pre-service teachers to explore innovative ways for teaching mathematics.

Finally, the construction of pedagogical and mathematical knowledge was analysed when preservice teachers design a task; this fact does not allow us to know how future teachers put into play their pedagogical content knowledge in the practice. This aspect is been explored in parallel studies.

Acknowledgment

I thank to VREA-PUCV and to the research project EDU2017-84276-R.


domain: spi phys

text:
M.-T Velluet

C Bell

J.-F Daigle

J Dijk

S Gladysz

A Kanaev

A Lambert

D Lemaster

G Potvin

M Vorontsov

Data collection and preliminary results on turbulence characterisation and mitigation techniques

Keywords: Turbulence, propagation, experiments, image processing, laser beam delivery, adaptive optics

In the framework of NATO task group SET 226 on turbulence mitigation techniques for OA systems, a trial was conducted in the premises of RDDC-Valcartier, using indoor and outdoor facilities in September 2016. Images data sets were collected under various turbulence conditions, both controllable (indoor) and natural (outdoor). The imagery of this trial was used in the Grand Challenge, where different experts were asked to process identical input data with state-ofthe-art algorithms. The trial also provided a data-base to validate theoretical and numerical models. The paper will give an overview of the experiment set-up (target, sensors, turbulence screens generators…) and present some preliminary results obtained with the collected data in terms of effectiveness of image processing techniques, new methods for turbulence characterisation, modelling of laser beam propagation.

INTRODUCTION

High performance optical systems suffer from atmospheric turbulence present near the ground that limits their range and resolution. Because the need of target recognition, tracking or threat defeat at longer distance increases, Electro-optics (EO) and laser systems are more and more sensitive to turbulence effects. Illustrations of turbulence effects are presented in the figures below. The decrease of resolution can be seen in Figure 1 and the decrease of focused beam quality with the increase of turbulence strength expressed by C n ² in Figure 2. Turbulence near the ground depends strongly on many variables, including sunlight, soil nature and humidity of the air. Over long distances, a great variability of environmental conditions can be found. This leads to a large spatial as well as temporal variability of turbulence conditions and lead to a varying impact and degradation level on the EO and laser (EOL) system performance. A better knowledge of turbulence strength and its distribution along the line of sight (LOS) can help to implement the most relevant turbulence mitigation techniques and to assess system performance models.

To overcome turbulence-induced limitations, mitigation techniques can be applied. Adaptive Optics (AO) systems are more efficient for laser beam delivery systems, while data processing is more suitable for visible/IR imaging sensors. However, higher the turbulence between target and receiver planes, lower the efficiency of these techniques.

The objective of the NATO SET 226 group is to improve the knowledge on turbulence features (strength, statistics, distribution …) and the impact on EOL systems performance in order to propose adequate mitigation techniques and then test them. Here we describe the experiments which were conducted in the framework of NATO group SET-226 and present some interesting preliminary results.

EXPERIMENTAL SET-UP

A joint NATO trial was conducted at the premises located in DRDC -Valcartier Research Centre, located 20 km from Quebec City, Canada. Two distinct test sites were selected to perform the different experiments: a 220 m indoor range with allowing for systematic measurements in controlled and reproducible conditions and a 2.5 km outdoor range for real-world conditions.

Assessing the impact of atmospheric turbulence on EOL systems requires extensive testing at various ranges. An indoor environment allows control of, to a certain extent, individual parameters of the atmosphere, while outdoor ranges provide a good insight into the impact of realistic and dynamic weather conditions. In an indoor environment, the quasi-absence of turbulence can be circumvented by adding controlled heat sources along the path to generate artificial turbulence. The drawbacks of such a facility include a limited engagement range, and the lack of transverse wind profiles.

As compared to indoor tunnels, outdoor ranges generally offer significantly longer paths. No control is possible over the test conditions; however it is possible to record strong correlations in between the EOL system performance and specific weather and turbulence parameters. A short propagation range increases the probability of uniformity of conditions (turbulence, humidity, wind) along the entire propagation path while longer ranges usually exhibit a stronger variability of atmospheric parameters along the path due to the increased possibility of variations in beam height, wind conditions and terrain (e.g grass, roads or rivers).

