Mathematics 317 Solutions to Assignment3

Mathematics 317 Solutions to Assignment3 - of the particle...

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Mathematics 317: Solutions to Assignment #3 1. Let 2 1 , T T be the respective periods for the orbits of Neptune and Earth and let 2 1 , a a be the respective lengths of the semi-major axes of the elliptical orbits of Neptune and Earth. Then 2 1 , a a can approximate respective distances from Neptune and Earth to the sun. Thus . 30 ) 164 ( 3 / 2 3 / 2 2 1 2 1 2 2 1 3 2 1 2245 = 2245 2245 T T a a T T a a 2. Let m be the mass of the particle. Then . / 3 r A m r r - = Then . / m A = μ As shown in class, the period of the orbit is given by 2 / 3 2 a T π = where a is the length of the semi- major axis of the orbit. The aim is to show that 1 2 0 0 2 - - = v r a where 0 v is the speed of the particle and 0 r is the distance from the particle to the centre of force measured at some time . 0 t For the ellipse, 2 1 ε - = p a where is the eccentricity of the elliptical orbit. In class, it was shown that Am L p / 2 = in terms of the constant total angular momentum
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Unformatted text preview: of the particle where . L = L In the next problem, it is proven that 2 2 2 2 1 mA E L + = where E is the total energy of the particle. Using simple algebra, one obtains . / 2 1 E A a-= But as shown in class, r A mv E / 2 2 1-= where here v is the speed of the particle at time t and r is the position of the particle at the same time t . Consequently, at time , t one obtains =-= 2 2 mv r A A a . 2 1 2- -v r 3. . 1 2 2 2 2 2 + -= - - = Ar A L v r A r A L r r r L r r L r & & & But . 2 m L = = r r L L r r & & Hence = -+ = -+ = r A mv mA L mr A v A L 2 2 1 2 2 2 2 2 2 1 2 1 . 2 1 2 2 mA E L +...
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This note was uploaded on 04/01/2010 for the course MATH 317 MATH 317 taught by Professor Bluman during the Spring '10 term at The University of British Columbia.

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