(a) Explanation Given information: The radius of the earth ( R ) is 3 , 960 mi . The altitude of point A ( h A ) from the surface of the earth is 40.3. The altitude of point B ( h B ) from the surface of the earth is 336 mi. Calculation: Find the product of the constant of gravitation G and the mass M of the earth ( G M ) using the equation: G M = g R 2 Here, G is the gravitational constant. Substitute 32.2 ft/s 2 for g and 3 , 960 mi for R . G M = 32.2 × ( 3 , 960 mi × 5 , 280 ft 1 mi ) 2 = 14.077 × 10 15 ft 3 /s 2 Find the radius of the point A ( r A ) using the equation: r A = R + h A Substitute 3 , 960 mi for R and 40.3 mi for h A . r A = 3 , 960 + 40.3 = 4 , 000.3 mi × 5 , 280 ft 1 mi = 21.1216 × 10 6 ft Find the radius of the point A ( r B ) using the equation: r B = R + h B Substitute 3 , 960 mi for R and 336 mi for h A . r B = 3 , 960 + 336 = 4 , 296 mi × 5 , 280 ft 1 mi = 22.683 × 10 6 ft Consider the minimum radius Write the equation of motion satellite and the planet at A of elliptical trajectory. 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. Write the equation of motion satellite and the planet at B of elliptical trajectory. 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 ) (b) To determine Find the increase in speed required at B to insert the shuttle into the circular orbit.

12th Edition

ISBN: 9781265566296

Author: BEER

Publisher: MCG

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Chapter 12, Problem 12.131RP

(a)

To determine

Find the speed v0 of the shuttle at A.

(b)

To determine

Find the increase in speed required at B to insert the shuttle into the circular orbit.

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Chapter 12, Problem 12.130RP

Chapter 12, Problem 12.132RP

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Find the speed v 0 of the shuttle at A .

Solution Summary: The author calculates the constant of gravitation G and the mass M of the earth using the equation: GM=gR2.