## Abstract

We study the trajectory of dense projectiles subjected to gravity and drag at large Reynolds number. We show that two types of trajectories can be observed: if the initial velocity is smaller than the terminal velocity of free fall, we observe the classical Galilean parabola: if it is larger, the projectile decelerates with an asymmetric trajectory first drawn by Tartaglia, which ends with a nearly vertical fall, as if a wall impeded the movement. This regime is often observed in sports, fireworks, watering, etc. and we study its main characteristics.

## 1. Introduction

Particles moving in air usually exhibit a curved trajectory with a strong asymmetry with respect to the maximum. Three examples are presented in figure 1, which illustrate (*a*) grinding, (*b*) fireworks and (*c*) field watering. This asymmetry historically led to the singular limit of ‘triangular’ trajectories reported in studies of artillery and illustrated in figure 2. According to Tartaglia [1,4], the trajectory was composed of two main phases, a straight violent motion at the exit of the canon (segment AB in figure 2*a*) and a final vertical path called natural (segment EF). The two limits were connected via a circular path.

Owing to its application in the military context [5] and more recently in sports [6], this ballistic problem has also been studied in detail and geometrical constructions [7], numerical solutions [8] and theoretical discussions [9,10] have been proposed to account for the trajectory. One of the more famous works on the subject is probably the ‘dialogues concerning the two new sciences’ [2] first published in 1638, one century after Tartaglia. The fourth and last day of the dialogues is dedicated to the motion of projectiles. For the first time, it describes the parabolic trajectory, an example of which is reproduced in figure 2*b*. Contrary to the Tartaglia curve which has well-defined asymptotes, the parabola does not have any.

Here, we study experimentally and theoretically the ballistic problem and show that, depending on the initial velocity, the two types of trajectories can be observed.

## 2. A brief overview of external ballistics

While the internal ballistics focuses on the physics of launching (inside the bore of the cannon), the general topic of external ballistics is to determine the trajectory of a projectile while moving in air [5,9]. For a particle of mass *M* and velocity ** U**, the whole problem is to solve the equation of motion:
2.1where

*F*_{G}=

*M*

**g**is the weight and

*F*_{A}the aerodynamic force. This equation must be solved with the initial condition

**(**

*U**t*=0)=

*U*_{0}. In the plane (

*U*_{0},

**g**), this problem can be discussed with the conventions presented in figure 3. Apart from the actual shape

*y*(

*x*), the points of interest for the applications are usually the range

*x*

_{0}, the maximal height

*h*and the optimal launching angle

*θ*

^{⋆}which maximizes the range.

Without air (*F*_{A}=0), the solution is the classical parabola, first found by Galileo [2], for which *θ*^{⋆}=*π*/4, and where *θ*_{0} is the launching angle. With air, equation (2.1) is not closed in the sense that the aerodynamic force depends on the velocity *F*_{A}(** U**).

To illustrate the diversity of the ballistic problem in this limit, we briefly discuss the relations *F*_{A}(** U**) obtained with spheres. In this discussion, we use the classical drag–lift decomposition of the aerodynamical force

*F*_{A}=

*F*_{D}+

*F*_{L}, where

*F*_{D}stands for the drag, that is the part of the force aligned with the velocity and

*F*_{L}the part which is perpendicular to it.

For incompressible Newtonian fluids (density *ρ*, viscosity *η*), the motion of non-spinning spheres (radius *R*, velocity ** U**) in an infinite domain has early become a classical subject [11]. In the low Reynolds number limit (

*Re*≡

*ρU*2

*R*/

*η*≪1 with

*U*=|

**|), Stokes [12] has established theoretically that the drag force experienced by the sphere during its motion is**

*U*

*F*_{D}=−6

*πη*

*U**R*. Experimentally, this result has been verified by several authors in the range

*Re*≤1 [13,14]. In the high Reynolds number domain, besides the classical d’Alembert paradox [15], Newton [16] is probably the first to propose an heuristic expression for the drag:

*F*_{D}=−1/2

*ρC*

_{D}

*πR*

^{2}

*U*

**. According to the early experiments performed by Eiffel [17],**

*U**C*

_{D}≈0.4−0.5. This value has since been confirmed by several authors for the range: 10

^{3}<

*Re*<2.10

^{5}. In the intermediate range, the asymptotic expansion method proposed by Oseen [18] has led to many theoretical developments [19]. The drag crisis experienced by the sphere once the boundary layer becomes turbulent (around

*Re*≈3.10

^{5}) has also been deeply studied [20–22].

