The Aerodynamics of Model Rocket Flight
Estimating the altitude of a model rocket requires solving the classical differential equations of motion under gravity and aerodynamic drag. A model rocket flight is divided into two major phases: the boost phase (where the motor is burning, providing thrust) and the coast phase (from motor burnout up to the highest point, the apogee).
During the boost phase, the forces acting on the rocket are thrust, gravity, and drag. Drag increases with the square of velocity: \(D = \frac{1}{2} \rho A C_d v^2\) (where \(\rho\) is air density, A is cross-sectional area, \(C_d\) is the drag coefficient, and v is velocity). The net acceleration during boost is: \(a = \frac{F_{\text{thrust}} - D}{m} - g\).
Because drag changes with speed, calculating velocity at burnout and boost height requires integrating this equation over the burn time of the motor. A well-designed rocket with a smooth nose cone and fins has a low drag coefficient (typically 0.45 to 0.75), whereas rough construction or mismatched body tubes can raise this significantly, drastically reducing peak altitude.