Explanation Calculation: Find the radius ( a t r ) of semi-major axis of an elliptic transfer orbit. a t r = 1 2 ( r E + r v ) Here, r E is the radius of the orbit of earth and r v is the radius of the orbit of Venus. Substitute 93.0 × 10 6 mi for r E and 67.2 × 10 6 mi for r v . a t r = 1 2 ( 93.0 × 10 6 + 67.2 × 10 6 ) = 80.1 × 10 6 mi Find the radius ( a earth ) of semi-major axis of an elliptic orbit of earth. a earth ≃ r E Substitute 93.0 × 10 6 mi for r E a earth ≃ 93.0 × 10 6 mi Take the periodic time of earth ( τ earth ) is 365.25 days. Find the periodic time ( τ t r ) of probe in the elliptic transfer orbit using Kepler’s Third Law: τ t r 2 τ earth 2 = a t r 3 a earth 3 τ t r 2 = ( a t r a earth ) 3 τ earth 2 Substitute 80.1 × 10 6 mi for a t r , 93.0 × 10 6 mi for a earth , and 365.25 days for τ earth . τ t r 2 = ( 80.1 × 10 6 93.0 × 10 6 ) ( 365.25 ) 2 τ t r = 291.954 days Find the time ( t probe ) for the probe to travel from the earth to Venus using the equation: t probe = 1 2 τ t r Substitute 291.954 days for τ t r . t probe = 1 2 ( 291.954 ) = 145.977 days Find the product of the constant of gravitation G and the mass M of the earth ( G M earth ) using the equation: G M earth = g R 2 Here, G is the gravitational constant and R is the radius of the earth. Substitute 32.2 ft/s 2 for g and 3 , 960 mi for R . G M earth = 32.2 × ( 3 , 960 mi × 5 , 280 ft 1 mi ) 2 = 14

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 12, Problem 12.132RP

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Find the angle ϕ which defines the relative position of Venus with respect to the earth at the time the probe is inserted into the transfer orbit.

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Chapter 12, Problem 12.131RP

Chapter 12, Problem 12.133RP

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Find the angle ϕ which defines the relative position of Venus with respect to the earth at the time the probe is inserted into the transfer orbit.

Solution Summary: The author calculates the radius of an elliptic transfer orbit using the angle varphi which defines the relative position of Venus with respect to the earth.

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Find the angle ϕ which defines the relative position of Venus with respect to the earth at the time the probe is inserted into the transfer orbit.

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