In real-world projectile motion, especially at high speeds, the resistive force exerted by air is more accurately modeled as being proportional to the square of the velocity. This quadratic drag model is well-supported in classical mechanics and is essential for accurate predictions in atmospheric flight, sports physics, and ballistic trajectories.

As established by Marion and Thornton [13] and further elaborated by Kane and Levinson [29], the drag force F d acting on a body moving through a fluid at high Reynolds number is given in Equation (1) below.

where C d is the drag coefficient, ρ the fluid density, A the cross-sectional area, and v the velocity. This quadratic dependence becomes dominant when inertial forces transcend viscous effects, typically at speeds exceeding a few m/s for small projectiles in air.

Lubarda [30] and Carmago [31] provided complete analytical and numerical treatments of projectile motion under quadratic drag, demonstrating its nonlinearity and sensitivity to both initial conditions and mass. Their work aligns with our simulation results showing that lighter projectiles are significantly more affected by air resistance, due to their lower momentum and higher deceleration rates under equal drag forces.

Specifically, since the acceleration due to drag is inversely proportional to mass, as in Equation (2):

A smaller mass results in a larger deceleration for the same velocity. This explains the reduced vertical displacement observed in our simulations for the 0.046 kg projectile, as compared to the 0.198 kg projectile, over equal time intervals.

This study presents a numerical investigation of vertical and horizontal components under the influence of gravity and quadratic air resistance. The motion is modeled using classical Newtonian mechanics, with particular attention to nonlinear drag forces that significantly affect objects moving at high velocities through a fluid medium. Due to the complexity of the resulting differential equations, a computational approach is adopted [32]. All simulations and numerical computations were performed using FreeMat, a high-level interactive environment for numerical computation and data visualization [33].

The use of FreeMat allowed for efficient coding, execution, and visualization of the results, including time-series plots of velocity and displacement [34][35]. The numerical accuracy and stability of the simulation were ensured through careful selection of the integration time step and validation against known theoretical behavior in limiting cases (e.g., motion in vacuum or low-speed regimes) [36][37]. Given the nonlinearity of the differential equations, closed-form analytical solutions are generally not obtainable. Therefore, a numerical solution is implemented using FreeMat. The simulation employed a time-stepping algorithm based on the classical fourth-order Runge-Kutta (RK4) method to approximate the velocity and displacement of the object as functions of time [21][38].

Key simulation parameters, such as initial velocity, object mass, drag coefficient, and time step, were defined and systematically varied to evaluate their influence on the motion. The computational model tracks the object’s velocity and position over time [39][40], enabling the analysis of critical phenomena such as terminal velocity and time to peak height. However, Simulations used a fixed time step of Δt = 0.01 s, chosen based on a convergence analysis showing that halving Δt altered the final range by less than 0.1%, confirming solution insensitivity to further refinement and ensuring both accuracy and computational efficiency.

The aerodynamic parameters governing drag, namely the drag coefficient ( C d ), air density ( ρ ) was taken as ρ = 1.225   kg / m 3 [41], and the cross-sectional area ( A ), directly influence the drag constant c , which appears in the quadratic drag formulation in Equation (3) below.

Table 1 summarizes these parameters for the three projectiles examined. The values of C d were selected from experimentally established ranges for spherical bodies moving at moderate to high Reynolds numbers ( R e ~ 10 4   -   10 5 ) [42][43], ensuring that the assumptions of Reynolds numbers independent drag is appropriate for the velocities analyzed. The diameters correspond to regulation sporting equipment and were used to compute the frontal area via A = π d 2 / 4 .

Table 1. Aerodynamic parameters used.

The hierarchy of drag constants and c m ratios in Table 1 is consistent with classical aerodynamic scaling: drag force increases with frontal area and C d , whereas resistance to deceleration increases with mass [10][44][45]. However, heavier projectiles preserve momentum longer and exhibit less curvature distortion and range suppression under quadratic drag.

Moreover, the aerodynamic parameters in Table 1 create the physical foundation for the observed mass-dependent behavior in subsequent results. By accurately representing realistic projectile geometries and aerodynamic loading conditions, these parameters ensure that the simulated trajectories faithfully capture the nonlinear dynamics of projectiles moving through atmospheric flows.