For spinning spheres, according to Barkla [23], the work seems to go back to Robins [24] and then Magnus [25], who is credited for the associated lift force. Besides these academic studies, the widespread use of balls in sports has also motivated many studies, in baseball [26] and golf [27] in particular, a review of which can be found in reference [6]. These studies show that the sphere experiences a lift force *F*_{L}=1/2*ρC*_{R}*πR*^{3}*ω*_{0}∧** U**, where

*ω*_{0}is the spin vector and

*C*

_{R}a coefficient which can depend on the Reynolds number, the Spin number (

*Sp*=

*Rω*

_{0}/

*U*) and surface roughness [28]. For sports balls, the dependency of

*C*

_{R}with

*Re*and

*Sp*is modest [29]. In baseball for example, Nathan concludes that in the range of spin number between 0.1 and 0.6 and Reynolds number between 1.1×10

^{5}and 2.4×10

^{5}, the coefficient

*C*

_{R}remains almost constant and equal to 0.5±0.1 [26].

What we retain from this discussion is that the expression of the aerodynamical force *F*_{A}(** U**) depends on both the Reynolds number and spin. In this study, we first focus on the large Reynolds number pure drag limit where the equation of motion for a dense particle takes the form:
2.2In the steady state (d

**/d**

*U**t*=

**0**), the velocity is equal to the terminal velocity with .

## 3. Experimental results

### (a) Measurement of the terminal velocity

The projectiles we use are sport balls. To measure their terminal velocity, , we conducted experiments in the vertical wind tunnel SV4 of ONERA (the French Aerospace Laboratory) in Lille. This unique facility blows air at 20^{°}C up to 50 m s^{−1} in a 4 m cylindrical vein [30]. The protocol consists of increasing the velocity step by step up to levitation. At each step, we wait 1 min for the flow to stabilize in the vein. The particle is then introduced in the centre part of the channel. If it falls, we keep increasing the velocity up to the point where it starts to levitate. When this levitation state is reached, we measure the velocity with a VT 200 Kimo anemometer and a pitot tube. Two examples of levitation are presented in figure 4, one for a Jabulani soccer ball leading to (*a*) and the other for a feather shuttlecock leading to (*b*). All the results are presented in table 1. For each projectile (first column), we measure its diameter (2*R* in second column), its mass (*M* in third column) and the terminal velocity ( in fourth column). The corresponding Reynolds number and drag coefficient are calculated in columns 5 and 6 using for the density and viscosity, the tabulated values at 20^{°} [31]: *ρ*=1.20 kg m^{−3} and *ν*=*η*/*ρ*=15.10^{−6} m^{2} s^{−1}. We observe in these columns that the Reynolds number is of the order of 10^{5} and that the drag coefficient takes values between 0.1 and 0.6. Focusing on spherical particles, we note that their drag coefficient for Reynolds numbers smaller than 10^{5} take values close to the classical 0.45, whereas their values decrease to approximately 0.2 for Reynolds numbers larger than 10^{5}. This behaviour is consistent with the behaviour of the drag coefficient in the region of the drag crisis [6,32].

The last two columns in table 1 respectively indicate the fastest hit recorded on fields, , and the ratio between this maximal velocity of the game and the terminal velocity of the ball. We observe in this last column that the record velocity is larger than the terminal velocity except for handball and basketball.

### (b) Qualitative observations

For the two extreme sports reported in table 1, namely basketball and badminton, we present in figure 5 two chronophotographies showing the trajectories of the particles obtained respectively, with (*a*) and (*b*). The trajectory of the basketball presents a left–right symmetry with respect to the maximum, with a continuous evolution of the velocity. For the badminton shuttlecock, the symmetry is broken, and the velocity first decreases from the hit location, and then reaches an almost constant value. The evolution of the trajectory with the hit velocity and initial angle are discussed in the §2*c*.

