## 2 Materials and Methods  
In this work, we derived \(C_{D}\) from the measured trajectories of silica microspheres (\(D_{\text{p}}\) = 3.7 and 7.6 \(\upmu\)m) launched at initial velocities ranging from 90 to 770 m/s in two different air pressures (\(P\) = 760 and 76 Torr) using LIPIT. Experiments were conducted in 25 \({}^{\circ}\)C air, corresponding to a 346 m/s speed of sound [16] and air viscosity [16] of 1.8 \(\times\) 10\({}^{-5}\)Ns/m\({}^{2}\). Air density [16], \(\rho\), was 1.2 kg/m\({}^{3}\) in \(P\) = 760 Torr and 0.12 kg/m\({}^{3}\) in \(P\) = 76 Torr. In LIPIT, which is illustrated in Figure 1a, microspheres are dispersed on the surface of a launch pad consisting of glass, gold, and crosslinked polydimethylsiloxane (PDMS) layers at respective thicknesses of 210 \(\upmu\)m, 70 nm, and 30 \(\upmu\)m. A 1064 nm laser pulse incident upon the underside of the launch pad locally ablates the gold film, causing a rapid expansion of the PDMS layer above it. This ablation launches the microsphere, whose flight is illuminated by ultrafast sub-picosecond white-light pulses at even time intervals (noted as \(\Delta t\) in Figure 1a) and captured in a multi-exposure stroboscopic image, as in Figure 1b. \(v_{0}\) is the microsphere's velocity at the first exposure within the field of view, as noted in Figure 1a. Each individual microsphere was launched through a single 115 \(\times\) 115 \(\upmu\)m opening in a solid baffle positioned between 1 and 2 mm from the launch pad. The baffle blocked the propagation of shock waves produced by the ablation event, thereby ensuring that the air behind the baffle would remain undisturbed until the microsphere's arrival through the opening. The exposures at \(t=t_{0}\) and \(t=t_{n}\) in Figure 1a represent the first and last exposures within the camera's field of view, where \(n\) ranged from 5 to 17. After calibrating the multi-exposure image using a calibration target in the focal plane, the positions of all exposures from \(t=t_{0}\) to \(t=t_{n}\) were measured to establish the microsphere's trajectory. The distance and time between the \(t=t_{0}\) and \(t=t_{n}\) exposures ranged from 330 to 1175 \(\upmu\)m and from 0.98 to 9.20 \(\upmu\)s, respectively. To illustrate the nonlinearity of the microspheres' trajectories, the positional deviation from a microsphere of constant velocity is presented in Figure 2a for microspheres at \(v_{0}\approx 520\) m/s for each of the three experimental conditions. We used a central difference approximation of the first derivative to convert the microsphere's position-time trajectory into velocity-position data, as shown in Figure 2b. We then performed a regression fit of the velocity-position data using Equation (1), achieving an average \(R^{2}\) value of 0.96 across all experiments. Equation (1) is an exponential decay function giving velocity as a function of position and assuming an exponential loss of kinetic energy. The fitting parameters of Equation (1) are \(v_{0}\) and \(\beta\), the exponential decay parameter. Despite the sub-10 \(\upmu\)s timescales of the measured portion of flight, the microspheres exhibited sufficient velocity drop to assess deceleration with confidence. A velocity drop of at least 1% of the initial velocity (\(\Delta v/v_{0}\geq 1\%\)) over the measured range is necessary in order to accurately calculate a microsphere's deceleration [17]. This condition was satisfied in all tests conducted in the present work. Microspheres launched in the three conditions (\(D_{\text{p}}=3.7\) um, \(P=760\) Torr; \(D_{\text{p}}=7.6\) um, \(P=760\) Torr; \(D_{\text{p}}=3.7\) um, \(P=76\) Torr) at \(0.3<M<2.2\) exhibited \(\Delta v/v_{0}\) ranges of 7.2-22.3%, 5.2-10.5%, and 1.6-6.0% and Knudsen numbers (\(kn\))  
Figure 1: (**a**) Schematic of LIPIT experiment (**b**) Ultrafast stroboscopic image of microsphere trajectory (\(\Delta t=640\) ns, \(D_{\text{p}}=7.6\) μm).