Historically, the study of aerodynamic forces acting on rotating bodies in a liquid or gas flow originates in Isaac Newton's seventeenth-century observations of tennis balls and Heinrich Gustav Magnus's classical mid-nineteenth-century experiments (Milner & Scobie, 2025). For a long time, this phenomenon, termed the Magnus effect, was considered primarily through the lens of sports physics to explain anomalous curvatures in ball trajectories (Philip et al., 2025). However, as technological systems have become more complex, the focus of contemporary research has shifted: today, a profound understanding of the aerodynamics of rotating bodies is required to solve highly complex engineering problems (Zhao et al., 2025).

At the same time, the transition from general physical concepts to precise engineering calculations remains difficult. A rotating spherical or cylindrical body in a viscous medium generates an unsteady and nonlinear flow pattern (Wu & Kinnas, 2021). A substantial role is played by the asymmetry of boundary-layer separation, wake structure, and the dependence of aerodynamic coefficients on the flow regime. Under certain conditions, anomalous regimes are also observed, including the reverse Magnus effect (Jiang et al., 2024). For this reason, even conceptually rich theoretical schemes often prove insufficient for confident prediction of trajectory and force impact.

An additional difficulty is that full-scale tests in wind tunnels and detailed computational modeling require substantial financial and computational resources. For engineering practice and educational-research work, methods are especially important that permit rapid, reproducible, and acceptably accurate verification of mathematical models without involving expensive infrastructure. It is precisely in this that the applied orientation of the present study resides.

The present study proposes an accessible computational-experimental algorithm for modeling the aerodynamics of low-speed rotating objects. A standard tennis ball of known mass and diameter serves as the physical model. Verification of the calculations is performed using optical tracking from video data. Such an approach may be regarded as a proxy model suitable for primary engineering assessment, rapid parameter tuning, and verification of numerical calculations under resource-limited conditions. The aim of the work is to develop and verify a compact methodology for the numerical calculation of the motion of a rotating spherical body, accounting for the Magnus and drag forces, and to validate it experimentally using optical tracking.

To achieve this aim, the following tasks are addressed

• To formalize a mathematical model of the motion of a rotating spherical body with allowance for the principal aerodynamic forces.
• To implement a numerical trajectory-calculation algorithm based on the fourth-order Runge-Kutta method.
• To conduct experimental verification of the model using optical tracking of the flight of calibrated spherical bodies.
• To assess the accuracy of agreement between calculated and experimental data.
• To analyze the engineering significance of rotation as a factor of trajectory control for the purposes of preliminary design and prototyping.

The novelty of the study lies in the development and verification of an accessible algorithm that integrates computational and experimental approaches to examine the Magnus effect in low-speed rotating bodies. Earlier treatments mostly probe the effect through wind-tunnel and water-channel measurement of isolated spinning spheres (Milner & Scobie, 2025; Sareen et al., 2024) or through full computational fluid dynamics (Jiang et al., 2024; Zhao et al., 2025), where instrumentation cost and meshing effort dominate the workflow, whereas the present route pairs lightweight numerical integration with optical tracking of a thrown body, a combination that the cited works do not pursue. It is shown that combining numerical integration with optical tracking of physical models enables high-accuracy prediction of the aerodynamic behavior of rotating bodies without costly test-bench equipment.

The obtained result has practical value for primary engineering design, parametric model adjustment, and accelerated prototyping. Concrete uses span the ballistics of spin-stabilized projectiles, trajectory shaping for spinning sports implements, the sizing of Magnus-type rotor blades in wind energy (Dyusembaeva et al., 2024) and early-stage tuning of guidance schemes for small unmanned aerial vehicles, each of which benefits from a cheap forecast before any wind-tunnel campaign begins.

The methodological foundation of the present study is built on the principle of scientific triangulation: the integration of theoretical deterministic modeling, computational analysis, and experimental optical validation. Such an approach ensures mutual verification of the results at each stage of the research. The theoretical part of the work is grounded in the classical differential equations of the kinematics and dynamics of a rigid body in a viscous medium, modified to account for nonlinear empirical aerodynamic coefficients. The determination of drag and Magnus forces was based on the equations of ideal fluid motion and empirical approximations from hydrodynamic literature.