### (c) Trajectories

Among the different sports projectiles that we have characterized in table 1 the shuttlecock is the one which enables (owing to its low terminal velocity) to maximize the range of the ratio . We thus use it to study how this ratio affects the shape of the trajectory. The shuttlecock for these experiments is presented in figure 6*a*: it is a MAVIS 370, composed of a plastic skirt (S) fixed on a cork (C). The length of the skirt is *L*=60 mm and its radius *R*=34 mm. The whole mass is *M*=5.3 g, 3 g for the cork and 2.3 g for the skirt. Using a Deltalab EA600 wind tunnel, we have measured the drag of shuttlecocks free to rotate. The evolution of the drag coefficient with the Reynolds number is presented in figure 6*b*. We observe that *C*_{D}≈0.65±7%, independent of the Reynolds number. This value is consistent with the one measured in the vertical wind tunnel and reported in table 1 [33].

Concerning the trajectory, we present in figure 7 several experiments (circles) obtained with different initial velocities *U*_{0} and inclination angle *θ*_{0}. The timestep between two data points is 100 ms. We observe that the asymmetry increases with both the hit velocity and initial angle.

More quantitatively, the numerical integrations of equation (2.2) are also presented with solid lines in figure 7. Without any adjustable parameter, the comparison with the experimental data is convincing for all initial conditions. We underline that in addition to their constant drag coefficient (figure 6*b*), shuttlecocks also present the advantage (compared with spheres) of having no additional Magnus lift.

### (d) Saturation of the range

The range *x*_{0} is defined in figures 3 and 7 as the location on the horizontal axis where the particle returns to its initial height [*y*(*x*_{0})=0]. As noted in figure 7, the range changes with both the initial angle and velocity. We report in figure 8*a*, the evolution of its reduced value as a function of the reduced speed . The characteristic length is linked to the terminal velocity via the relation . In the parabolic limit, one expects the linear relationship between the two quantities: . This linear dependency is presented with a dashed black line in figure 8*a*. It only fits the data obtained in the low velocity limit: . For larger velocities, the range strongly deviates from the gravitational limit and almost saturates: for instance, with , we measure instead of 60. To characterize the saturation, we present in figure 8*b* the same set of data in a logline plot. This reveals the logarithmic saturation of the range: .

## 4. Analysis of the trajectory

### (a) Two exact solutions

The vertical launching limit (*U*_{0}∧**g**=**0**) can be solved analytically. If *U*_{0} is orientated downwards (*U*_{0}=−*U*_{0}*e*_{y}, *θ*_{0}=−*π*/2), equation (2.2) leads to: . The velocity simply relaxes exponentially from its initial value *U*_{0} to its final value over the characteristic length . When the initial velocity is positive, (*U*_{0}=+*U*_{0}*e*_{y}, *θ*_{0}=+*π*/2) the solution of equation (2.2) is . In this limit, the particle first decelerates and stops at the maximal height *h*, before falling downwards with the solution obtained for *θ*_{0}=−*π*/2 and *U*_{0}=0. The maximum height can be written as
4.1In the low velocity limit (), this equation reduces to the Galilean result, , with a quadratic dependency on the launching speed. However, in the large velocity limit (), this expression reveals that *h* mainly depends on its aerodynamic length with a weak logarithmic dependency on the launching speed.

### (b) Origin of the aerodynamical wall

We continue the analysis of equation (2.2) in the general case (*U*_{0}∧**g**≠**0**) by discussing the origin of the aerodynamical wall: without drag (*C*_{D}=0), equation (2.2) shows that the particle never reaches a steady state but always accelerates owing to gravity. With drag (*C*_{D}≠0), a steady state (d** U**/d

*t*=0) appears in equation (2.2) where the velocity of the particle is 4.2In this final stage (4.2), the velocity is aligned with the gravitational acceleration

**g**which means that the trajectory is vertical. As soon as the drag appears, there is thus a vertical asymptote on the trajectory. This vertical asymptote is reached within the characteristic distance . This point can be shown by rewriting the equation of motion in the form: 4.3where

*s*is the arc length along the trajectory. The projection of this equation along the

*e*_{x}direction leads to , where

*U*

_{x}is the horizontal component of the velocity. This equation can be integrated with the initial condition where and we find 4.4If

*U*

_{x0}≠0, this equation states that the horizontal component of the velocity vanishes exponentially within the distance . Beyond this distance,

**and are expected to be aligned. The distance is thus expected to characterize the location of the aerodynamical wall. This point is further discussed in §4**

*U**d*.