In many real-world scenarios, especially at higher velocities, objects moving through a fluid (such as air or water) experience a drag force that is proportional to the square of their velocity. This type of resistance is referred to as quadraticdrag, and it plays a significant role in the dynamics of projectiles, vehicles, and objects in free fall. In this section, we consider the horizontal motion of an object subject to such a drag force and derive analytical expressions for its velocity and displacement as functions of time.

To begin the analysis, we apply Newton’s second law in the horizontal direction, considering only the drag force, which acts to oppose the motion. The force due to quadratic drag is given by F = − c v 2 , where v is the velocity, and ccc is the drag coefficient, which depends on the shape, surface area, and properties of the medium. However, for a standard size, it is the quadratic drag force that affects the motion. For this reason, the equation of motion (in terms of v ) becomes as in Equation (3).

Equation (4) can also be written with components from vertical and horizontal as.

As we noted earlier, these Equations (5) and (6) are coupled and are generally not solvable analytically (in terms of equations); they can be solved numerically.

Applying Newton’s second law F = m a , we obtain the differential equation governing the motion:

This is a first-order nonlinear differential equation due to the v 2 term. Nonlinear equations are generally more complex to solve than linear ones, but this equation is separable, which simplifies the process. By rearranging terms, we separate the variables v and t :

We then integrate both sides of the equation. The left-hand side is integrated with respect to velocity from the initial velocity v 0 to v ( t ) , and the right-hand side with respect to time from 0 to t :

The integration yields:

Solving for v ( t ) , we obtain

Where we define a characteristic time constant τ as:

This characteristic time τ indicates how quickly the velocity decays due to drag. Physically, it represents the time it takes for the object’s velocity to reduce significantly from its value.

Next, to find the position x ( t ) of the object as a function of time, we integrate the velocity:

This integral evaluates to:

Assuming the initial position x 0 = 0 , the final expression for displacement becomes:

This function, as described in Equations (14) and (11), represents the horizontal velocity and position of an object experiencing quadratic drag as it decelerates over time, respectively. Unlike motion in a vacuum, where velocity remains constant in the absence of horizontal forces, the quadratic drag causes a rapid initial deceleration, which gradually slows as the object continues moving and its velocity decreases. The logarithmic nature of the position function reflects the diminishing displacement over time as drag dominates the motion.

In the vertical direction, the motion of an object subjected to both gravitational force and quadratic air resistance is governed by the following equation of motion for downward (15) and upward motion (16):

where m = The mass of the projectile (kg);

g = Acceleration due to gravity (m/s²);

v = The instantaneous velocity (m/s);

c = Gravity coefficient (kg/m).

The term k v 2 represents the quadratic drag force, which becomes increasingly significant at higher velocities and is a more accurate representation of air resistance for many real-world objects compared to linear models.

If we consider the spherical projectile shape (baseball) thrown vertically upward with speed v 0 and is subjected to a quadratic drag with magnitude f ( v ) = c v 2 , and the velocity reaches its terminal ( v t e r m ), then the separation from the vertical equation of the motion becomes as in Equation (17).

In this separation form, we can integrate both sides directly (assuming v 0 = 0 the condition is met).

Consider the left-hand side of Equation (18):

The simplification of (20) yields Equation (21) as the vertical velocity of the projectile motion under quadratic drag.

To get the maximum vertical displacement of the projection under quadratic drag motion, we have to integrate the other form of equation v = − g [ 1 + ( v v t e r m ) 2 ] , which yields Equation (22).

The Equations (21) and (22) characterize the vertical motion of a body subjected to gravitational acceleration in the presence of a quadratic drag force, where the resistive force is proportional to the square of the velocity. These models assume motion through a uniform medium with constant physical properties, neglecting all forces except gravity and quadratic drag on the form F d r a g = − c v 2 . The velocity-time relation, given by Equation (21), describes the temporal evolution of velocity for a body released from rest, where v t e r m = m g / c denotes the terminal velocity attained as t → ∞ . The associated displacement, expressed in Equation (22), represents the vertical position as a function of time for initial upward motion and symmetric ascent-descent scenarios, respectively. These analytical solutions are critical for accurately modelling projectile trajectories in real-world applications, particularly where air resistance cannot be neglected, such as in aerospace vehicle dynamics, sports ballistics, and environmental physics.

