Understanding 2-Body Equations of Motion and Solutions
Explore the derivation, assumptions, and solutions to the 2-body equation of motion in physics, along with insights on apogee, perigee, eccentricity, and flight path angles.
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Presentation Transcript
2 Body Equation of Motion 2 Body Equation of Motion (EOM) Solution 1
2 BODY EQUATION OF MOTION (EOM) ? Derivation not shown here ? ?2 ? = 0 ? + ?? ? = position vector ? = ? = velocity vector ? = ? = acceleration vector ? = unit vector in ? direction = G?? Alternate form: ? ? + ?3? = 0 5
ASSUMPTIONS ??
ASSUMPTIONS 1)Only 2 bodies earth and satellite 2)Bodies are spherically symmetrical acts through center of bodies 3) ?? ?? 4) >> any other forces 5) Coordinate system is sufficiently inertial (so Newton s Laws apply) 6) Mass is constant so ? = 0 ?? ?? ?? 7
SOLUTION TO 2 BODY EOM ?(1 ?2) 1 + ???? ? = ? ? ? apogee perigee F F 8 2a
SOLUTION TO 2 BODY EOM ? ? = ? ? a = semi-major axis 2a = ??+ ?? ? apogee perigee Circle: a = R Ellipse: a > 0 Parabola: a = Hyperbola: a < 0 F F = true anomaly 2a Measured from ? ?? ? ?? ?? c 9
SOLUTION TO 2 BODY EOM ? = flight path angle + outward bound (perigee to apogee) - Inbound (apogee to perigee) ? ? apogee perigee e = eccentricity ?? ?? ??+ ?? F F ? = circle: e = 0 ellipse 0 < e < 1 parabola e = 1 hyperbola e > 1 2a ?? ?? c 10
SOLUTION TO 2 BODY EOM y ? = flight path angle ? x apogee perigee F F 2a ?? ?? 11
SOLUTION TO 2 BODY EOM ? c = distance between foci = ?? ??= ?? ? ??= ? ? = ? ?? ? apogee perigee F F ??= ? 1 ? 2a ??= ? + ? = a + ae ?? ?? ??= ?(1 + ?) c 12