250 m indoor range

The Aero-ballistic Range test site consists of a 250 m long indoor firing range located on the southern site of DRDC-Valcartier. The corridor, has been (and still remains) a test ground to characterize the ballistics and aerodynamics of projectiles used by the CAF. This fully equipped tunnel is also an ideal place to perform systematic EOL experiments in a controlled environment. An aerial view of the test site is presented in Figure 3(a). Figure 3(b) shows the interior of the tunnel which is fully equipped with cameras and imaging screens for characterization of the ballistics of various projectiles.

Description and characterization of the Turbulence Generators

The turbulence generators consisted of six 250 Watt bulbs placed under a 4 cm by 183 cm vent formed from aluminum sheet, as depicted in Figure 4. The bulbs were arranged in two banks of three which could be individually turned on or off and the bulbs' temperature could be controlled using a rheostat. There were five sets in total. The turbulence generators were placed under the line of sight of the various cameras and lasers to produce a screen of optical turbulence. For some of the measurements the turbulence generators were placed at an angle across the beam to increase the turbulent path-length. Also, for some measurements, so that all the targets could be imaged simultaneously, two screens were placed next to each other, one in front of each target. The turbulence generated by these devices was characterized using both optical means and temperature measurements from thermocouples. In this later case, turbulence feature has been derived and reported in Table 1. The turbulence strength is quite high, a few 10 -10 m -2/3 , which leads to Fried parameter (r0) lower than 2 cm (2 cm at λ = 0.6 µm, C n ² = 10 -10 m -2/3 and four screens at 10, 110, 165 and 210 m from the target). This value is representative of the following case: propagation distance = 2500m, C n ² constant along the LOS = 1.5 10 -14 m -2/3 which can be considered as medium turbulence conditions.

Targets

There were two main targets used in the tunnel: a LED panel provided by AFRL and a projection screen on which various targets could be projected. The projector used was an EPSON PowerLite Pro Z9900WNL This had a 1280 by 800 screen resolution with an output of 9200 lumens.

Both still and moving images were projected onto the screen. The still images included test charts e.g. the US Air Force 1951, vertical or horizontal sinusoidal bars. In addition, there were also images of natural scenes. The LED panel consisted of a 10x10 array of individually controllable sources with a10 cm spacing between them. The wavelength of the LEDs was 627 nm with an output of 12 cd and an 8° output angle. Sample images of these targets are shown in Figure 5; an image through turbulence is also presented for the horizontal bars case.

2500 m Outdoor Range

Outdoor testing was conducted on a 2.5 km-long range available as part of DRDC infrastructures located at the nearby Canadian Forces Base in Valcartier. Figure 6(a) presents an aerial view of the outdoor range while Figure 6(b) presents the view of the range as seen from the position of the EOL systems.

During the experiments EO systems tested and the Canadian HEL platform were located at the lower-left extremity of the range depicted in Figure 6 (a) while all the targets and laser propagation diagnostics were installed at the other end of the range 2.5 km away. A garage was available to accommodate and protect all the EO sensors and related hardware from rain and humidity.

The trailer-mounted HEL system was installed beside the garage for the laser propagation tests and was aiming towards an interception screen installed in a sea container positioned at the other end of the range.

Instrumentation

Different instruments were installed to monitor turbulence (2 scintillometers and a 2D sonic anemometer) and environmental conditions (a weather station).

Targets

There were two types of static test charts used: a standard USAF 1951 resolution chart and a triangle orientation discrimination (TOD) method test target. Also, the LED array was used again, but because of the increased range every other row and column was turned off, so producing a 5x5 array of LEDs.

In addition, live targets were used in the form of a moving van and a person walking for some of the serials.

Description of cameras used

For conventional imaging, there were four different visible (VNIR) cameras used for the image acquisition with different resolution and pixel size at the target (see Table 2) both for indoor and outdoor ranges. In front of the Baumer camera a plenoptic system was implemented. The projected pixel size quoted is if it were used as a conventional camera. As it was used as a plenoptic camera, the real projected pixel size would be different.

EXPERIMENTAL RESULTS

Turbulence characterization

It is in principle possible to estimate path-integrated from image motion (angle-of-arrival method, Equation (1) [START_REF] Smith | The infrared and electro-optical systems handbook[END_REF]) but this approach is susceptible to vibrations, drifts and thermal effects on optics of the transmitter and the receiver.