### (c) Two different regimes

To identify the parameters which govern the whole trajectory, we use the reduced variables and . The above equation of motion (4.3) becomes
4.5Equation (4.5) must be solved with the initial condition: , where *t*_{0} is the unit vector tangent to the trajectory at the origin. The whole system is thus governed by only two parameters, namely the initial angle *θ*_{0} and the ratio , between the terminal and the initial velocities. In equation (4.5), the last term is initially equal to 1. This value allows us to identify two different regimes:

#### (i) The parabola

In the low-launching velocity regime (), the second term in equation (4.5) is initially much larger than the third and the equation of motion reduces to the classical parabola:
4.6In this regime, the velocity increases from its initial value *U*_{0} to its final value .

#### (ii) The Tartaglia curve

In the high-launching speed regime (), the second term in equation (4.5) is initially much smaller than the third and the equation of motion reduces to:
4.7which can be integrated as . The initial part of the trajectory is thus a straight line along which the particle decelerates over the characteristic distance . The final state is also a straight line in which the second and the last term in equation (4.5) compensate: . In between these two regimes, the three terms in equation (4.5) have to be considered in order to connect the two straight lines. These features of the trajectory are very close to the one depicted by Tartaglia and presented in figure 2*a*. We will thus refer to the trajectories obtained in this large velocity domain as Tartaglias. It is important to underline that in this specific regime the equation of motion never reduces to a parabola. Indeed, the first two terms in equation (4.5) never compensate. These two regimes are illustrated in figure 9 with the low velocity () in (*a*) and the high velocity () in (*b*).

### (d) Location of the wall

The aerodynamical wall (or vertical asymptote) is visible in figure 9*b* but not in figure 9*a*. To show the existence of the wall in both cases, we zoom out and present in figure 10 the trajectory obtained with the same conditions. We observe that the range and the location of the wall are distinct in the low-velocity regime (figure 10*a*), whereas they almost coincide in the high speed limit (figure 10*b*).

We study analytically the two locations with dimensional quantities. As , the location of the wall is defined by . Using and (from equation (4.4)), we deduce
4.8To evaluate *U*(*s*), we consider separately the two different regimes.

#### (i) The low-velocity limit:

In this regime, the equation of motion initially reduces to d** U**/d

*t*=

**g**. Because the velocity is very small at the beginning compared with the terminal velocity, we assume that most of the trajectory which leads to the wall is governed by the equation

*U*(

*t*)≈

*gt*or

*s*=1/2

*gt*

^{2}. The equation for wall location (4.8) thus takes the form: , where . This equation leads to the expression: 4.9Equation (4.9) is presented with a dashed line in figure 11. The comparison with the numerical results is fair in the domain . In this low-velocity regime, the wall location increases linearly with the velocity.

#### (ii) The large-velocity limit:

In this limit, the equation of motion (4.3) initially reduces to . The velocity thus decreases exponentially over the characteristic distance : . This regime holds up to where . For larger curvilinear locations, the maximal value of the velocity is the terminal velocity, which leads to the approximation for *s*>*s*^{⋆}. In this large velocity regime, the wall location (4.8) thus takes the form: which leads to:
4.10This solution is presented with a solid line in figure 11 and compared with the numerical results obtained through the integration of equation (4.5) with different initial conditions. The comparison is satisfactory for velocity ratio larger than unity (). In this large-velocity regime, the wall location *x*_{w} is mainly fixed by the product .

### (e) An analytical expression for the range

The range, *x*_{0}, is defined in figures 3, 7 and 10 as the distance from the origin where the particle returns to its initial height [*y*(*x*_{0})=0]. This distance differs from 0 only for positive values of *θ*_{0}. To obtain the expression of the range in this regime, we use the projection of the equation of motion (4.3) along the direction : . Using both the geometrical relation and equation (4.4) for *U*_{x}, we find
4.12Equation (4.12) can first be integrated numerically and we present in figure 12 the trajectories obtained with *θ*_{0}=*π*/4 and two extreme values of the reduced velocity: in figure 12*a* and in figure 12*b*. This latter case corresponds to the parabola limit, whereas the first one (strong velocity limit) presents almost a triangular shape. Equation (4.12) can also be integrated by parts which leads to:
4.13

