## 3 Results and Discussion  
By varying microsphere velocity, microsphere diameter, and air pressure, we quantified \(C_{D}\) at a range of flow conditions, reflected by \(Re\) and \(M\), spanning from 5 to 300 and 0.3 to 2.2, respectively. Figure 3 presents \(C_{D}\) vs. \(Re\) colored according to \(M\) for the three combinations of \(D_{\text{p}}\) and \(P\). It is apparent from Figure 3 that in \(P=760\) Torr, \(C_{D}\) changes non-monotonically near \(M=1\). In the supersonic regime, \(C_{D}\) increases with decreasing \(M\). Figure 3 illustrates how pressure drag mainly contributes to drag force in this regime due to the large low-pressure region in the wake of the detached bow shock, which contains a toroidal recirculation region downstream of the sphere [22; 23]. As the supersonic velocity decreases, the angle of the shock wave increases, causing the low-pressure region in its wake to expand [23; 24]. Expansion of the low-pressure region incurs greater pressure drag and results in an increase in \(C_{D}\). As the microsphere decelerates through the transonic regime (estimated to be 0.8 < \(M\) < 1.2), the additional drag created by the shock wave is lost, resulting in a reduction in \(C_{D}\). Due to differing microsphere diameters, this transition occurs at lower \(Re\) values for the \(D_{\text{p}}=3.7\) um, \(P=760\) Torr case than the \(D_{\text{p}}=7.6\) um, \(P=760\) Torr case. As the microsphere's velocity continues to decrease below \(M=0.8\), the flow separation point moves to the downstream side of the microsphere and the low-pressure zone in the microsphere's wake reduces in volume [22; 23; 25; 26]. While the reduction of the low-pressure region induces a decrease in pressure drag, the concurrent increase in friction drags results in a monotonic net increase in \(C_{D}\). In the \(P=76\) Torr case, we do not observe a distinct change in \(C_{D}\) during the transonic regime due to the weakened contribution of the bow shock in reduced pressure. The error bars [27], representing a 95% confidence interval of two standard deviations, are the largest for the \(P=76\) Torr condition. By implementing the method of sequential perturbation to calculate error [27], it was found that the uncertainty in deceleration accounted for most of the error in \(C_{D}\). As noted previously, the \(\Delta v/v_{0}\) values are smallest for the \(P=76\) Torr condition, yielding the least accurate measurement of deceleration among the three experimental conditions. The \(C_{D}\) vs. \(Re\) results in Figure 3 can be used to model microsphere velocity for micro-ballistic mechanical characterization and for projectile delivery of genes and drugs, ultimately determining impact velocity in both applications. In the former, the specimen impact velocity provides the microsphere's kinetic energy upon impact. In the latter, the skin impact velocity [10] may be utilized in a theoretical penetration model [28] to estimate the tissue depth reached by the penetrating microsphere [9; 10; 29]. Future studies of projectile gene and drug delivery may achieve greater efficacy by including the present work's quantification of \(C_{D}\) in such penetration models. The \"standard drag curve\" for spheres illustrates \(C_{D}\) vs. \(Re\) in subsonic, incompressible flow [30]. Many empirical and semi-empirical models, which are functions of \(Re\) only, have been developed to approximate the standard drag curve [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. The model given by Equations (4) and (5) is recommended by [PERSON], [PERSON], and [PERSON] [30] as the most accurate approximation. This approximation is provided in Figure 3 and shows reasonable agreement with the subsonic \(P=760\) Torr results from the present work for \(M<0.8\).