Module 2 — Physics of flight

Four spinning props, one rigid body, Newton in charge. Master this module and every tuning mystery in Module 9 becomes mechanical intuition.

🟢 Foundations. A propeller throws air downward; Newton's third law pushes the drone up. Hovering means total thrust $T$ equals weight $W = mg$. Moving means tilting: lean the thrust vector forward and its horizontal component accelerates you. A quad steers by making its four thrusts unequal:

• Roll/pitch: speed up motors on one side, slow the other side → torque tilts the frame.
• Yaw: every spinning prop applies a reaction torque opposite its rotation. Speed up the clockwise pair, slow the counter-clockwise pair → the body twists.

            FRONT
     M4 (CW)      M2 (CCW)
        \            /
         \          /
          [ flight ]
          [  ctrl  ]
         /          \
        /            \
     M3 (CCW)     M1 (CW)
            REAR
  Betaflight default numbering & "props-in" rotation —
  always verify in your configurator.

🟡 Practitioner. The mixer turns four pilot commands — throttle $T$, roll $R$ (+ = right), pitch $P$ (+ = nose up), yaw $Y$ (+ = clockwise from above) — into four motor outputs. For the layout above:

$\begin{aligned} M_4\ (\text{FL, CW}) &= T + R + P - Y\\ M_2\ (\text{FR, CCW}) &= T - R + P + Y\\ M_1\ (\text{RR, CW}) &= T - R - P - Y\\ M_3\ (\text{RL, CCW}) &= T + R - P + Y \end{aligned}$

(Sign conventions differ between firmwares; the pattern — which motors move together — is the invariant.) Propeller performance follows scaling laws with rotation speed $n$ (rev/s) and diameter $D$:

$T = C_T\,\rho\,n^2 D^4 \qquad P_{\text{shaft}} = C_P\,\rho\,n^3 D^5$

Thrust grows with the square of RPM but power with the cube — the fundamental reason bigger, slower props are more efficient. Air density $\rho \approx 1.225$ kg/m³ at sea level (and less at altitude or in heat — your quad genuinely flies worse in summer mountains).

🔴 Advanced. Momentum (actuator-disk) theory gives the floor on hover power. With total disk area $A$:

$v_h = \sqrt{\frac{T}{2\rho A}} \qquad P_{\text{ideal}} = T\,v_h = \frac{T^{3/2}}{\sqrt{2\rho A}}$

Worked example — 7-inch long-range quad. AUW 1.5 kg → $T = 14.7$ N. Four 7″ props: $A = 4\pi(0.089)^2 \approx 0.099$ m². Then $v_h \approx 7.8$ m/s and $P_{\text{ideal}} \approx 114$ W. Real electrical power divides by total efficiency (figure of merit × motor × ESC ≈ 0.55): $P_{\text{hover}} \approx 205$ W, i.e. hover efficiency ≈ 7.3 g/W — right in the real-world range for this class. Module 6 turns this into flight time.

Full rigid-body dynamics (used by every simulator and autopilot):

$m\dot{\mathbf v} = m\mathbf g + \mathbf R(q)\begin{bmatrix}0\\0\\T\end{bmatrix} - \mathbf F_{\text{drag}},\qquad \mathbf I\dot{\boldsymbol\omega} = \boldsymbol\tau - \boldsymbol\omega\times(\mathbf I\boldsymbol\omega)$

with drag $F_d = \tfrac12\rho v^2 C_d A_{\text{ref}}$, per-motor thrust $T_i = k_f\omega_i^2$ and yaw torque $\tau_i = \pm k_m\omega_i^2$. The gyroscopic cross-term $\boldsymbol\omega\times \mathbf I\boldsymbol\omega$ is why fast rolls couple into pitch on stretched frames.

⚫ Master. You reason about vortex ring state (descending into your own downwash — why fast vertical drops get mushy), translational lift, blade flapping, prop inertia vs control bandwidth, and you can build a numeric model of your exact quad (mass, inertia tensor, $k_f$, $k_m$ from thrust-stand data) that predicts blackbox logs before you fly.

Mastery checklist

• Derive the mixer table for a hexacopter from the same three principles.
• Estimate hover current of any quad within ~20 % from mass and prop size alone.
• Explain propwash oscillation using momentum theory, not folklore.

🖼️ Image ideas: your own thrust-stand photos; NASA public-domain rotor/airflow illustrations; Wikimedia Commons "Quadcopter yaw/roll/pitch" diagrams (check PD/CC0).

📚 Free resources: MIT OCW 16.07 Dynamics; ArduPilot & PX4 docs on multicopter dynamics; open thrust databases (miniquad test data spreadsheets).