The function can be approximated by (cf. appendix). The integral of trajectory (4.13) relates the local angle *θ* to the curvilinear coordinate *s*. In particular, for the maximum (*θ*=0), one finds . The location of *s*_{0} is presented with a full circle in figure 12. In the triangular shape limit (*a*), one could use the expression to evaluate the range. However, this evaluation fails by a factor of 2 in the parabolic limit presented in (*b*). Instead, we use , where *s*_{−θ0} is the curvilinear location at which the local angle gets to the value *θ*(*s*_{−θ0})=−*θ*_{0}.

This location is indicated with a white dot in figure 12. This approximation leads to the following expression for the range: 4.14

At small velocities (), the logarithmic term can be expanded and equation (4.14) reduces to the classical gravitational result . In this domain, the range is very sensitive to the initial velocity (). This scaling is different from the one obtained for the wall location, which increases linearly with the velocity. Remarkably, this sensitivity disappears at large velocities: in that limit (), the logarithmic term weakly increases with the velocity. In that domain, the range is mainly fixed by the product, , which does not depend on the velocity but only on the ball and fluid characteristics.

Analytical expression (4.14) is compared with the numerical results obtained through the integration of equation (4.12) in figure 13. This comparison reveals that the range *x*_{0} is well predicted by equation (4.14) for all initial conditions. The maximum deviation observed in figure 13 between the range computed numerically and the theoretical expression (4.14) is 25% in the very large velocity limit.

In figure 14*a*, we plot the predicted range *x*_{0th} (full symbols) against the measured one *x*_{0,mes} for different projectiles (shuttlecock, table tennis ball, balloon and plastic sphere underwater) thrown with various initial speeds and initial angles. The model agrees well with the data. To emphasize the role of the aerodynamic drag, we also plot (empty symbols), the predicted range in the case of no drag (parabolic range). The measurements are much shorter than the parabolic prediction.

We can also compare our work with previous studies. Lamb [9] proposed an approximation of the trajectory *y*(*x*), whereas Chudinov gave a different approximation of the range *x*_{0} [10]. For various *θ*_{0} between 10^{°} and 80^{°} and various between 0.1 and 10, we compare in figure 14*b* our model with these previous studies. We plot for each model, the predicted range versus the one calculated by integrating numerically the equation of motion . In the parabolic limit (), the three models recover the classical values. However, for , both previous studies are less accurate than our model.

### (f) An analytical expression for the height

To obtain the analytical expression for the height of the trajectory, *h*, we follow the same steps as for the range: the first integral of the equation of motion (4.13) provides an exact relation between the local angle *θ* and the curvilinear coordinate *s*. If we choose the location of the maximum, *s*_{0} (*θ*=0), to evaluate the height with the relation we obtain . In the limit of small velocities, the log term can be expanded and leads to the expression for the height: , which is twice the expected value in this limit.

Instead of *θ*=0, we choose an intermediate value *θ*_{1} between *θ*_{0} and 0 such that . This intermediate value leads to the expression:
4.15In the limit of small velocity (), this expression for the height reduces to the exact value of the parabola. For , we also recover the exact solution (4.1) derived in §4*a*. For different velocities and initial angles, we present in figure 15 the comparison between the height, *h*, calculated numerically through the integration of equation (4.12) and the theoretical expression (4.15). Over the whole range of initial conditions, the analytical expression of the height (4.15) is in good agreement with the numerical calculation.

### (g) The optimal angle *θ*^{⋆}

Given an initial velocity, the optimal angle *θ*^{⋆} is that for which the range is maximized. As the range vanishes for *θ*_{0}=0^{°} and *θ*_{0}=90^{°}, this optimal angle is expected to exist. In the limit of small velocities (), the parabolic solution leads to *θ*^{⋆}=45^{°}. This optimum is known as Tartaglia’s law [1] and an illustration is presented in figure 16*a*. Using the numerical integration of equation (4.12), we present in figure 16*b* the evolution of the optimal angle *θ*^{⋆} as a function of the reduced velocity . We observe that Tartaglia’s law is obeyed in the range . Above this limit, *θ*^{⋆} slowly decreases with increasing velocity: It reaches 30^{°} for and 20^{°} for .

