MATHEMATICAL MODELS Constant velocity motion A function model: p(t) = V*t + P0 A 1^st order differential equation model: p’(t) = V A 2^nd order differential equation model: p’’(t) = 0 A falling object Newton’s model: r’’(t) = -G*M*MEarth/r(t)^2 A simplified model: r’’(t) = -10 Motion of a rigid object F=MA T=a’’/I Atmospheric drag for an object Air density: D = 1.4*(1 – altitude/30,000) for altitude < 30,000 Force of atmospheric drag: Fdrag = Cdrag * D * v^2 * cs / 2 Airplane lift and drag CL = 0.5 + 0.1*AoA Flift = CL*D** v^2 * s CD = (5 - AoA )*1 + 0.7 for AoA <= 5 CD = (AoA - 5 )*1 + 0.7 for AoA > 5 Fdrag = CD*D** v^2 * s Electrical components A resistor: vR(t) = i(t)R A capacitor: vC’(t) = i(t)/C An inductor vI(t) = i’(t)L Mechanical components A spring: Fs(t) = -K * p(t) A hydraulic damper: Fd(t) = -cp’(t)

Sliding friction Coulomb or sliding friction: FC(t) = -mNsign(p’(t)) Programmed models We have programmed models for rocket thrusters, airplane thrust, aircraft elevator torque, electrical impulses, and rocky roads, by directly specifying their values at each subinterval in the computation. PHYSICAL LAWS Laws of physics The law of gravity: F = GM1M2/R^2 The 2^nd law of motion: F=MA The 2^nd law of rotation: T=a’’/I Kirchoff’s Electrical Circuit Laws Kirchoff’s voltage law: The sum of the voltages around a loop in a circuit is zero. Kirchoff’s current law: The sum of currents entering a node in a circuit is zero.