〈 〉 2.91 / / (1)
Measurement of differential image motion is not affected by any of these effects [START_REF] Gladysz | Nearly Complete Characterization of Optical Turbulence with an LED Array[END_REF]. Assuming a pure Kolmogorov model, the angular correlation of the beam centroid of a grid of point sources can be expressed as a function of the profile [START_REF] Chassat | Propagation optique à travers la turbulence atmosphérique. Etude modale de l'anisoplanétisme et application à l'optique adaptative[END_REF] and is defined by eq. ( 2). For different position of the phase screens, this angular correlation has been studied. As presented in Figure 7 we have compared angular correlation given by eq. ( 2) assuming that the value is the same for each screen and between each phase screen the turbulence is very low and can be neglected.

, ! " " (2)
Figure 7: Left: configuration of the experiment; Right:

, as a function of the point sources position, white curve is the mean of the correlation figure obtained for each horizontal line and blue curve is given by the analytical expression [START_REF] Chassat | Propagation optique à travers la turbulence atmosphérique. Etude modale de l'anisoplanétisme et application à l'optique adaptative[END_REF].

We find a very good agreement between the measurement (line curve) and the theory (blue curve). This result is promising. In fact, the study of angular correlation should make it possible to estimate the relative strength of a few turbulence layers along the LOS.

Recordings of the LED array were also processed at Fraunhofer IOSB using the method outlined in Ref. [START_REF] Gladysz | Nearly Complete Characterization of Optical Turbulence with an LED Array[END_REF]. Differential variances (structure functions) of motion were fitted with the Kolmogorov or the von Kármán models in order to extract and L 0 values. Processing of the same variances in the time domain is used to obtain the value of the transverse wind speed, or flow velocity for the case at hand. A "discrete" model of turbulence along the path can be easily inserted into the equations by replacing the integral # $ % with a sum ∑ $ % ' ' ∆ )

* )+
, where $ % is the path weighting function for this experiment, m is the number of heaters used and ∆ ) is the longitudinal extent of the turbulence "slab" created above each heater.

Laser propagation

Thanks to the indoor facilities, the images of the beam in the focal plane can be recorded with different turbulence profile (number and location of the phase screens). The distortion of the beam takes a different structure depending on where in the path the phase screen generators are placed. Typical beam profiles are shown in Figure 9. For an ideal TEM00 beam in the absence of phase distortion we would expect a Gaussian focal image.

Phase screen at 20 m from the target

Phase screen at 165 m from the target Figure 9: Influence of the turbulence on the laser beam quality.

Laser beam propagation has also been studied over longer distance (outdoor range) and the images of the beam recorded in the observation plane have compared with images obtained with the same propagation condition with numerical models. The size of the long exposure beam and the standard deviation of the beam motion were compared for both images. The results are presented in Figure 10 for different turbulence strengths. Experiments (back) and simulation (blue) are in good agreement except for very weak turbulence conditions where the experimental system is dominated by vibrations.

Plenoptic camera

There are at least two kinds of plenoptic cameras: standard and focused plenoptic cameras [START_REF] Wu | Determining the phase and amplitude distortion of a wavefront using a plenoptic sensor[END_REF]. In the standard plenoptic camera, the microlens array is coincident with the focal plane of the prime lens. The spatial resolution of the final image is equal to the number of lenses in the array and the number of 'directions' of the rays equals the number of pixels behind each microlens.

Figure 11: Left: principle of the focused plenoptic camera; Right: an example of recorded image.

In the focused plenoptic camera, the microlens array can be either in front or behind the focal plane of the prime [START_REF] Wu | Determining the phase and amplitude distortion of a wavefront using a plenoptic sensor[END_REF]. In this camera, the microlens array was behind the prime lens's focal plane (Figure 11 However, for each point in a micro image, all the rays come from a single circular patch on the lens of diameter, * -⁄ . If this circular patch is smaller than r 0 , then the major effect of turbulence on the image from the microlens will be shift and distortion.

Plenoptic camera was envisioned to measure phase and amplitude of disturbed wavefront of a laser beam after propagation through turbulence. The authors claim that it is potentially possible to reconstruct the turbulence volume with this type of sensor [START_REF] Wu | Determining the phase and amplitude distortion of a wavefront using a plenoptic sensor[END_REF]. They also proposed this type of camera to overcome high image distortion given by conventional imaging systems [START_REF] Wu | Imaging through strong turbulence with a light field approach[END_REF]. Further studies must be carried-on in the future to evaluate limitations and advantages of plenoptic camera for target recognition through turbulence.

