(a) Explanation Given information: The velocity of Saturn to reach the point A ( v A ) is 68.8 × 10 3 ft / s . The distance from center of Saturn to point A ( r A ) is 115 × 10 3 mi . The distance from center of Saturn to point B ( r B ) is 183 × 10 3 mi . The speed of the Tethys ( v c i r c ) is 37.2 × 10 3 ft / s . The radius of the earth ( R ) is 3960 mi . Calculation: Convert the distance from center of Saturn to point A ( r A ) from miles to feet: r A = 115 × 10 3 mi = 115 × 10 3 mi × ( 5280 ft 1 mi ) = 607.2 × 10 6 ft Convert the distance from center of Saturn to point B ( r B ) from miles to feet: r B = 183 × 10 3 mi = 183 × 10 3 mi × ( 5280 ft 1 mi ) = 966.2 × 10 6 ft Consider the point A . Calculate the product of gravitational constant and mass of the Saturn ( G M s a t ) using the relation: v c i r c = G M s a t r B G M s a t = v c i r c 2 r B Here, G is the gravitational constant and M s a t is the mass of the Saturn. Substitute 37.2 × 10 3 ft / s for v c i r c and 966.2 × 10 6 ft for r B . G M s a t = ( 37.2 × 10 3 ) 2 ( 966.2 × 10 6 ft ) = 1.337 × 10 18 ft 3 / s 2 Write the expression for the kinetic energy at point A ( T A ) as follows: T A = 1 2 m v A ' 2 Here, v A ' is the speed of the spacecraft in the elliptical orbit after its speed has been decreased. Calculate the potential energy at point A ( V A ) using the formula: V A = − G M s a t m r A Substitute 1.337 × 10 18 ft 3 / s 2 for G M s a t and 607.2 × 10 6 ft for r A . V A = − ( 1.337 × 10 18 ft 3 / s 2 ) m 607.2 × 10 6 ft = − 2.202 × 10 9 m Consider the point B . Write the expression for the kinetic energy at the point B ( T B ) as follows: T B = 1 2 m v B 2 Calculate the potential energy at the point B ( V B ) as follows; V B = − G M s a t m r B Substitute 1.337 × 10 18 ft 3 / s 2 for G M s a t and 966.2 × 10 6 ft for r B . V B = − ( 1.337 × 10 18 ft 3 / s 2 ) m 966.2 × 10 6 ft = − 1.384 × 10 9 m Use the principle of conservation of energy states that sum of the kinetic and potential energy of a particle remains constant (b) To determine Find the speed of the spacecraft ( v B ) when it reaches the orbit of Tethys at B .

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.102P

(a)

To determine

Find the decrease in speed required by the spacecraft at A (ΔvA) to achieve the desired orbit.

(b)

To determine

Find the speed of the spacecraft (vB) when it reaches the orbit of Tethys at B.

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Chapter 13.2, Problem 13.101P

Chapter 13.2, Problem 13.103P

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Find the decrease in speed required by the spacecraft at A ( Δ v A ) to achieve the desired orbit.

Solution Summary: The author calculates the distance from center of Saturn to point A (r_A) from miles to feet.