An algorithmic approach was used for numerical implementation. Since the resulting system of differential equations of motion lacks an exact analytical solution due to the complex dependence of the aerodynamic coefficients on velocity and rotation frequency, a numerical integration method was employed. Scilab, an open-source platform for engineering and mathematical calculations, was used as the development tool. The fourth-order Runge-Kutta method (RKM4) was selected with an integration step size of h = 0.001 seconds. RKM4 is an iterative single-step scheme that advances the solution of an ordinary differential equation by sampling the derivative at four points across each interval and combining them through a weighted average, which suppresses the truncation error that a single-slope estimate would accumulate. The choice of this method was driven by the need to minimize approximation error when calculating steep trajectory curves, since RKM4 computes the average tangent slope at four intermediate points at each step.

The experimental part of the study, a practical case study, used high-speed dual-camera motion tracking. The test objects were pressureless Tretorn tennis balls of known mass and diameter. This choice was dictated by the fact that the empirical formulas for the lift and drag coefficients in the model had been derived specifically for this geometry and surface texture. The rough felt cover matters in its own right, because it trips the boundary layer into turbulence at moderate Reynolds numbers, and a turbulent layer clings to the rotating surface longer before separating, which amplifies the wake asymmetry that produces the Magnus force and keeps the deflection large enough to track on video, where a polished sphere would yield a far fainter and less repeatable signal. The experiment involved the use of two synchronized high-speed optical sensors, providing 240 frames per second of recording. One camera fixed the planar projection of the trajectory, i.e., the x and z axes; the second, installed along the axis of motion, registered the angular velocity of the marked projectile. Specialized Logger Pro software was used to digitize spatial-temporal coordinates and extract the initial vectors. To minimize measurement noise, the initial velocity and launch angle were calculated as the arithmetic mean of the derivatives over the first four coordinate points.

Theoretical Kinematic and Dynamic Model

The foundation of engineering prediction of the trajectory of any flying body is the accurate resolution of the force vectors acting on it in space. In the physical-mathematical model under consideration, the flight of a rotating spherical body is analyzed in a Cartesian coordinate system. Under the assumption of the absence of side wind and sidespin, the motion may be considered in a two-dimensional plane. Three principal forces act on the projectile in flight: the gravitational force directed vertically downward; the aerodynamic drag force, whose vector is antiparallel to the velocity vector of the body; and the aerodynamic lift force, namely the Magnus force (Chudinov, 2024).

The physical mechanism responsible for the emergence of the Magnus force is determined by the interaction of the rotating surface with the oncoming flow of a viscous medium. If the spherical body possesses topspin, the lower hemisphere moves in the same direction as the airflow, which increases the local flow velocity and, according to Bernoulli's principle, creates a region of reduced static pressure (Gladkov, 2022). The upper hemisphere moves against the flow, encountering streamlines, which leads to air deceleration and the formation of a zone of elevated pressure (Jiang et al., 2024). The resulting pressure gradient generates a Magnus force directed downward. The interaction of these forces is shown in Fig. 1.

Fig. 1 Vector decomposition of aerodynamic forces acting on a spherical projectile, taking into account the spin axis orientation. Mathematical formalization in accordance with Newton's second law makes it possible to express the acceleration of the body through the sum of the acting forces:

Here m is the mass of the body, a is its acceleration vector, g is the gravitational acceleration vector, D is the drag force vector, and M is the Magnus force vector.

The drag force is expressed by the formula:

where is the magnitude of the drag force, C_D is the dimensionless drag coefficient, d is the diameter, and ρ is air density. Here v is the velocity vector of the body, |v|=√(v_x^2+v_z^2 ) is its magnitude, that is, the instantaneous speed, and v_x and v_z      are its components along the horizontal and vertical axes.

The Magnus force is described by the vector product:

where F_M=C_L  (πd^2)/8 ρ|v|^2 is the magnitude of the Magnus force, C_L is the lift coefficient, ω is the angular velocity vector. 

