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HW7_prob3_tangential impulse

HW7_prob3_tangential impulse - Problem Tangential Impulse...

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Problem - Tangential Impulse Orbit Change (a) From the result of GPS Problem 2-11 (2.9 in 2 nd ed.) we can relate the change in angular momentum incurred by the comet-star system to the strength of the tangential impulse S , ( μr 2 · θ ) f ( r 2 · θ ) i = l l 0 = S , (1) where the subscripts i and f refer to values just before and after the impulse, respectively. Since the impulse occurs at perihelion, r = r p , where · r = 0 , we can use the energy equation for the parabolic orbit to relate r p and l 0 . The general expression for the energy of the orbit is E = μ · r 2 2 k r + l 2 2 μr 2 . (2) For the parabolic orbit, E = 0 , and we have 0 = k r p + l 2 0 2 μr 2 p , (3) which yields r p = l 2 0 2 μk . (4) The new circular orbit has radius r p , and we can use the circular orbit condition to relate r p and l , f ef ( r p ) = 0 = k r 2 p + l 2 μr 3 p , (5) which yields, r p = l 2 μk . (6) Comparing Eqs.(4) and (6), we fi nd l 2 = l 2 0 2 , (7) implying two possible values for l . When l = l 0 / 2 we have S = l 0 (1 2 / 2) , (8)
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