"Grand challenge"

Previously there have only been a few turbulence mitigation algorithms compared simultaneously. It was decided to run a 'Grand Challenge' to compare the performance of as many algorithms as possible against the static USAF 1951 chart. This was done by inviting leading academic, military, and defense industry researchers to participate. As an enticement, results were to be shared anonymously across all participants. The restoration algorithms represented in the challenge included a mix of published and proprietary approaches. Among the published approaches, the bibliography contains a cross-section of applicable papers (References [START_REF] Wu | Determining the phase and amplitude distortion of a wavefront using a plenoptic sensor[END_REF]- [START_REF] Lou | Video Stabilization of Atmospheric Turbulence[END_REF]). In this way, each participant would be able to evaluate their competition without risking anything if their submission compared poorly. For the SET-226 group, the outcome allows for a broad survey of the state-of-the-art from experts at very little cost. In total there were 14 different participants

Data collection

The static scene used in the restoration challenge includes a USAF 1951 resolution target, a 5x5 array of 20 cm spaced LEDs, a scintillometer, and the TNO triangle orientation/acuity target, among other things. The range from telescope to target was 2.54 km with a path-averaged refractive index structure parameter, C n 2 , of 2.33e-15 m -2/3 as measured by the scintillometer. This scenario corresponds to a band-averaged coherence diameter of 0.071 m. The telescope used in this test was a Celestron StarBright XLT with a focal length of 3.91 m and 0.356 m aperture (~35% linear obscuration). The imagery was recorded using a PCO.edge camera with 6.5 µm detectors (unfiltered silicon) operated with 2 ms integration time at 37 frames per second. The first frame in the image sequence is shown in Figure 12.

Scoring

Restoration algorithm performance is quantified by determining the smallest resolvable bar pattern on the USAF 1951 test target in each restored submission. A bar pattern is resolved if all three horizontal and vertical bars are apparent. For the results shown in Figure 13, this determination was made by group consensus of the NATO SET-226 members during a meeting of the group (June 2017) at FFI in Kjeller, Norway. The results are reported in Table 3 as well the limited resolution in terms of line pairs/mrad. Simple criteria were also used to evaluate the quality of the restored image for two resolutions (see Figure 14 CNR is not exactly the same measure as the algorithm performance evaluation done by the group looking for the smallest resolvable bar pattern on the USAF 1951 test target in each restored submission including smaller bars. However, if we compare the results of CNR on the small bar pattern with the group evaluation, we see that the best image is also image 2B, followed by image 1 and image 3. An improvement of this method would be the automatic selection of all bar patterns in the images, and then evaluate either the improvement for all different frequencies, or select the smallest bar pattern for which still three bars can be discerned. For this, the method should also be changed so that the difference value will depend on three bars instead of the maximum and minimum in the area.

CONCLUSION

During the trial, a large amount of data was recorded under a variety of turbulent conditions in both controllable indoor and natural outdoor environments. A data-base was developed and completed with relevant information (ground truth data, optical sensors characteristics…) to validate theoretical and numerical models.

The images collected were used for a Restoration Challenge that brought together experts in the field and allowed them to process identical input data using state-of-the-art algorithms.

In parallel, new turbulence characterization instruments and methods have been developed and tested in particular to achieve the turbulence profile along the line of sight and have thus extended the path averaged measurement given by current instruments such as the scintillometer or local measurement given by eddy covariance or anemometers sensors.

Emerging capabilities have been tested and evaluated like the plenoptic camera which simultaneously records multiple images of the same object from slightly different angles of view through differing paths in the turbulent atmosphere. Unfortunately, the recorded images were not sufficiently resolved to conclude but this can be further investigated in a follow-on group.

We plan to further develop turbulence characterization methods to provide a simple and effective way to evaluate the actual performance of optical systems during operational use. We also highlighted a lack of relevant metrics to assess the effects of turbulence on system performance and the effectiveness of correction techniques. This point will be addressed in the next group. We will also extend the comparison of image processing algorithms to other turbulence conditions and kind of images in a "Signal Processing Cup Challenge" using the metrics to score the results.

Figure 1 :

 1 Figure 1 : Left target area; right one target collected image at 2.5 km.

Figure 2 :

 2 Figure 2 : Beam focusing at 2.5 km for (a) C n ² = 10 -15 m -2/3 and (b) C n ² = 7.5 x10 -14 m -2/3 .

Figure 3 :

 3 Figure 3: (a) Aerial of the 250 m long Aero-Ballistic Range firing site. (b) Interior of the Aero-Ballistic Range. The corridor is fully equipped with cameras used to characterize the ballistics and aerodynamics of projectiles.