The horizontal displacement data expose a clear nonlinear relationship with both projectile mass and initial launch velocity. As shown in Table 2, increasing the initial speed from 10 m/s to 50 m/s results in substantial gains in horizontal range. However, the magnitude of these gains is disproportionately larger for heavier projectiles. Figure 1(c) shows that the horizontal displacement for the softball (0.189 kg) at 50 m/s is approximately 698.81 m, compared to just 209.92 m for the golf ball (0.046 kg) in Figure 1(a). This disparity accentuates the inverse scaling of the quadratic drag force with mass, as articulated by the ratio c v 2 m , where lighter bodies experience more pronounced deceleration.

Table 2. The variations of initial velocities and masses with corresponding displacements (m).

Figure 1. The horizontal displacement of different mass projectiles: (a) golf ball (0.046 kg); (b) baseball (0.145 kg); (c) softball (0.198 kg).

The simulations further demonstrate that at lower initial velocities, the displacement curves tend to flatten, indicative of a saturation behavior commonly observed in drag-dominated regimes. The diminished marginal gains in displacement at lower speeds reinforce the quadratic nature of the resistive force, which imposes exponentially increasing energy losses with velocity. This highlights the inadequacy of ideal projectile models in accurately predicting flight range in realistic aerodynamic environments, particularly for lightweight projectiles (see Figure 1).

Table 3 shows the horizontal velocity of each projectile mass after a flight duration of 40 seconds. Figure 2 shows the results of a steep decay in horizontal velocity for all cases, with the rate of decay inversely correlated with mass. At 50 m/s initial velocity, the golf ball’s horizontal speed drops to 0.80 m/s, while the baseball and softball maintain significantly higher terminal velocities of 2.45 m/s and 3.15 m/s, respectively. This is consistent with the analytical expectation that terminal velocity in the presence of quadratic drag is attained faster in lighter bodies due to their reduced inertial resistance to deceleration.

The decay trends observed are highly consistent with the analytical expression in Equation (11), where Equation (12) represents the characteristic decay time. The slower decay observed in heavier projectiles reflects their longer characteristic times, reaffirming the theoretical model and verifying the numerical accuracy of the simulation. Notably, the approach to terminal velocity is asymptotic, demonstrating the diminishing impact of acceleration over time due to drag dominance (Table 3).

Table 3. The final velocities (m/s) with mass variations at time reference 40 s.

Figure 2. The horizontal final velocities of different mass projectiles: (a) golf ball (0.046 kg); (b) baseball (0.145 kg); (c) softball (0.198 kg).

The vertical displacement of a projectile subjected to quadratic air drag was analyzed for varying projectile masses (0.046 kg, 0.145 kg, and 0.198 kg) over different time intervals. The results, presented in Figures 3(a)-(d) and summarised in Table 4, demonstrate the significant influence of both mass and time on the vertical displacement trajectory under drag effects.

As shown in Figure 3, the displacement-time curves disclose nonlinear behaviour due to the velocity-squared dependence of the drag force. For all cases, the projectile with the highest mass (0.198 kg) consistently attained the most significant vertical displacement, while the lightest projectile (0.046 kg) experienced the most pronounced reduction in displacement. This behaviour is expected, as the decelerating effect of drag is inversely related to mass, causing lighter objects to lose vertical momentum more rapidly.

It is observed that at 5 seconds, the vertical displacement ranges from 67.03 m (for 0.046 kg) to 97.94 m (for 0.198 kg). By 20 seconds, the displacement increases significantly, reaching 645.72 m for the 0.198 kg projectile. This progression, detailed in Table 4, further underscores the nonlinear growth of displacement and the mass-dependent nature of drag deceleration.

Moreover, the velocities shown in each subfigure highlight how drag reduces the effective vertical velocity more significantly for lighter projectiles. The declining slope gradient over time, especially for the 0.046 kg mass, confirms that energy dissipation due to drag becomes increasingly dominant as time progresses. The corresponding velocities at the end of each time interval are also indicated in the legends. The results illustrate that greater projectile mass leads to reduced sensitivity to drag force, resulting in higher vertical displacements over time.