For engineering calculations, the aerodynamic coefficients C_D and C_L are of critical importance. They are not constants, but depend on dimensionless similarity criteria, the Reynolds number, and the spin parameter. In the present study, empirical relationships approximated for spherical bodies with rough surface texture were used, following the approximations obtained for tennis balls by Štĕpánek, (1988):

These formulas are valid in the velocity intervals 13.6≤|v|≤28 m/s and rotation frequencies 800≤n≤3250 rpm. Resolving the vector equation with respect to the X and Z axes yields a system of ordinary differential equations describing acceleration along the axes:

The projection of the vector equation onto the axes and the resulting pair of scalar equations follow the derivation given by Gander and Hřebíček, (2004) where v_x^' and v_z^' denote the time derivatives of the horizontal and vertical velocity components, that is, the corresponding components of acceleration. The plus signs on the Magnus terms in these equations carry the geometric sense of the cross product, so that with topspin and the chosen axis orientation the resulting force presses the body toward the ground, which matches the downward Magnus direction described above.

Numerical Modeling and the RKM4 Algorithm

The obtained system of nonlinear ODEs does not admit an analytical solution. In this connection, a numerical integration algorithm was developed in the Scilab environment. The key decision at the simulation design stage was to use the fourth-order Runge-Kutta method (Mehdi & Kareem, 2017).

Unlike the simplest Euler method, which approximates the next point of the trajectory curve relying exclusively on the derivative at the current point, thereby leading to an accumulation of truncation error over extended flight segments, RKM4 computes four intermediate tangent-slope coefficients (k) and averages them with weighting coefficients:

where h is the integration time step; in the Scilab code, h=0.001 seconds. The Scilab code was structured as a modular architecture including a function for calculating derivatives, such as dynamic forces depending on instantaneous velocity and height, and a main iteration loop calculating the x and z coordinates until the instant of intersection with the ground plane. In parallel, a script was implemented to calculate the trajectory without accounting for rotation (CL=0, CD=0.508), enabling direct A/B testing of the Magnus effect's influence on body aerodynamics. This code is given in Appendix 1.

Experimental Validation

To verify the developed numerical model, a full-scale experiment was conducted. Such tests are standard practice in engineering: before transitioning to costly wind-tunnel tests or ballistic-range trials, macroscopic models are tested on accessible analogues. The test site was a standard tennis court, whose full length of 23.77 m defined the measured section, while the net at the midpoint, standing 0.914 m high, served as the intermediate obstacle. Spherical projectiles were launched with different initial parameters: the Low Spin group with a low rotation frequency of approximately 800-1300 rpm, and the High Spin group with rotation of approximately 1700-2500 rpm. Calibration color markings on the projectiles enabled the Logger Pro software package to determine optically the rotation frequency frame by frame from the frontal high-speed camera. Table 1 presents the key extracted data for the full set of five low-spin and five high-spin trajectories.

Table 1: Extracted kinematic parameters of experimental launches.

Comparison of the coordinate arrays extracted from the video analysis with the curves generated by the Scilab program under identical initial conditions showed high convergence. Uncertainty analysis revealed that the maximum error along the horizontal axis was 0.3 m over a distance of 20.1 m, which amounts to approximately 1.5%, while the maximum vertical error was 0.11 m at an apogee of 4.2 m, which equals approximately 2.6% (see Fig. 2).

Fig. 2 Comparison of trajectories for launch Low Spin 2 calculated in Scilab (blue dots) and extracted from LoggerPro experimental data (black dots), one of the ten trajectories used for the uncertainty analysis. The error stayed below 3% across all ten trajectories, and the single launch shown here reflects the same bound obtained for the remaining nine. This margin verifies the selected mathematical model, the reliability of the RKM4 algorithm, and the applicability of the empirical functions within the specified Reynolds-number range. Minor deviations are explained by stochastic disturbances of the medium, such as micro-gusts of wind, the discreteness of video capture, and the assumption of constant angular velocity throughout the entire flight, i.e., the neglect of viscous damping of rotation. Fig. 3 shows a comparison for high spin, demonstrating analogous results.

Fig. 3 Comparison of trajectories of launch High Spin 3 calculated in Scilab (blue dots) and extracted from LoggerPro experimental data (black dots)

Calculation of Tolerance Ranges for Initial Speed and Launch Angle under Different Spin Rates

A practical application of the developed and verified model is the analysis of error tolerance, i.e., the range of variability of input parameters within which the system fulfills the assigned task. In the context of ballistics and guidance systems, the task is for the projectile to pass over a specified obstacle and land within a prescribed target zone. Two limits bound every admissible shot: the far boundary at the court baseline, which a shot must not overfly, and the obstacle boundary at the net, which a shot must clear. Initial speed and launch angle are the two parameters swept here, since they are the inputs a thrower controls directly while spin and geometry stay fixed. To assess the influence of the Magnus effect on trajectory stability, parametric sweep was performed in Scilab.

