Explanation Given information: The radius of semi-major axis of ellipse is a . The radius of semi-minor axis of ellipse is b . Calculation: Write the general equation for radius (a) of semi-major axis of ellipse. a = r A + r B 2 2 a = r A + r B (1) Write the general equation for radius (b) of semi-minor axis of ellipse. b = r A r B b 2 = r A r B (2) Consider the minimum radius 1 r A = G M h 2 + C cos θ A (3) Here, G is the gravitational constant, M is the planet, h is the angular momentum per unit mass of the satellite, C is the constant, and θ A is the trajectory angle at A . Consider the maximum radius. 1 r B = G M h 2 + C cos θ B (4) Here, θ B is the trajectory angle at B . Write the equation of angle at B . θ B = θ A + 180 ° Apply cosine on both sides. cos θ B = cos ( θ A + 180 ° ) cos θ B = − cos θ A Find the angular momentum per unit mass of the satellite ( h ). Adding Equation (3) and (4). 1 r A + 1 r B = G M h 2 + C cos θ A + G M h 2 + C cos θ B Substitute cos θ B for − cos θ A . 1 r A + 1 r B = G M h 2 + C cos θ A + G M h 2 + C ( − cos θ A ) r A + r B r A r B = 2 G M h 2 (5) Substitute Equation (1) and (2) in Equation (5)

VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)

12th Edition

ISBN: 9781260916942

Author: BEER

Publisher: MCG

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Chapter 12.3, Problem 12.120P

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Derive Kepler’s third law of planetary motion.

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Chapter 12.3, Problem 12.119P

Chapter 12.3, Problem 12.121P

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VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)

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