Because we have an analytical expression for the range (equation (4.14)), the optimal angle can also be determined via the condition . This condition leads to: , where . This implicit equation is difficult to invert. Instead, we use the approximation which leads to the expression for *θ*^{⋆}:
4.16In the small-velocity limit (), this expression reduces to *θ*^{⋆}=45^{°}. In the large-velocity domain, it leads to , which implies a slow decrease in the optimal angle with the velocity. More quantitatively, equation (4.16) is presented in figure 16*b* with a solid line. Over nine decades, it shows a fair agreement with the results obtained numerically through the integration of equation (4.12). For cannonballs, Hélie [35] noted that ‘there is an angle which gives the highest range. Experiments show that the angle is always smaller than 45^{°}’. He extracted abacus from experiments performed at Gâvre between 1830 and 1864. For massive cannonballs (2*R*=8 cm, *M*=15.1 kg) launched at *U*_{0}=485 m s^{−1}, the large-velocity regime is reached as . The range is maximized and equal to 5690 m for an angle *θ*^{⋆}=37.5^{°}. The solid square reported on figure 16*b* shows that Hélie’s data are in agreement with the theoretical and numerical results.

## 5. Applications and perspectives

### (a) Application in fire hoses

Firemen use water guns that produce jets such as those presented in figure 17*a*. The shape of the jet is far from a parabola and exhibits a dissymmetry which appears similar to that of the shuttlecock (figure 5*b*). More quantitatively, we have studied the evolution of the range as a function of the exit velocity using different water guns. The experiments were conducted in the test centre of the company POK S.A. which produces fire equipment. We used a converging water gun similar to the one presented in figure 17*b*. The inclination angle was kept constant at *θ*_{0}=30^{°} and we used two different exit diameters, *D*=6 mm (squares) and *D*=12 mm (circles). The evolution of the reduced range, , is presented in figure 17*c* as a function of the reduced exit velocity . The two sets of data collapse and exhibit a nonlinear evolution. We have used the analytical expression of the range (4.14) to fit these data: . As *θ*_{0}=30^{°} and *U*_{0} is measured, the only free parameter of the fit is . The fits are presented as continuous lines in figure 17*c*. We find and , respectively, for the diameter 6 and 12 mm. To understand the order of magnitude of this characteristic length scale, one can make a simplified model in which the water gun produces water balls of diameter *D*. With this model, the characteristic length scale would write (where *ρ*_{w} is the water density). Considering measurements of water drops terminal velocities [36], we deduce the value of the drag coefficient associated with those particles: *C*_{D}=0.44. This would lead to the value of 18 m for the 6 mm water gun and 36 m for the 12 mm one. So, even if the water jet breaks, it seems that the approximate range of water guns can be predicted by the analytical expression (4.14). A more refined model should consider the break up of the water jet.

### (b) Spin effect

If the sphere rotates, it undergoes a side force owing to a Magnus effect. In order to study the modification caused by this effect on the previous study, we observed a soccer ball trajectory in the case of a long clearance. It is known that goalkeepers always put an important backspin in those conditions. Figure 18 reports an example of long clearance recorded from the side of the field.

It is interesting to compare this experimental trajectory with the one expected by solving equation (2.2) numerically with the same initial conditions (dashed line). This resolution includes the experimental value of the aerodynamic length of the soccer ball: . The difference between the observed trajectory and that predicted on the basis of our model without spin is significant. The range is about 50% larger experimentally, and the maximal height is doubled. To understand this difference, we need to consider the dynamics of a spinning ball.

In the case of a constant spin *ω*_{0} along the *z* direction (*ω*_{0}=*ω*_{0}*e*_{z}), the trajectory stays in a vertical plane but its shape and range are modified. Taking into account the expression of the side force in the equation of motion, we obtain
5.1Along the ** t** and

**directions (cf. figure 3), the previous equation can be written in a non-dimensional way with , and . This provides the two following equations: 5.2and 5.3where is a typical distance over which the spin curves the trajectory. The two non-dimensional parameters which characterize the effect of ball rotation on the trajectory are and**

*n**Sp*

_{0}=

*Rω*

_{0}/

*U*

_{0}. This latter parameter, also called ‘spin number’, compares the rotation and translation speeds.