Figure 4 :

 4 Figure 4: A turbulence generator consisting of six 250 Watt bulbs under an aluminium vent that is 4 cm wide and 183 cm long.

Figure 5 :

 5 Figure 5: From left to right: Sample images of the horizontal sinusoidal bars from the pristine and the turbulent sequences, projected video of a man walking in a circle, LED array panel.

Figure 6 :

 6 Figure 6: (a) Aerial of the 2.5 m outdoor range. (b) 2.5 km range view as seen from the interior of the garage accommodating EO sensors and related hardware.

Figure 8

 8 Figure 8 illustrates some of the results. The top panels show the agreement between the theoretical and the experimental differential motions for two chosen experiments (the agreement was similarly good for the other cases). The bottom left panel shows the extracted outer scale values and the bottom right panel shows the obtained values of flow velocity.

Figure 8 :

 8 Figure 8: Top left: parallel and perpendicular differential motions, theory (solid lines) and experiment (symbols), data taken on September 15 at 11:28 local time. Top right: data taken on September 15 at 11:33 local time. Bottom left: outer scale values for the same time period. Top right: flow velocity measurements for the same time period.

Figure 10 :

 10 Figure 10: Statistical analysis of the mean beam spread (a) and the mean beam centroid displacement (b) measured as a function of Cn2.

  -left). Each microlens refocuses a subsection of the prime's focal plane (i.e. the prime's image) onto a second focal plane, forming an array of micro images (Figure 11 -right). The spatial resolution of the focused plenoptic camera is b/a times the resolution of the sensor and the number of angular samples is a/b.

Figure 12 :

 12 Figure 12: The first frame in the static target challenge.The static target test was useful because comparison between restoration algorithms can be quantified in terms of limiting resolution. In the static target challenge, participants were asked to produce the best possible restoration given 200 sequential images of the turbulence degraded scene. No time limit was placed on the effort and the definition of "best possible restoration" was left open to individual interpretation.

Figure 14 :

 14 Figure 14: Left: Image region used for evaluation; Right: Evaluated bar patterns for the different images. Top row, demagnified large bar patterns; bottom row, magnified small bar patterns.

Figure 15 :

 15 Figure 15: CNR evaluation for the large (left) and small (right) bar values. Note that the difference is multiplied by 10 and the std by 50 to obtain values in the same range as the CNR.

Table 1 : Estimates from thermocouples measurements of outer scale, CT2 and C n 2 . H (cm) Outer scale (cm) CT2 (K 2 m -2/3 ) C n 2 (m -2/3 )

 1

	48	5.88	459.2	3.63 x 10 -10
	63	10.0	189.4	1.50 x 10 -10
	71	7.14	168.6	1.33 x 10 -10
	76	8.33	143.5	1.14 x 10 -10

Table 2 :

 2 Cameras used for image acquisition

Camera Resolution Pixel size at the target (cm)

			Tunnel	Outdoor
	pco.edge 5.5	2560 x 2140	1.55	4.22
	Mikrotron EOsens mini camera	1280 x 1024	2.37	17.5
	Optronis CamPerform CP80-3-M/C-540	640 x 480	2.20	25.4
	AVT, Mako G-125C IRC PoE	1,024 x 964	1.03	11.9
	Baumer HXC13	1280x1024	3.12	36.0 1
	1

Table 3 :

 3 Summary of the static target challenge results.

	Method	1	2A	2B	3	4	5	9	11	12	13
	Resolved	G4	G4	G4	G4	G4	G4	G4	G4	G5	G4
	(Group/element)	E5/6	E2/3	E6	E4/5	E1	E2	E1	E1/2	E6	E3/4
	Limiting Resolution	97.25 68.8 102.8 86.7	57.7	64.9	57.7	61.3	51.4	77.2
	(line pairs / mrad)

ACKNOWLEDGEMENTS

The trial was partially supported by NATO RTO SET Panel. The NATO SET 226 field trial was expertly organized and hosted by Jean-Francois Daigle and his colleagues at DRDC Valcatier. Special thanks to Pascal Duchesne and Gaston Nadeau for their support and patience during the trial. Thanks to Lt Matt Trippel (USAF) for enthusiastically serving as camera operator and technician during the field trial.