Fixing the launch angle at 17° and searching for the boundary speed values v_1, corresponding to contact with the far boundary, and v_2, corresponding to contact with the upper edge of the obstacle, where the far boundary is the court baseline at 23.77 m and the obstacle is the net at 0.914 m. The width of the interval Δv=v_1-v_2 determines the tolerance in initial launch speed. Fixing the speed at 26 m/s and searching for the boundary angles and α and β, where α is the obstacle-boundary angle at which the trajectory just clears the net and β is the far-boundary angle at which it just reaches the baseline. The width of the interval Δθ=β-α determines the tolerance in aiming angle. Table 2 shows the boundary values of initial speeds and launch angles for different rotation frequencies.

Table 2: Boundary values of initial speeds and launch angles for different rotation frequencies.

The results show an expansion of operational tolerances (Rᵥ and Rα). Rotation at 3200 rpm increases the admissible speed window by 50.8%, and the admissible launch-angle window by 95.1% compared with a non-rotating body. The Magnus force, pressing the projectile toward the ground along a steep trajectory, compensates for excessive kinetic energy or excessively high elevation angles. The boundary sweep at 3200 rpm reaches an initial speed of about 28 m/s, which sits at the upper edge of the interval where the empirical coefficient fits hold, so predictions at that corner should be read as a limiting estimate. In artillery systems, UAV launch systems, or in the design of robotic manipulators, ensuring absolute precision of the initial impulse in both vector and magnitude is technically difficult and expensive. Imparting intense rotation to the projectile acts as a passive stabilization system, offsetting hardware errors of launching devices. The data show diminishing returns, since the rate of tolerance growth falls as rotational speed rises. Raising the spin from 1400 to 2000 rpm widens the speed window by 9.2 percentage points, while raising it from 2600 to 3200 rpm adds only 7.2 percentage points. Consequently, an infinite increase in rotation speed is energetically and mechanically impractical.

Reverse Magnus Effect

The linear paradigm of the classical Magnus effect is violated in two critical scenarios, which poses a significant risk in the design of aeronautical and space systems. The first scenario is the transitional Reynolds-number regime. In the near-critical range around Re≈3×10^5 for a smooth sphere, the direction of the Magnus lift force may invert (Nagahiro & Hayakawa, 2023). The physical nature of this phenomenon, termed the reverse Magnus effect, lies in the asymmetric transition of the boundary layer from laminar to turbulent flow. On the advancing side of the sphere, rotation destabilizes the flow, initiating an early transition to turbulence. A turbulent boundary layer possesses greater kinetic energy and can resist a positive pressure gradient for a longer time, thereby shifting the separation point farther downstream and forming a laminar separation bubble (Sareen et al., 2024). On the retreating side, the flow remains laminar and separates earlier. As a result, the wake deflects in the opposite direction, and the lift force vector inverts and points upward under topspin, which reverses the downward direction that the normal Magnus effect produces for the same spin.

The second scenario is hypersonic rarefied flow. In the design of reentry vehicles and in modeling the entry of space debris into dense atmospheric layers, the continuum hypothesis is no longer valid (Chen & Zhou, 2021). For such calculations, the direct simulation Monte Carlo method is used. Numerical studies of hypersonic rarefied-gas flows show the stable formation of a reverse Magnus force. Under these conditions, the inversion is already explained by direct tangential momentum transfer between individual incident gas molecules and the windward side of the sphere (Jiang et al., 2024). For aerospace design engineers, this means that aerodynamic control algorithms designed for a dense atmosphere may lead to the fatal loss of a vehicle in the stratosphere (Jiang et al., 2024). Large-scale implementation is hindered by technical limitations. First, there is a need for active mechanical drives to maintain the rotation of cylinders or spheres, which reduces the overall system efficiency. Second, gyroscopic moments from heavy rotating masses complicate maneuvering. Third, seam asymmetry or manufacturing defects may cause unpredictable asymmetry of the boundary layer. Deviations in seam geometry from the axis of rotation may introduce irregular shifts of the force vector, rendering precise numerical calculations, such as those provided by the RKM4 model, inapplicable in that regime, though the present study does not test this case and the extent of the effect remains to be quantified.