*Sp*

_{0}depends on the initial launching conditions whereas only depends on ball and fluid properties. We solve numerically equations (5.2) and (5.3) with the initial conditions of the experimental soccer long clearance. This calculation is performed with , the experimental value of the initial ball spin

*ω*

_{0}and considering as an adjustable parameter. We find that is the value which minimizes the error between ten different experimental trajectories and numerical ones. In the case of the clearance shown in figure 18, this approach provides the numerical trajectory drawn with a solid black line. The agreement between this trajectory and the experimental one validates our assumptions, that is to say a constant rotation rate and no dependency of

*C*

_{R}with

*Sp*and

*Re*. The first assumption is driven by the fact that the spin rate decreases on a longer timescale than the translation velocity [37]. The second assumption is consistent with the conclusions drawn by Nathan for baseballs in the range of spin number and Reynolds number experienced during the game [26]. Moreover, from we deduce

*C*

_{R}=0.52 which is in the range of values determined by Nathan for baseballs (

*C*

_{R}≃0.5±0.1).

Solving equations (5.2) and (5.3) numerically allows us to evaluate the modified range *x*_{0} for a wide range of initial conditions and parameters. Typical examples of numerical solutions are reported in figure 19.

Quantitatively, the effect of the spin and initial launching angle on the range is shown in figure 20. The numerical study is conducted with and (soccer conditions).

Focusing on figure 20*a*, we observe a non-monotonic evolution of the range with the spin number. The optimum value depends on the launching angle. The smaller the angle, the larger the range at the optimum and the larger the spin.

For a given spin number, the evolution of the range with the launching angle is shown in figure 20*b*. Again, the evolution presents an optimum, the value of which increases with the spin number. The larger the backspin effect, the larger the range at the maximum and the smaller the optimal launching angle. For example, with *Sp*_{0}=+0.2, the maximal range is and is obtained with *θ*_{0}=20^{°}. With such a backspin, the goal keeper thus increases the range of his clearance by 40% compared with the non-spinning limit.

## 6. Conclusion

The trajectories of particles under the influence of gravity and drag at large Reynolds numbers are studied. Each particle is characterized by its terminal velocity for which the drag balances the weight. We show that depending on the launching speed, two different types of trajectories can be observed: when the initial velocity is smaller than the terminal one, the particle describes the classical Galilean parabola. However, when the launching speed exceeds the terminal velocity the trajectory is never a parabola but an asymmetric curve that we have called a Tartaglia. In both limits, the trajectory exhibits a vertical asymptote (aerodynamical wall) for which we have provided an analytical expression. Apart from the wall location, we also study the range and the height. These three quantities exhibit a logarithmic saturation at large velocities. This saturation has implications in several domains and we have discussed more precisely its influence in the determination of water hose performance and how spin can make a difference in the flight of sports balls.

## Acknowledgements

We thank M. Phomsoupha for the hours spent to perform badminton trajectories. We are grateful to F. Moisy, M. Rabaud and T. Faure for the access to the wind tunnel at the FAST laboratory. We also thank L. Jacquin and D. Sipp for the connection with ONERA Lille. The experiments in the vertical wind tunnel have involved Olivier Renier, Cecile Fatien, Dominique Farcy and Pierre Olivier. The study of fire hoses was made possible thanks to Bruno Grandpierre and his company POK. S.A. For the experiments on goalkeeper clearance, we have worked with semiprofessional players under the supervision of their coach, C. Puxel, in Longjumeau. May all of them find, here, the expression of our gratitude.

## Appendix A: function *F*(*θ*)

The theoretical development of the equation of the motion leads to the function *F*(*θ*) defined on the interval *I*= ]−*π*/2;*π*/2[:
A1We introduce and so that *F*=1/2[*G*(*θ*)+*H*(*θ*)]. Studying those two functions, we see that for , *G*∼*H*, and for , *G* and *H* diverge with . We thus approximate *F*(*θ*) by *G*(*θ*):
A2Figure 21 shows the functions *F* and *G*. We see that the approximation of *F* by *G* is quite good.

- Received July 26, 2013.
- Accepted September 30, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.