Explanation Given information: The velocity of Jupiter to reach the point A ( v A ) is 26.9 km / s . The distance from center of Jupiter to point A ( r A ) is 350 × 10 3 km . The distance from center of Jupiter to point B ( r B ) is 100 × 10 3 km . The radius of the earth ( R ) is 6370 km . The mass of the Jupiter is 319 times the mass of the earth. The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Convert the radius of the earth ( R ) from kilometer to meter using the relation: R = R E × 1000 m km Here, R E is the radius of the earth in kilometer. Substitute 6370 km for R E . R = 6370 × 1000 m km = 6.370 × 10 6 m Consider the point A . The expression for the decrease in speed required by the spacecraft at A ( Δ v A ) to achieve the desired orbit using the relation: Δ v A = v A − ( v A ) ' ( v A ) ' = v A − Δ v A Calculate the kinetic energy at the point A ( T A ) using the formula: T A = 1 2 m ( v A ' ) 2 Substitute v A − Δ v A for ( v A ) ' . T A = 1 2 m ( v A − Δ v A ) 2 Calculate the product of gravitational constant and mass of the Jupiter ( G M J ) using the relation: Since the mass of the Jupiter is 319 times the mass of the earth. G M J = 319 G M E Here, G is the gravitational constant, M J is the mass of the Jupiter, and M E is the mass of the earth. Substitute g R 2 for G M E . G M J = 319 g R 2 Substitute 9.81 m / s 2 for g and 6.370 × 10 6 m for R . G M J = 319 ( 9.81 ) ( 6.370 × 10 6 m ) 2 = 126.98 × 10 15 m 3 / s 2 Calculate the potential energy at point A ( V A ) using the formula: V A = − G M J m r A Substitute 126.98 × 10 15 m 3 / s 2 for G M J and 350 × 10 3 km for r A . V A = − 126.98 × 10 15 m 3 / s 2 ( m ) 350 × 10 6 m = − 362.8 × 10 6 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 point B ( V B ) using the formula: V B = − G M J m r B Substitute 126.98 × 10 15 m 3 / s 2 for G M J and 100 × 10 3 km for r B . V B = − 126.98 × 10 15 m 3 / s 2 ( m ) 100 × 10 6 m = − 1269.8 × 10 6 m Use the principle of conservation of energy states that sum of the kinetic and potential energy of a particle remains constant. Write the expression for the conservation of energy as follows: T A + V A = T B + V B Substitute 1 2 m ( v A − Δ v A ) 2 for T A , − 362.8 × 10 6 m for V A , 1 2 m v B 2 for T B , and − 1269

Loose Leaf for Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977206

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

Publisher: McGraw-Hill Education

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Chapter 13.2, Problem 13.103P

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Find the decrease in speed (ΔvA) at point A which will place the spacecraft into the required orbit.

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Chapter 13.2, Problem 13.102P

Chapter 13.2, Problem 13.104P

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Loose Leaf for Vector Mechanics for Engineers: Statics and Dynamics

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Find the decrease in speed ( Δ v A ) at point A which will place the spacecraft into the required orbit.

Solution Summary: The author calculates the radius of the earth (R) from kilometer to meter using the relation: R_E.

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Find the decrease in speed (ΔvA) at point A which will place the spacecraft into the required orbit.

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Chapter 13 Solutions