Engineering Implication

The proposed methodology combines numerical modeling with accessible experimental verification, which removes the need for a wind tunnel, a ballistic range, or high-cost measurement benches. A computational model in a standard calculation environment and optical registration of a physical sample over a limited section are sufficient to assess how speed, launch angle, and rotation shape a trajectory before large-scale testing begins. This makes preliminary aerodynamic assessment workable for small research groups and educational laboratories, where each full-scale test consumes materials, time, and measurement resources. The methodology is particularly valuable for the design of devices that use rotation to enhance flight stability. Such systems include sports launching mechanisms, educational ballistic installations, experimental launching complexes, and compact flying objects with passive stabilization. The model allows calculating in advance how the range, trajectory height, and impact-point position will change as the angular velocity varies. This gives the engineer the opportunity to select a rotation regime under which admissible deviations in speed and launch angle are expanded. In practical terms, this means a reduction in the requirements for the actuation mechanism's accuracy and a simplification of the launching unit's design.

A separate direction of application concerns problems of tolerances and reliability. In engineering practice, it is insufficient to calculate a single trajectory for ideal initial conditions. It is necessary to understand within what limits a system remains operable despite unavoidable variations in speed, launch angle, mass, and rotation frequency. The proposed algorithm enables parametric scanning and the determination of the boundaries of the stable regime in which the body passes over the obstacle and enters the specified region. This is especially useful in the design of systems that involve technological manufacturing errors and instability in the launch impulse. As a result, the engineer obtains the calculation of the nominal regime and a quantitative picture of the operational margin. A natural extension of this model would be its use in tuning guidance or launch-correction algorithms, though this would require validating the model under closed-loop conditions not tested here.

Finally, the methodology is useful as a tool for engineering selection of the model's applicability and for identifying its validity limits. The text shows that, in a number of regimes, the direction of the lift force may change, and the body's behavior ceases to follow the customary pattern. For the designer, this is of fundamental significance, since an error in physical interpretation leads to an error in design, material selection, and control logic. The use of a computational-experimental algorithm enables sufficiently early detection of which parameter range yields a reliable forecast and where a more complex description of the medium, rotation, and flow structure is required. Precisely for this reason, the methodology is useful both as a computational instrument and as a means of engineering judgment, enabling the distinction between steadily predictable regimes and potentially dangerous, poorly formalizable scenarios.

The conducted study confirms that the proposed computational-experimental methodology is a valid instrument for analyzing the trajectories of rotating spherical bodies in the low-speed regime. The constructed mathematical model, incorporating drag and Magnus forces, was implemented numerically in Scilab using the fourth-order Runge–Kutta method, ensuring stable calculation of the nonlinear trajectory with a small integration step. Experi-mental validation based on video data demonstrated a high degree of agreement between calculated and measured coordinates. The recorded discrepancy level of less than 3% indicates that the model is sufficiently accurate for primary engineering assessment, parametric tuning, and applied analysis of the motion of rotating bodies. The work acquires particular significance because verification of the numerical calculation was carried out using an inexpensive optical tracking method based on high-speed video recording and subsequent trajectory digitization. Such an approach substantially lowers the entry threshold to experimental aerodynamics and makes model verification accessible under conditions of limited budget, shortage of specialized infrastructure, and compressed engineering-development timelines. In combination with the open computational environment Scilab, this scheme forms a practical and reproducible research framework in which the cost of the instrumentation does not suppress the investigative value of the result. The results of numerical modeling and experimental verification also show that rotation is a substantial factor in determining the trajectory's character and the range of admissible initial conditions. Parametric scanning revealed an expansion of operational tolerances in speed and launch angle with increasing rotation frequency, suggesting the possibility of using the Magnus effect to improve flight stability and reduce the system's sensitivity to initial parameter dispersion. For the tasks of preliminary design, accelerated prototyping, and tuning of computational schemes, this is of direct importance, as it enables assessment of operating regimes at an early stage without recourse to costly testing facilities.

The author thanks the anonymous reviewers for comments that improved the clarity of the manuscript. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The author declares no conflicts of interest regarding the publication of this article.