# spl2 - Trajectory Calculation Lab 2 Lecture Notes...

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V Trajectory Calculation Lab 2 Lecture Notes Nomenclature t time ρ air density h altitude g gravitational acceleration V velocity, positive upwards m mass F total force, positive upwards C D drag coeﬃcient D aerodynamic drag A drag reference area ˙ ( ) time derivative ( = d ( ) /dt ) i time index Trajectory equations The vertical trajectory of a rocket is described by the altitude and velocity, h ( t ), V ( t ), which are functions of time. These are called state variables of the rocket. Figure 1 shows plots of these functions for a typical ballistic trajectory. In this case, the initial values for the two state variables h 0 and V 0 are prescribed. h 0 h t V 0 t V h Figure 1: Time traces of altitude and velocity for a ballistic rocket trajectory. The trajectories are governed by Ordinary Differential Equations (ODEs) which give the time rate of change of each state variable. These are obtained from the definition of velocity, and from Newton’s 2nd Law. h ˙ = V (1) V ˙ = F/m (2) Here, F is the total force on the rocket. For the ballistic case with no thrust, F consists only of the gravity force and the aerodynamic drag force. mg D , if V > 0 F = (3) mg + D , if V < 0 The two cases in (3) are required because F is defined positive up, so the drag D can subtract or add to F depending in the sign of V . In contrast, the gravity force contribution mg is always negative.

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