Notes_E%20BP - Solution: Relative Motion of Two Bodies...

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Unformatted text preview: Solution: Relative Motion of Two Bodies Solve 3 r r r + = Method 1 : Classical Derivation I. Observations from angular momentum 1. n-body problem angular momentum of system ( ) 3 1 i n i i i m r r C = = constant vector System linear momentum conserved cm v = constant any frame moving with center of mass is inertial so assume cm fixed in the new inertial frame ; use cm as the inertially fixed base point E1 Let 2 n = ( ) ( ) 3 1 1 1 2 2 2 C m r r m r r = + Sub back into equation for 3 C 2 2 1 1 3 1 2 1 2 1 2 1 2 1 2 m m m m C m r r m r r m m m m m m m m = + + + + + ( ) 1 2 3 1 2 m m C r r m m = + ( ) 1 2 3 1 2 m m C m m = + r r = h Specific Angular Momentum Note : i d r r = dt relative velocity ( ) 1 2 2 2 1 2 cm cm m m r m r m r r r m m + = = = + 2 2 1 1 2 1 1 2 m r r m m m r r m m = + = + Use m 1 to locate cm E2 2. h r r = = constant plane of motion constant (motion known to be 2D) Invariable plane plane containing c.m. whose normal coincides with h 3. Represent h in scalar component / magnitude form E3 , h r r 2 r r r r r r r h r r r = = + = = 2 h = r Actually already known from h 4. h related to areal velocity [Kepler III. Line joining planet to Sun sweeps out equal areas in equal times.] (Assume motion in a plane)...
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This note was uploaded on 12/22/2010 for the course A&AE 532 taught by Professor Kathleenhowell during the Spring '10 term at Purdue University-West Lafayette.

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Notes_E%20BP - Solution: Relative Motion of Two Bodies...

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