Table 4. Vertical displacement (m) for varying projectile masses and time intervals under quadratic drag.

Figure 3. Vertical displacement of a projectile under the influence of quadratic air drag for three different masses: 46 g (blue), 145 g (red), and 198 g (black). Subfigures (a) - (d) represent displacement profiles over increasing time intervals: (a) 0 - 5 s, (b) 0 - 10 s, (c) 0 - 15 s, and (d) 0 - 20 s.

These results validate the theoretical model of quadratic drag and suggest that mass is a critical parameter when predicting vertical motion in resistive media. The consistency between simulated trajectories and physical expectations indicates that the numerical model reliably captures drag-induced motion suppression.

A consistent and central finding across both horizontal and vertical analyses is the pivotal role of mass in modulating drag effects. Heavier projectiles exhibit greater mechanical robustness against the retarding influence of air resistance, maintaining higher velocities and displacements throughout their trajectories. This is not merely a scaling effect; instead, it reveals a multiscale interaction between inertial and aerodynamic forces that profoundly alters trajectory characteristics. The differential acceleration profiles suggest that optimising projectile design for distance or stability in real-world applications must consider the nuanced interplay between mass, drag coefficient, and initial kinetic energy (Figure 4). Three masses 0.046 kg (blue), 0.145 kg (red), and 0.198 kg (black) are simulated, with corresponding terminal velocities of 17.9 m/s, 31.8 m/s, and 37.2 m/s, respectively. The curves clearly show how increased mass results in higher terminal velocity due to greater gravitational force overcoming the quadratic drag, consistent with the theoretical dependence v t e r m ∝ m as in Equation (20).

Figure 4. Vertical velocity as a function of time for projectiles with different masses falling under gravity with quadratic air resistance.

Such mass-dependent behavior has implications beyond simple range estimation. In high Reynolds number flows, as simulated here, the projectile’s response to aerodynamic damping can inform predictive modeling in fields as diverse as aerospace re-entry, sports physics, and environmental dispersion of aerosols and debris.

To directly visualize the combined effect of horizontal and vertical motion under quadratic drag and to assess the overall shape of the projectile path, we present the full 2D trajectories for the golf ball (0.046 kg), baseball (0.145 kg), and softball (0.198 kg) in Figure 5. All projectiles were launched from ground level with an initial velocity of 50 m/s at a launch angle of 45˚, which maximizes range in a vacuum and provides a clear baseline for comparison.

Figure 5. 2D trajectories for projectiles of different masses launched at 50 m/s and 45˚ under quadratic air drag.

The trajectories exhibit two key features predicted by theory but not visible in separate 1D analyses:

1)TrajectoryFlattening: Compared to the ideal parabolic path (black dashed line, if added, or described as “vacuum trajectory”), all three trajectories are visibly flattened. The apex height is significantly reduced, and the range is shortened. This effect is most pronounced for the lightest projectile, the golf ball, whose peak height is less than half that of the vacuum trajectory. This demonstrates the dominant energy dissipation due to drag, particularly during the ascent phase when both gravity and drag act to decelerate the projectile.

2)AsymmetricPath: A crucial observation is the distinct asymmetry between the ascent and descent phases. The descent is steeper and more rapid than the ascent. This is because during ascent, the drag force acts downward, adding to gravity, resulting in a large negative acceleration. During descent, the drag force acts upward, opposing gravity, leading to a smaller net downward acceleration. This results in a trajectory that is skewed to the left of its apex compared to the symmetric parabola. The degree of asymmetry increases for lighter masses, as seen in the golf ball’s trajectory, which has a much sharper drop-off after its peak compared to the more gradual descent of the heavier softball.

This 2D visualization directly connects our component analyses to the overall projectile path. The reduced range observed in Table 4 and the lower maximum height noted in Section 3.3 are now seen as consequences of the distorted, asymmetric, and flattened trajectory caused by quadratic drag. The mass dependence is also evident: the heavier softball follows a path closest to the ideal parabola, while the lighter golf ball deviates most significantly, confirming that increased mass provides greater resistance to the distorting effects of aerodynamic drag